Abstract
This paper aims to study time periodic solutions for 3D inviscid quasi-geostrophic model. We show the existence of non trivial rotating patches by suitable perturbation of stationary solutions given by generic revolution shapes around the vertical axis. The construction of those special solutions are done through bifurcation theory. In general, the spectral problem is very delicate and strongly depends on the shape of the initial stationary solutions. More specifically, the spectral study can be related to an eigenvalue problem of a self-adjoint compact operator. We are able to implement the bifurcation only from the largest eigenvalues of the operator, which are simple. Additional difficulties generated by the singularities of the poles are solved through the use of suitable function spaces with Dirichlet boundary condition type and refined potential theory with anisotropic kernels.
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1 Introduction
The large scale dynamics of an inviscid three-dimensional fluid subject to rapid background rotation and strong stratification can be described through the so-called quasi-geostrophic model. It is an asymptotic model derived from the Boussinesq system for vanishing Rossby and Froud numbers, for more details about its formal derivation we refer to [36]. Rigorous derivation can be found in [3, 10, 30].
We point out that this system is a pertinent model commonly used in the ocean and atmosphere circulations to describe the vortices and to track the emergence of long-lived structures. The quasi-geostrophic system is described by the potential vorticity q which is merely advected by the fluid,
The second equation involving the standard Laplacian of \({{\,\mathrm{{\mathbb {R}}}\,}}^3\) can be formally inverted using Green’s function leading to the following representation of the stream function \(\psi \),
where dA denotes the usual Lebesgue measure. The velocity field (u, v, 0) is solenoidal and can be recovered from q through the Biot–Savart law,
Notice that the velocity field is planar but its components depends on the all spatial variables and the potential vorticity is transported by the associated flow. The incompressibility of the velocity allows us to adapt without any difficulties the classical results known for 2D Euler equations. For instance, see [34], one may get global unique strong solutions when the initial data \(q_0\) belongs to Hölder class \({\mathscr {C}}^\alpha \), for \(\alpha \ge 0\). Yudovich theory [44] can also be implemented and one gets global unique solution when \(q_0\in L^1\cap L^\infty \). This latter context allows to deal with discontinuous vortices of the patch form, meaning a characteristic function of a bounded domain. This structure is preserved in time and the vortex patch problem consists of studying the regularity of the boundary and to analyze whether singularities can be formed in finite time on the boundary.
For the 2D Euler equations, the \({\mathscr {C}}^{1,\alpha }\) regularity of the boundary of the patch, with \(\alpha \in (0,1)\), is preserved in time, see [4, 11, 41]. The contour dynamics of the patch is in general hard to track and filamentation may occur. Therefore it is of important interest to look for ordered structure in turbulent flows like relative equilibria. It seems that only a few explicit examples are known in the literature in the patch form: the circular patches which are stationary and the elliptic ones which rotate uniformly with a constant angular velocity. This latter example is known as the Kirchhoff ellipses. However, a lot of implicit examples with higher symmetry have been constructed during the last decades and the first ones were discovered numerically by Deem and Zabusky [13]. Having this kind of V-state solutions in mind, Burbea [5] designed a rigorous approach to generate them close to Rankine vortices through complex analysis tools and bifurcation theory. Later this idea was fruitfully improved and extended in various directions, generating a lot of contributions dealing, for instance, with interesting topics like the regularity problem of the relative equilibria, their existence with different topological structure or for different active scalar equations and so forth. For more details about this active area we refer the reader to the works [6,7,8,9, 14,15,16, 19, 21,22,29] and the references therein.
Coming back to the 3D quasi-geostrophic system, it seems that stationary solutions in the patch form are more abundant than the planar case. Indeed, any domain with a revolution shape about the z-axis generates a stationary solution. The analogues to Kirchhoff ellipses still surprisingly survive in the 3d case. In [35] it is shown that a standing ellipsoid of arbitrary semi-axis lengths a, b and c rotates steadily about the z-axis with the angular velocity
where \(\lambda =\frac{a}{b}\) is the horizontal aspect ratio, \(\mu := \frac{c}{\sqrt{ab}}\) the vertical aspect ratio and \(R_D\) denotes the elliptic integral of second order
For more details about the stability of those ellipsoids we refer to [17, 18, 20].
The main concern of this paper is to investigate the existence of non trivial relative equilibria close to the stationary revolution shapes. In our context, we mean by relative equilibria periodic solutions in the patch form, rotating uniformly about the vertical axis without any deformation. Very recently, Reinaud has explored numerically in [40] the existence and the linear stability of finite volume relative equilibria distributed around circular point vortex arrays. Similar analysis has been implemented in [39] for toroidal vortices. Apart from the numerical experiments, no analytical results had been yet developed and the main inquiry of this paper is to design some technical material allowing us to construct relative equilibria close to general smooth stationary revolution shapes. The basic tool is bifurcation theory but as we shall see its implementation is an involved task which requires refined and careful analysis. Let us explain more our strategy and how to proceed. First, we start with deriving the contour dynamic equation for rotating finite volume patches \({\mathbf{1}}_D\). To do so, we look for smooth domains D with the following parametrization,
where the shape is sufficiently close to a revolution shape domain, meaning that
with small perturbation f. Since the domain is assumed to be smooth then we should prescribe the Dirichlet boundary conditions,
Notice that without any perturbation, that is, \(f\equiv 0\), the initial data \(q_0=\mathbf{1}_D\) defines a stationary solution for (1.1), as we will prove in Lemma 2.1. Now a rotating solution about the vertical axis is a time-dependent solution taking the form,
We shall see later that it is equivalent to check that
for any \((\phi ,\theta )\in [0,\pi ]\times [0,2\pi ]\), where
where \(\psi _0\) stands for the stream function associated to \(q_0\). With this reformulation we visualize the smooth rotating surface as a collection of interacting stratified horizontal sections rotating with the same angular velocity but their size degenerates when we approach the north and south poles corresponding to \(\phi \in \{0,\pi \}\).
In order to apply a bifurcation argument, one has to deal with the linearized operator of F around \(f=0\). From Proposition 3.2 such linearized operator has a compact expression in terms of hypergeometric functions. Indeed, for \(h(\phi ,\theta )=\sum _{n\ge 1}h_n(\phi )\cos (n\theta )\), one gets
where
with
and for \(n\ge 1\),
Here \(F_n\) denotes the hypergeometric function
An important observation is that the kernel study of \(\partial _{f} F(\Omega ,0)\) amounts to checking whether 1 is an eigenvalue for \({\mathcal {K}}_{n}^{\Omega }\). To do that we first start with symmetrizing this operator by working on suitable weighted Hilbert spaces. A natural candidate for that is the Hilbert space \(L^2_{\mu _\Omega }(0,\pi )\) of square integrable functions with respect to the measure
In general this defines a signed measure and to get a positive one we should restrict the values of \(\Omega \) to the set \((-\infty ,\kappa )\), where \(\displaystyle {\kappa :=\inf _{\phi \in (0,\pi )}\int _0^\pi H_1(\phi ,\varphi )d\varphi }\).
In the next step we prove that for any \(n\ge 1\), the operator \({\mathcal {K}}_{n}^{\Omega }:L^2_{\mu _\Omega }\rightarrow L^2_{\mu _\Omega }\) acts as a compact self-adjoint integral operator. This gives us the structure of the eigenvalues which is a discrete set and we establish from the positivity of the kernel that the largest eigenvalue \(\lambda _n(\Omega )\) giving the spectral radius is positive and simple. For given integer \(n\ge 1\), we define the set
and in Proposition 4.2 we shall describe some basic properties of \(\lambda _n\) through precise study of the kernel. Those properties show in particular that the set \({\mathscr {S}}_n\) is formed by a single point denoted by \(\Omega _n\), see Proposition 4.3 for more details. In addition, we show that the sequence \(n\in N^\star \mapsto \Omega _n\) is strictly increasing which ensures that the kernel of the linearized operator is a one-dimensional vector space, see Proposition 4.6. Notice that the weighted space \(L^2_{\mu _\Omega }\) is chosen to be so weak in order to have it be stable under the nonlinear functional F. So we need to reinforce the regularity by selecting the standard Hölder spaces \({\mathscr {C}}^{1,\alpha }\) with Dirichlet boundary condition and \(\alpha \in (0,1)\). However this choice generates two delicate problems. The first one is to check that the eigenfunctions constructed in \(L^2_{\mu _\Omega }\) are sufficient smooth and belong to the new spaces. To reach this regularity we need to check that the function \(\nu _\Omega \) is \({\mathscr {C}}^{1,\alpha }\) and this requires more careful analysis due to the logarithmic singularity, see Proposition 4.1. Notice that the eigenfunctions satisfy the boundary condition provided that \(n\ge 2\) and which fails for \(n=1\). The second difficulty concerns the stability of the Hölder spaces by the nonlinear functional F, in fact not F but another modified functional \({\tilde{F}}\) deduced from the preceding one by removing the singularities coming from of the north and south poles, see (2.13). The deformation of the Euclidean kernel through the cylindrical coordinates generates singularities on the poles because the size of horizontal sections degenerates at those points. That is the central difficulty when we try to implement potential theory arguments to get the stability of the function spaces and will be discussed in Sect. 5.
Before stating our result, we need to make the following assumptions on the initial profile \(r_0\) and denoted throughout this paper by (H) :
- (H1):
-
\(r_0\in {\mathscr {C}}^{2}([0,\pi ])\), with \(r_0(0)=r_0(\pi )=0\) and \(r_0(\phi )>0\) for \(\phi \in (0,\pi )\).
- (H2):
-
There exists \(C>0\) such that
$$\begin{aligned} \forall \,\phi \in [0,\pi ],\quad C^{-1}\sin \phi \le r_0(\phi )\le C\sin (\phi ). \end{aligned}$$ - (H3):
-
\(r_0\) is symmetric with respect to \(\phi =\frac{\pi }{2}\), i.e., \(r_0\left( \frac{\pi }{2}-\phi \right) =r_0\left( \frac{\pi }{2}+\phi \right) \), for any \(\phi \in [0,\frac{\pi }{2}]\).
Now we are ready to give a short version of the main result of this paper and the precise one is detailed in Theorem 6.1.
Theorem 1.1
Assume that \(r_0\) satisfies the assumptions (H). Then for any \(m\ge 2\), there exists a curve of non trivial rotating solutions with m-fold symmetry to the Eq. (1.1) bifurcating from the trivial revolution shape associated to \(r_0\) at the angular velocity \(\Omega _m\), the unique point of the set \({\mathscr {S}}_m\).
We specify that by m-fold shape symmetry of \({{\,\mathrm{{\mathbb {R}}}\,}}^3\), we mean a surface invariant under rotation with the vertical axis and angle \(\frac{2\pi }{m}\cdot \)
There is the particular case of \(r_0(\phi )=\sin (\phi )\) defining the unit sphere. Here, its associated stream function can be explicitly computed (see [32]) and it is quadratic inside the shape, that is,
That gives us some interesting properties on the eigenvalues \(\Omega _m\) of the above Theorem 1.1. In particular, we achieve that the above eigenvalues \(\Omega _m\) belong to \((0,\frac{1}{3})\). The same properties occur also in the case of an ellipsoid of equal x and y axes defining a revolution shape around the z-axis. In this case, the associated stream function is also quadratic. See Sect. 6.1 for a more detailed discussion about those cases.
The paper is structured as follows. In Sect. 2, we provide different reformulations for the rotating patch problem and we introduce the appropriate function spaces. Section 3 is devoted to different useful expressions of the linearized operator around a stationary solution. The spectral study of the linearized operator will be developed in Sect. 4. In Sect. 5, we shall discuss the well-definition of the nonlinear functional and its regularity. In Sect. 6, we give the general statement of our result and provide its proof. We end this paper with three appendices concerning special functions, bifurcation theory and potential theory.
2 Vortex Patch Equations
Take an initial data uniformly distributed in a bounded domain of \({{\,\mathrm{{\mathbb {R}}}\,}}^3\), that is, \( q_0={\mathbf{1}}_{D}\). Then, this structure is preserved by the evolution and one gets for any time \(t\ge 0\)
for some bounded domain D(t). To track the dynamics of the boundary (which is a surface here) we can implement the contour dynamics method introduced by Deem and Zabusky for Euler equations [13]. Indeed, let \(\gamma _t: (\phi ,\theta )\in {{\,\mathrm{{\mathbb {T}}}\,}}^2\mapsto \gamma _t(\phi ,\theta )\in {{\,\mathrm{{\mathbb {R}}}\,}}^3\) be any parametrization of the boundary \(\partial D_t\). Since the boundary is transported by the flow then
where \(U=(u,v,0)\) and \(n(\gamma _t)\) is a normal vector to the boundary at the point \(\gamma _t\). There is a special parametrization called Lagrangian parametrization given by
which is commonly used to follow the boundary motion. From the Biot–Savart law we deduce that
where \(d\sigma \) denotes the Lebesgue surface measure of \(\partial D_t\). We have used the notation \(x^\perp =(-x_2,x_1,0)\in {{\,\mathrm{{\mathbb {R}}}\,}}^3\) for \(x=(x_1,x_2,x_3)\in {{\,\mathrm{{\mathbb {R}}}\,}}^3\).
2.1 Stationary patches
Our next goal is to check that any initial patch with revolution shape around the vertical axis generates a stationary solution. More precisely, we have the following result.
Lemma 2.1
Let \(r:[-1,1]\rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}_+\) be a continuous function with \(r(-1)=r(1)=0\) and let D be the domain enclosed by the surface \(\left\{ (r(z)e^{i\theta },z),\ \theta \in [0,2\pi ], z\in [-1,1]\right\} \), then \(q(t,x)={\mathbf{1}}_{D}(x)\) defines a stationary solution for (1.1).
Proof
Recall from (2.3) that
Define
and let us prove that \(G\equiv 0\). Take \(\theta \in {{\,\mathrm{{\mathbb {R}}}\,}}\) and denote by \({\mathcal {R}}_\theta \) the rotation: \( x=(x_h,x_3)\mapsto (e^{i\theta } x_h, x_3)\). Since D is invariant by \({\mathcal {R}}_\theta \), changing variables leads to
Therefore \(G(x)=G(|x_h|,0,x_3)\), which means that
Since D is invariant by the reflexion: \(y\mapsto (y_1,-y_2,y_3)\) then a change of variables implies that \(G(x_1,x_2,x_3)=G(x_1,-x_2,x_ 3)=-G(x_1,x_2,x_3)\) and thus \(G(x)=0\). Consequently we get in particular that
On the other hand, we get from the revolution shape property of D that the horizontal component of the normal vector is \({n}_h(x)=(x_1,x_2)\), which implies
This implies that \({\mathbf{1}}_{D}\) is a stationary solution in the weak sense. \(\quad \square \)
2.2 Reformulations for periodic patches
In this section, we shall give two ways to write down rotating patches using respectively the velocity field and the stream function. Assume that we have a rotating patch around the \(x_3\) axis with constant angular velocity \(\Omega \in {{\,\mathrm{{\mathbb {R}}}\,}}\), that is \(D_t={\mathcal {R}}_{\Omega t}D\), with \({\mathcal {R}}_{\Omega t}\) being the rotation of angle \(\Omega t\) around the vertical axis. Inserting this expression into the Eq. (2.2) we get
Since U is horizontal then this equation means also that each horizontal section \(D_{x_3}:=\{ y\in {{\,\mathrm{{\mathbb {R}}}\,}}^2,\, (y,x_3)\in D\}\) rotates with the same angular velocity \(\Omega \). Hence the horizontal sections satisfy the equation
where \({n}_{D_{x_3}}\) denotes a normal vector to the planar curve \(\partial D_{x_3}\). Next we shall write down this equation in the particular case of simply connected domains that can be described through polar parametrization in the following way:
Notice that we have assumed in this description, and without any loss of generality, that the orthogonal projection onto the vertical axis is the segment \([-1,1]\). The horizontal sections are indexed by \(\phi \) and parametrized by the polar coordinates as \(\theta \mapsto r(\phi ,\theta )\) and it is obvious that
Then, the equation of the sections reduces to
with, according to (2.3) and the change of variable \(y_3=\cos \varphi \),
We shall look for a rotating solution close to a stationary one described by a given revolution shape \((\theta ,\phi )\mapsto (r_0(\phi ) e^{i\theta },\cos (\phi ))\). This means that we are looking for a parametrization in the form
Implicitly, we have assumed that the domain D is symmetric with respect to the plane \(x_2=0\). In addition, we ask the following boundary conditions,
meaning that the domain D intersects the vertical axis at the points \((0,0,-1)\) and (0, 0, 1).
Define the functionals
with
The subscript \(\mathbf{v}\) refers to the velocity formulation and we use it to compare it later to the stream function formulation. Hence, we need to study the equation:
By Lemma 2.1, one has \(F_\mathbf{v}(\Omega ,0)(\phi , \theta )\equiv 0\), for any \(\Omega \in {{\,\mathrm{{\mathbb {R}}}\,}}\).
2.3 Stream function formulation
There is another way to characterize the rotating solutions described in the previous subsection by virtue of the stream function formulation.
For \(\phi \in [0,\pi ]\), let \( \theta \in [0,2\pi ]\mapsto \gamma _\phi (\theta ):=r(\phi ,\theta )e^{i\theta }, \) be the parametrization of \(\partial D_z\), where \(z=\cos (\phi )\). Then one can check without difficulties that (2.7) agrees with
Then, the equation can be integrated obtaining
where \(m(\Omega ,f)(\phi )\) is a function depending only on \(\phi \) and given by
Let us consider the functional
where
and the stream function is given by
Then, finding a rotating solution amounts to solving in f, for some specific angular velocity constant \(\Omega \), the equation
Remark that one may check directly from this reformulation that any revolution shape is a solution for any angular velocity \(\Omega \), meaning that, \({F}_\mathbf{s}(\Omega ,0)=0\), for any \(\Omega \). Motivated by the Sect. 3 on the structure of the linearized operator, we find it better to get rid of the singularities of the poles and work with the modified functional
Therefore, we get
with
and
2.4 Functions spaces
First we shall recall the Hölder spaces defined on an open nonempty set \({\mathscr {O}}\subset {{\,\mathrm{{\mathbb {R}}}\,}}^d\). Let \(\alpha \in (0,1)\) then
with
It is known that \({\mathscr {C}}^{1,\alpha }({\mathscr {O}})\) is a Banach algebra, meaning a complete space satisfying
Denote by \({{\,\mathrm{{\mathbb {T}}}\,}}\) the one-dimensional torus and we identify the space \({\mathscr {C}}^{1,\alpha }({{\,\mathrm{{\mathbb {T}}}\,}})\) with the space \({\mathscr {C}}_{2\pi }^{1,\alpha }({{\,\mathrm{{\mathbb {R}}}\,}})\) of \(2\pi \)-periodic functions that belongs to \({\mathscr {C}}^{1,\alpha }({{\,\mathrm{{\mathbb {R}}}\,}})\). Next, we shall introduce the function spaces that we use in a crucial way to study the stability of the functional \({\tilde{F}}\) defined in (2.13). For \(\alpha \in (0,1)\) and \(m\in {{\,\mathrm{{\mathbb {N}}}\,}}^\star \), set
supplemented with the conditions
This space is equipped with the same norm as \({\mathscr {C}}^{1,\alpha }((0,\pi )\times {{{\,\mathrm{{\mathbb {T}}}\,}}})\). The first assumption in (2.16) is a kind of partial Dirichlet condition and the second one is a symmetry property with respect to the equatorial \(\phi =\frac{\pi }{2}\). Notice that any function \(f\in {\mathscr {C}}^{1,\alpha }\big ((0,\pi )\times {{\,\mathrm{{\mathbb {T}}}\,}}\big ) \) admits a continuous extension up to the boundary, so the foregoing conditions are meaningful. Furthermore, the Dirichlet boundary conditions allow us to use Taylor’s formula to get a constant \(C>0\) such that for any \(f\in X_m^\alpha \)
The notation \(B_{X_m^\alpha }(\varepsilon )\) means the ball in \(X_m^\alpha \) centered in 0 with radius \(\varepsilon \).
Next we shall discuss quickly some consequences needed for later purposes and following from the assumptions \({\mathbf{(H)}}\) on \(r_0\), given in the Introduction before our main statement.
-
From (H2) we have that \(r_0'(0)>0\) and by continuity of the derivative there exists \(\delta >0\) such that \(r_0'(\phi )>0\) for \(\phi \in [0,\delta ]\). Combining this with the mean value theorem, we deduce the arc-chord estimate: there exists \(C>0\) such that
$$\begin{aligned} C^{-1}(\phi -\varphi )^2\le (r_0(\varphi )-r_0(\phi ))^2+(\cos (\phi )-\cos (\varphi ))^2\le C(\phi -\varphi )^2, \end{aligned}$$(2.18)for any \(\phi ,\varphi \in [0,\pi ]\).
-
We have that \(\frac{\sin (\cdot )}{r_0(\cdot )}\in {\mathscr {C}}^\alpha ([0,\pi ])\), and then \(\phi \in [0,\frac{\pi }{2}]\mapsto \frac{\phi }{r_0(\phi )}\) \(\in {\mathscr {C}}^\alpha ([0,\frac{\pi }{2}])\).
3 Linearized Operator
This section is devoted to show different expressions of the linearized operator around a revolution shape. We can find an useful one in terms of hypergeometric functions. See “Appendix A” for details about these special functions.
From now on, we will use the stream function formulation and then we omit the subscript \(\mathbf{s}\) from \(F_\mathbf{s}\) in order to alleviate the notation. The linearized operator of the velocity formulation is closely related to this one, see the previous section.
3.1 First representation
In the following, we provide the structure of the linearized operator of F around the trivial solution \((\Omega ,0)\).
Proposition 3.1
Let \({\tilde{F}}\) be as in (2.13) and \( (\phi ,\theta )\in [0,\pi ]\times [0,2\pi ]\mapsto h(\phi ,\theta )=\sum _{n\ge 1}h_n(\phi )\cos (n\theta )\) be a smooth function. Then,
Proof
First, note that
The linearized operator at a state \(r_0\) is defined by Gateaux derivative,
Thus straightforward computations yield
with
By expanding h in Fourier series we get
Let us analyze every term. For the first one, making the change of variable \(\theta -\eta \mapsto \eta \) we get using a symmetry argument,
Concerning the last integral term, we first use the identity
Consequently
Thus
Integrating by parts with respect to \(\eta \) gives
Putting together the preceding identities allows to get
Therefore we obtain
Now it is clear that
and so (3.1) is obtained. \(\quad \square \)
Remark 3.1
Notice that the local part of the linearized operator (3.1) can be directly related to the stream function \(\psi _0\) associated to the domain parametrized by \((\phi ,\theta )\mapsto (r_0(\phi )e^{i\theta },\cos (\phi ))\). Indeed, by differentiating the functional \({\widetilde{\psi }}_0:f\mapsto \psi _0((r_0(\phi )+f(\phi ,\theta ))e^{i\theta },\cos (\phi ))\) at \(f\equiv 0\) and in the direction h one gets
This form is useful later for spherical and ellipsoidal shapes where the stream functions admit explicit expressions inside these domains, see Sect. 6.1.
3.2 Second representation with hypergeometric functions
The main purpose of this subsection is to provide a suitable representation of the linearized operator. First we need to use some notations. For \(n\ge 1\), set
where the hypergeometric functions are defined in the “Appendix A”. Other useful notations are listed below,
and
Now we are ready to state the main result of this section.
Proposition 3.2
Let \({\tilde{F}}\) be as in (2.13) and \(h(\phi ,\theta )=\sum _{n\ge 1}h_n(\phi )\cos (n\theta ), (\phi ,\theta )\in [0,\pi ]\times [0,2\pi ]\), be a smooth function. Then,
where
Proof
With the help of Lemma A.1, we can simplify more the expression of the linearized operator given in Proposition 3.1. We shall first give another representation of the first integral of (3.1),
From Lemma A.1 we infer
Thus we deduce
Remark that the validity of Lemma A.1 is guaranteed since the inequality
is satisfied provided that \(\phi \ne \varphi \) which leads to a negligible set. For the last integral in (3.1), we apply once again Lemma A.1,
It follows that
which gives the announced result. \(\quad \square \)
Remark 3.2
By virtue of Remark 3.1 and the previous expression one has that
where \(\psi _0\) is the stream function associated to the domain parametrized by \((r_0(\phi )e^{i\theta } ,\cos (\phi ))\), for \((\phi ,\theta )\in [0,\pi ]\times [0,2\pi ]\).
3.3 Qualitative properties of some auxiliary functions
In the following lemma, we shall study some specific properties of the sequence of functions \(\{H_n\}_n\) introduced in (3.3). We shall study the monotonicity of the sequence \(n\mapsto H_n(\phi ,\varphi )\) which will be crucial later in the study of the monotonicity of the eigenvalues associated to the operators family \(\{{\mathcal {L}}_n, n\ge 1\}\). We will also study the decay rate of this sequence for large n.
Lemma 3.1
For any \(\varphi \ne \phi \in (0,\pi )\), the sequence \( n\in {{\,\mathrm{{\mathbb {N}}}\,}}^\star \mapsto H_n(\phi ,\varphi ) \) is strictly decreasing.
Moreover, if we assume that \(r_0\) satisfies (H2), then, for any \(0\le \alpha <\beta \le 1 \) there exists a constant \(C>0\) such that
Proof
By virtue of (3.3) we may write
where \(x:=\frac{4r_0(\phi )r_0(\varphi )}{R(\phi ,\varphi )}\) belongs to [0, 1) provided that \(\varphi \ne \phi \). Now using the integral representation of hypergeometric functions (A.2) we obtain
with the notation
Therefore the desired result amounts to checking that \(n\mapsto {\mathcal {H}}_n(x)\) is strictly decreasing for any \(x\in (0,1)\). This follows from the fact that \(n\mapsto x^{n+\frac{1}{2}}\) is strictly decreasing combined with the identity
which shows the strict decreasing of this sequence since \(0<\frac{t(1-t)}{1-tx}<1\), for any \(t,x\in (0,1)\).
It follows that for any \(\phi \ne \varphi \), the sequence \(n\mapsto H_n(\phi ,\varphi )\) is strictly decreasing.
It remains to prove the decay estimate of \(H_n\) for large n. It is an immediate consequence of the following more precise estimate: for any \(\alpha \in [0,1]\), we get
for \(n\ge 1\) and \(0\le x<1.\) To see the connection with (3.6) recall first from (3.7) that
Since \(0\le x\le 1\) then we obtain from (3.8) that for any \(1\ge \beta >\alpha \ge 0\)
According to (4.10) we deduce that
which is the desired inequality. Let us now turn to the proof of (3.8). We write
where we have used that
for any \(t\in [0,1]\) and \(0\le x<1\). Observe that we easily get the identity
which implies
and
By using interpolation, we obtain
which gives us
for \(n\in {\mathbb {N}}^\star \) and \(0\le x<1\). \(\quad \square \)
4 Spectral Study
In this section, we aim to investigate some fundamental spectral properties of the linearized operator \(\partial _f {\widetilde{F}}(\Omega ,0)\) in order to apply the Crandall–Rabinowitz theorem. For this goal one must check that the kernel and the co-image of the linearized operator are one dimensional vector spaces. Noting that the study of the kernel agrees with the eigenvalue problem of a Hilbert–Schmidt operator, we achieve that the dimension is one. Moreover, we will study the Fredholm structure of the linearized operator, which will imply that the codimension of the image is one. At the end of the section, we characterize also the transversal condition.
4.1 Symmetrization of the linearized operator
The main strategy to explore some spectral properties of the linearized operator at each frequency level n is to construct a suitable Hilbert space, basically an \(L^2\) space with respect to a special Borel measure, on which it acts as a self-adjoint compact operator. Later we investigate the eigenspace associated with the largest eigenvalue and prove in particular that this space is one-dimensional.
Let us explain how to symmetrize the operator. Recall from (3.4) that for any smooth function \(\displaystyle {h(\phi ,\theta )=\sum _{n\ge 1}h_n(\phi ) \cos (n\theta )}\), we may write the operator \({\mathcal {L}}_n\) under the form
with
and the signed measure
Define the quantity
We shall discuss in Proposition 4.1 below the existence of \(\kappa \) which allows to guarantee the positivity of the measure \(d\mu _\Omega \) provided that the parameter \(\Omega \) is restricted to lie in the interval \((-\infty ,\kappa )\). We shall also study the regularity of the function \(\nu _\Omega \) which is delicate and more involved. In particular, we prove that, under reasonable assumptions on the profile \(r_0\), this function is at least in the Hölder space \({\mathscr {C}}^{1,\alpha }\) for any \(\alpha \in (0,1)\).
Notice that the kernel \(K_ n\) is symmetric. Indeed, according to (3.3) we get the formula
which gives the desired property in view of the symmetry of R, that is, \(R(\phi ,\varphi )=R(\varphi ,\phi )\).
We shall explore in Sect. 4.3 more spectral properties of the symmetric operator associated to the kernel \(K_n\).
4.2 Regularity of \(\nu _\Omega \)
This section is devoted to the study of the regularity of the function \(\nu _{\Omega }\) that arises in (4.1), which turns to be a very delicate problem. This is needed for getting enough regularity for the kernel elements that should belong to the function spaces where bifurcation will be applicable. For lower regularities than Lipschitz class, this can be implemented in a standard way using some boundary behavior of the hypergeometric functions. However for higher regularity of type \({\mathscr {C}}^{1,\alpha }\), the problem turns out to be more delicate due to some logarithmic singularity induced by \(H_1\). To get rid of this singularity we use some specific cancellation coming from the structure of the kernel. We shall also develop the local structure of \(\nu _\Omega \) near its minimum which appears to be crucial later especially in Proposition 4.2.
The main result of this section reads as follows.
Proposition 4.1
Let \(r_0\) be a profile satisfying (H1) and (H2). Then the following properties hold true.
-
(1)
The function \(\phi \in [0,\pi ]\mapsto \nu _\Omega (\phi )\) belongs to \({\mathscr {C}}^\beta ([0,\pi ])\), for all \(\beta \in [0,1)\).
-
(2)
We have \(\kappa >0\) and for any \( \Omega \in (-\infty ,\kappa )\) we get
$$\begin{aligned} \forall \phi \in [0,\pi ],\quad \nu _{\Omega }(\phi )\ge \kappa -\Omega >0. \end{aligned}$$ -
(3)
The function \(\nu _\Omega \) belongs to \({\mathscr {C}}^{1,\alpha }([0,\pi ])\), for any \(\alpha \in (0,1)\), with
$$\begin{aligned} \nu _\Omega ^\prime (0)=\nu _\Omega ^\prime (\pi )=0. \end{aligned}$$ -
(4)
Let \(\Omega \in (-\infty ,\kappa ]\) and assume that \(\nu _\Omega \) reaches its minimum at a point \(\phi _0\in [0,\pi ]\) then there exists \(C>0\) independent of \(\Omega \) such that,
$$\begin{aligned} \forall \phi \in [0,\pi ],\quad 0\le \nu _\Omega \big (\phi \big )-\nu _\Omega \big (\phi _0\big )\le C|\phi -\phi _0|^{1+\alpha }. \end{aligned}$$Moreover, for \(\Omega =\kappa \) this result becomes
$$\begin{aligned} \forall \phi \in [0,\pi ],\quad 0\le \nu _\kappa \big (\phi \big )\le C|\phi -\phi _0|^{1+\alpha }. \end{aligned}$$
Proof
\({\mathbf{(1)}}\) To start, notice that according to (3.3)
where
and
Therefore we may write
Using the boundary behavior of hypergeometric functions stated in Proposition A.1 we deduce that
From the assumption (H2) on \(r_0\) we can write, using the mean value theorem
In view of (2.18) we get for all \(\phi \ne \varphi \in [0,\pi ]\),
Consequently, we get
On the other hand, it is obvious using the assumption (H2) on \(r_0\) that
It follows that
which ensures that \(\nu _\Omega \) is bounded.
Now let us check the Hölder continuity by estimating
Let us begin with \({\mathscr {H}}_1\). Notice that
which implies that
It follows that
Differentiating \({\mathscr {K}}_1\) with respect to \(\phi \) yields
Hence using (H2) we deduce that
From an interpolation argument using the boundedness of \({\mathscr {K}}_1\) we find, according to the mean value theorem,
Using (4.11), we obtain
for any \(\beta \in (0,1)\).
Next we shall proceed to the estimate \({\mathscr {H}}_2\). From (4.12), one finds that
We separate the last integral as follows:
where \(d=|\phi _1-\phi _2|\), \(B_{\phi }(r)=\{\varphi \in [0,\pi ]: |\varphi -\phi |<r\}\) and \(B^c_{\phi }(r)\) denotes its complement set. For the first term, \({\mathscr {H}}_{2,1}\), we simply use (4.11)
Since for \(\varphi \in B_{\phi _1}(d)\cup B_{\phi _2}(d)\) one has \(|\phi _1-\varphi |\le 2 |\phi _1-\phi _2|\) and \(|\phi _2-\varphi |\le 2 |\phi _1-\phi _2|\), then one achieves
for any \(\beta \in (0,1)\). For the second term of \({\mathscr {H}}_2\), we observe that for any \({\varphi \in B_{\phi _1}^c(d)\cap B_{\phi _2}^c(d)}\) one has
On the other hand, direct computations yield
We know that
and by virtue of the boundary behavior stated in Proposition A.1 we get
It follows that, using (2.18),
By explicit calculation using (4.13) we get
Then using the mean value theorem we find
Putting together the preceding estimates we find
Then applying once again the mean value theorem, we get for any \({\varphi \in B_{\phi _1}^c(d)\cap B_{\phi _2}^c(d)}\) a value \(\phi \in (\phi _1,\phi _2)\) such that
Combining this estimate with (4.11) and (4.18), and using an interpolation argument we get for any \(\varepsilon >0\),
Hence, we find that
for any \(\beta \in (0,1)\). Finally, putting together (4.17)–(4.21) we get
for any \(\beta \in (0,1)\). Then, we find that \(\nu _\Omega \in {\mathscr {C}}^\beta \), for any \(\beta \in (0,1)\).
\({\mathbf{(2)}}\) The function \(\phi \mapsto \Omega +\nu _\Omega (\phi )\) is continuous over the compact set \([0,\pi ]\) then it reaches its minimum at some point \(\phi _0\in [0,\pi ]\). Thus from the definition of \(\kappa \) in (4.5) we deduce that
which implies that
Hence we infer that for any \(\Omega \in (-\infty ,\kappa )\)
\({\mathbf{(3)}}\) The proof is long and technical and for clarity of presentation it will be divided into two steps. In the first one we prove that \(\nu _\Omega \) is \({\mathscr {C}}^1\) in the full closed interval \([0,\pi ]\). This is mainly based on two principal ingredients. The first one is an important algebraic cancellation in the integrals allowing us to get rid of the logarithmic singularity coming from the boundary and the second one is the boundary behavior of the hypergeometric functions allowing us to deal with the diagonal singularity lying inside the domain of integration. Notice that in order to apply Lebesgue theorem and recover the continuity of the derivative up to the boundary we use a rescaling argument. This rescaling argument shows in addition a surprising effect concerning the derivative at the boundary points \(\nu _\Omega ^\prime (0)\) and \(\nu _\Omega ^\prime (\pi )\): they are independent of the global structure of the profile \(r_0\) and they depend only on the derivative \(r_0^\prime (0)\). This property allows us to compute \(\nu _\Omega '(0)\) in the special case of \(r_0(\phi )=\sin (\phi )\) by using the special geometry of the sphere and observe that this derivative vanishes. As to the second step, it is devoted to the proof of \(\nu _\Omega ^\prime \in {\mathscr {C}}^\alpha (0,\pi )\) which is involved and requires more refined analysis.
\(\bullet \) Step 1: \(\nu _\Omega \in {\mathscr {C}}^1([0,\pi ])\). The first step is to check that \(\nu _\Omega \) is \({\mathscr {C}}^1\) on \([0,\pi ]\). Define
then we can check that
which implies after simple manipulations that
In addition using the identity
we find
with
Notice that \(F_1(0)=1,\,F_1^\prime (0)=\frac{3}{4}\). Assuming that the following functions are well-defined and using the boundary conditions then we can write
with
and
Direct computations show that
According to (4.13) and using some cancellation, it implies that
We point out that this simplification is crucial and allows to get rid of the logarithmic singularity.
Now we shall start with the regularity of the function
and prove first that it is continuous in \([0,\pi ]\). It is obvious from (4.24) that \(\varkappa _0\) is \({\mathscr {C}}^1\) over any compact set contained in \((0,\pi )\times [0,\pi ]\) and therefore \(\zeta _1\) is \({\mathscr {C}}^1\) over any compact set contained in \((0,\pi )\). Thus it remains to check that this function is continuous at the points 0 and \(\pi \). The proofs for both cases are quite similar and we shall only check the continuity at the origin. For this purpose it is enough to check that \(\zeta _1\) admits a limit at zero. Before that let us check that \(\zeta _1\) is bounded in \((0,\pi )\). From the definition of R stated in (3.2) and using elementary inequalities it is easy to verify the following estimates: for any \((\phi ,\varphi )\in (0,\pi )^2\)
In addition, the assumption (H2) implies that
Thus we find according to (4.14) and (H2)
Hence we deduce that
Making the change of variables \(\sin \varphi =x\) we get
Thus
Let us now prove that \(\zeta _1\) admits a limit at the origin and compute its value. For this goal, take \(0<\delta<<1\) small enough and write
The assumption \(\mathbf{(H2) }\) combined with standard trigonometric formula allow to get the estimate
From this we infer that
Thus we get from (4.25)
This implies that for given small parameter \(\delta \) one has
Therefore
Making the change of variables \(\varphi =\phi \theta \) we get
From (4.25) and (H2) one may write
which yields after simplification to the uniform bound on \(\phi \),
This gives a domination which is integrable over \((0,+\infty )\). In order to apply classical dominated Lebesgue theorem, it remains to check the convergence almost everywhere in \(\theta \) as \(\phi \) goes to zero. This can be done through the first-order Taylor expansion around zero. First one has the expansion
with \(c_0=r_0^\prime (0)\) and \(\displaystyle {\lim _{x\rightarrow 0}\epsilon (x)=0.}\) Thus, from the definitions (3.2) and (4.8) it is straightforward that
Hence
Similarly we get
and
Standard computations yield
Thus
Therefore
Plugging (4.30), (4.31), (4.32) and (4.35) into (4.24)
Using Lebesgue dominated theorem we deduce that
Computing the integrals we finally get
Let us now move to the regularity of the function \(\zeta _2\) defined in (4.23) through
where \(\varkappa _1\) is defined in (4.22). From direct computations using \(|\partial _\phi R|\lesssim R^\frac{1}{2}\), the boundedness of \({\mathscr {K}}_1\), the assumption (H2) and (4.33) one can check that
Using Proposition A.1 we get
Thus
Hence using the arc-chord property (2.18) we find
for some constant \(C>0\). In addition, using (4.27) we get
which leads to
This implies that
Now in the region \(\phi ,\varphi \in [0,\pi /2], \) we use the estimate (H2) leading to
Plugging this into (4.38) we find
Making the change of variables \(\varphi =\phi \theta \) we obtain
which implies that
Therefore we obtain by virtue of (4.40)
By symmetry we get similar estimate for \(\phi \in [\frac{\pi }{2},\pi ]\) and hence
Let us now calculate the limit when \(\phi \) goes to 0 of \(\zeta _2\). We shall proceed in a similar way to \(\zeta _1\). Let \(0<\delta \ll 1\) be enough small, then using (4.38) combined with (4.28) we obtain
Hence
Now we make the change of variables \(\varphi =\phi \theta \) and then
According to (4.22) one has
From the differentiating of the expression of R stated in (3.2) we get
Taking Taylor expansion to first order we deduce the pointwise convergence,
Combined with (4.29) it implies that
Plugging (4.34) and the preceding estimates into the expression of \(\varkappa _1\) given by (4.22) we find
From the result
we deduce the point-wise convergence
Consequently,
Therefore,
Note that we can apply the dominated convergence theorem in the previous integral since
which is integrable. Next, we shall implement a similar study for \(\zeta _3\) defined in (4.23). Straightforward computations yield
with
and
Since \(r_0^\prime \) is Lipschitz then using the mean value theorem we get
From Proposition A.1 combined with (2.18) we get
In addition
Consequently, we obtain in view of (H2)
As before, we can assume without any loss of generality that \(\phi \in [0,\,\pi /2]\), then by (H2)
By the change of variables \(\varphi =\phi \theta \) we get
Therefore
Consequently
Now, we shall calculate the limit of \(\zeta _3\) at the origin. Let \(0<\delta \ll 1\) be enough small, then using (4.48) combined with (4.28) and (H2) we obtain
It follows that
Making the change of variables \(\varphi =\phi \theta \) yields
Using Taylor expansion to first order one in (4.44) and (4.45) we can check that
Hence we get in view of the definition of \(\varkappa _2\) and (4.29) the point-wise limit
It follows that
Moreover, by (4.48)
so we can apply dominated convergence theorem obtaining
Putting together (4.23), (4.36), (4.42) and (4.50) we find
Notice that the real number \(\eta \) is well-defined since all the integrals converge. This shows the existence of the derivative of \(\nu _\Omega \) at the origin. It is important to emphasize that number \(\eta \) is independent of the profile \(r_0\) and we claim that the number \(\eta \) is zero. It is slightly difficult to check this result directly from the integral representation of \( \eta \), however we shall check it in a different way by calculating its value for the unit ball
whose boundary can be parametrized by \((\phi ,\theta )\mapsto (r_0(\phi ) e^{i\theta },\cos \phi )\) with \(r_0(\phi )=\sin \phi \). Now according to the identity (3.5) one has
However it is known [32] that the stream function \(\psi _0\) is radial and quadratic inside the domain taking the form
Consequently, with this special geometry the function \(\displaystyle {\nu _\Omega }\) is constant and therefore
\(\bullet \) Step 2: \(\nu _\Omega ^\prime \in {\mathscr {C}}^\alpha (0,\pi )\). We shall prove that \(\nu _\Omega ^\prime \) is \({\mathscr {C}}^\alpha (0,\pi )\) and for this purpose we start with the first term in (4.23), i.e., \(\zeta _1\). According to (4.24) it can be split into several terms and to fix the ideas let us describe how to proceed with the first term given by
and check that it belongs to \({\mathscr {C}}^\alpha (0,\pi )\). The remaining terms of \(\zeta _1\) can be treated in a similar way and to alleviate the discussion we leave them to the reader.
From the assumptions (H) on \(r_0\) we have \(r_0^\prime ,\,\phi \mapsto \frac{r_0(\phi )}{\sin \phi }\in {\mathscr {C}}^\alpha (0,\pi )\) , then using the fact that \({\mathscr {C}}^\alpha \) is an algebra, it suffices to verify that
This function is locally \({\mathscr {C}}^1\) in \((0,\pi )\) and so the problem reduces to check the regularity close to the boundary \(\{0,\pi \}\). By symmetry it suffices to check the regularity near the origin. Decompose the integral as follows
Since we are considering \(\phi \in (0,\pi /2)\), it is easy to check that the last integral term defines a \({\mathscr {C}}^1\) function in \([0,\pi /2]\). Since \(\sin (\phi )/\phi \) is \({\mathscr {C}}^\alpha (0,\pi /2)\), then the problem amounts to checking that the function
is \({\mathscr {C}}^\alpha \) close to zero. Making the change of variables \(\varphi =\phi \theta \) we get
Let us now define the following functions
and
We will show that \(T_{1,\phi }\in {\mathscr {C}}^\alpha [\phi ,\pi /2]\) and \(T_{2,\phi }\in {\mathscr {C}}^\alpha (0,\phi ]\) uniformly in \(\phi \in (0,\frac{\pi }{2})\). Thus we get in particular a constant \(C>0\) such that for any \(0<\phi _1\le \phi _2< \frac{\pi }{2}\),
and
By combining (4.52) and (4.53), since \(T_{1,\phi _1}(\phi _ 2)= T_{2,\phi _2}(\phi _1)\) we get
This ensures that \(\zeta _{1,1}\in {\mathscr {C}}^\alpha (0,\pi /2)\).
It remains to show that \(T_{1,\phi }\in {\mathscr {C}}^\alpha ([\phi ,\pi /2])\) and \(T_{2,\phi }\in {\mathscr {C}}^\alpha ((0,\phi ])\) uniformly in \(\phi \in (0,\frac{\pi }{2})\). We start with the term \(T_{1,\phi }\). Then straightforward computations imply
for any \(\phi ,s\in (0,\pi /2]\). Notice that we have used in the last line the following inequalities which follow from the assumptions (H2),
and
Hence \(T_{1,\phi }\in \text {Lip}([\phi ,\pi /2])\), uniformly with respect to \(\phi \in (0,\pi /2)\).
Let us move to the term \(T_{2,\phi }\). First, we write
and taking the derivative with respect to \(\phi \) we obtain
Hence,
By the mean value theorem we infer
Interpolating between (4.54), which is also true for \(r_0=\sin \), and (4.56) we obtain
Using Taylor’s formula
one finds that if \(0\le \phi \theta \le \pi /2\) then
As before, one gets that if \(0\le s_1 \theta ,\, s_2\theta \le \pi /2\) hence
Now, let us check that \(T_{2,\phi }\) is \({\mathscr {C}}^\alpha (0,\phi ]\) uniformly in \(\phi \in (0,\pi /2)\). Let \(s_1,s_2\in (0,\phi ]\), then using the estimates (4.56) and (4.54), we achieve for any \(s\in (0,\phi ]\),
for \(\alpha \in (0,1)\). In the same way
To analyze the difference of the denominator in \(T_{2,\phi }\) we first write that for any \(0\le s\theta \le \frac{\pi }{2}\),
Using an slight variant of the argument in (4.55) one gets
and
By differentiation, and using (4.61) and (4.58), we find that if \( 0\le s\theta \le \frac{\pi }{2}\) hence
This implies in view of the mean value theorem
Moreover
Then by interpolation we get
Therefore we obtain
which converges since \(\alpha \in (0,1)\).
Combining the preceding estimates one deduces that
uniformly in \(\phi \in (0,\pi /2)\). Hence, we conclude that \(\zeta _{1,1}\) is \({\mathscr {C}}^\alpha (0,\pi /2)\), for any \(\alpha \in (0,1)\).
The argument used to prove that the other terms in (4.24) are in \({\mathscr {C}}^\alpha (0,\pi /2)\), for any \(\alpha \in (0,1)\), are quite similar, but for the reader convenience we will sketch some details. The second term in the sum is
Let us consider the functions
and
To prove that this second term is in \({\mathscr {C}}^\alpha (0,\pi /2)\) we will see that \(T_{1,\phi }\in \text {Lip}([\phi ,\pi /2])\) and \(T_{2, \phi }\) is in \({\mathscr {C}}^\alpha (0,\phi )\). Using the mean value theorem and the estimate
we can easily see that \(T_{1,\phi }\) is in the desired space. Now we will check that \(T_{2,\phi }\) is \({\mathscr {C}}^\alpha (0,\phi ]\) uniformly in \(\phi \in (0,\pi /2)\). Let \(s_1,s_2\in (0,\phi ]\), then using the estimates (4.54) and (4.56), we achieve for any \(s\in (0,\phi ]\),
for \(\alpha \in (0,1)\). In the same way
Now by differentiation, and using (4.58) and (4.61) we find that if \( 0\le s\theta \le \frac{\pi }{2}\) then
Therefore, using interpolation argument we obtain
and the last integral converges since \(\alpha \in (0,1)\).
Another term to consider in (4.24) is given by
which will be treated exactly in the same way as the previous one. For this reason we will not repeat again the arguments. The next term in (4.24) can be written as
As in the previous cases we can consider the auxiliary functions
and
For this term it is enough to check that the auxiliary functions are in the \({\mathscr {C}}^\alpha \) space. For \(T_{1,\phi }\) we will use the mean value theorem. Thus the inequality
will be enough. To estimate \(T_{2,\phi }\) we will follow the same arguments developed in the previous cases. Hence, using inequalities (4.58), (4.59), (4.55), (4.56), (4.54), (4.61) and (4.60) one gets the following estimates for \(\alpha \in (0,1)\).
and
On the other hand, using (4.61), (4.60) and interpolation, one obtains
Then, by (4.64) we obtain
To estimate the last term of \(T_{2,\phi }\), using (4.63)
The next function to analyze in (4.24) is
We will not repeat the arguments for this function because they are quite similar to the preceding case. The last function to consider in (4.24) is given by
This term generates several functions. Some of them are similar to the functions estimated in the previous cases and the others are similar between them. For this reason we will only check the first one. Let us prove that the function
is in \({\mathscr {C}}^\alpha (0,\pi /2)\), for any \(\alpha \in (0,1)\). Since the integral in the interval \([\frac{\pi }{2},\pi ]\) provides a function in \({\mathscr {C}}^\alpha \), as in the above cases we can reduce the integral to the interval \([0,\pi /2]. \) Now the strategy is again to consider the auxilary functions
and
Since
the function \(T_{1,\phi }\) is in \({\mathscr {C}}^\alpha \), for any \(\alpha \in (0,1)\). To establish that \(T_{2,\phi }\) is in the same space we need the following estimates.
in the first inequality we have used that \(r_0'\) is a bounded function. The next estimate is
Using the inequality \(|r_0'(s_1\theta )-r_0'(s_2\theta )|\le C\theta ^{\alpha }|s_1-s_2|^{\alpha }\) we get
The last term to estimate is
Hence, we obtain the announced result. The remaining terms can be studied using the same inequalities and for this reason we will avoid them.
Let us now move to the regularity of \(\zeta _2\) defined in (4.23) which takes the form
and
The first function can be split into two parts as follows
As before, by evoking the symmetry property of r we can restrict the study to \(\phi \in [0,\frac{\pi }{2}]\). The second term is the easiest one and we claim that \(I_2\in W^{1,\infty }\). Indeed,
It can be transformed into
Integrating by parts yields
Notice that the last term is bounded uniformly on \(\phi \in [0,\pi /2]\). In fact, one has from the definition of R in (3.2)
Using (4.13) we get
Moreover, since \(r_0\) is symmetric with respect to \(\pi /2\) then we get \(r_0^\prime \left( \frac{\pi }{2}\right) =0\), which implies that \(\partial _\phi R(\pi /2,\pi /2)=0\) and by the mean value theorem,
Hence, combining (4.37) and (4.10) we find
Consequently
which ensures that this quantity is bounded in the interval \((0,\frac{\pi }{2})\).
Next, let us check the boundedness of the integral terms of \(I_2^\prime \). Inequality (4.39) allows to get
which implies
Therefore, (4.10) combined with (4.19) and (4.46) yield
Let us move to \(I_1\). First, we do the change of variables \(\varphi =\phi \theta \) leading to
We will check that \(I_1\) is \({\mathscr {C}}^\alpha (0,\pi /2)\), for any \(\alpha \in (0,1)\). Indeed, take \(\phi _1\le \phi _2\in (0,\frac{\pi }{2})\), then
where
We follow the ideas done for \(\zeta _1\). In order to estimate \(I_{1,1}\), define
Then
for \(\phi \in (0,\pi /2)\). Taking the derivative in \(\phi \) of the function R
we get
Moreover, proceeding as before in (4.37) combined with the assumptions (H) we find
Putting together the preceding estimates allows to get
It follows that
Now using this estimate combined with the mean value theorem we get for \(0<\phi _1\le \phi _2<\frac{\pi }{2}\)
Using Hölder inequality yields for any \(\alpha \in (0,1)\),
Notice that the constant \(C_\alpha \) blows up when \(\alpha \) approaches 1. Thus
Next, let us move to the estimate of \(I_{1,2}\). Using (4.19) we arrive at
Set
then differentiating with respect to \(\theta \) we get
Using the assumption (H2) we may check that
where C depends only on \(\Vert r_0^\prime \Vert _{L^\infty }\). Now by rewriting
and differentiating in \(\phi \) we get the estimate
where C depends only on \(\Vert r_0\Vert _{C^2}\). Taylor’s formula
combined with \({\mathscr {R}}(1,\phi )=1\) yields
This implies in turn that
Combining this estimate with (4.69) we deduce that
Following an interpolation argument combining the preceding estimate with (4.68) yields \(\hbox { for any} \alpha \in [0,1]\) and for \(0<\phi _1\le \phi _2\le \frac{\pi }{2\theta }\)
Plugging this estimate into the definition of \(I_{1,2}\) given in (4.66) implies
This integral converges, close to 1 and at \(\infty \), provided that \(0\le \alpha <1\). We mention that to get the integrability close to 1 we use the approximation
As to the estimate of the term \(I_{1,3}\) described in (4.66) we roughly implement similar ideas. For that purpose, we introduce the function
Then combining (4.56), (4.67) and (4.68), we deduce that
provided that \(\alpha \in (0,1)\). Implementing the same analysis for the remaining terms and using (4.59) and (4.62) as for \(\zeta _{1}\), we find
uniformly for \(\phi \in (0,\pi /2)\). Therefore from the definition (4.66) we obtain for any \(0\le \phi _1\le \phi _2\le \frac{\pi }{2}\),
Now let us consider the next term in \(\zeta _2(\phi )\)
The first function that we intend to study is
Since the function \(\frac{r_0(\phi )}{\phi }\) has bounded derivatives, it is enough to estimate the function I. Thus,
The arguments to estimate the terms \(I_1(\phi )\) and \(I_2(\phi )\) are similar to the case of the function \(\zeta _1\), but we will repeat them for the reader convenience. First we will prove that \(I_2'(\phi )\) is a bounded function. By direct computations we infer
The estimate of \(I_{2,2}\) can be done using (A.10), (4.39) and (4.12), leading to
For the term \(I_{2,1}\) we may apply (4.46), (2.18) and (4.39) in order to get
To estimate the term \(I_{2,3}\) we decompose it in different terms.
Since \(r_0'(\pi /2)=r_0(\pi )=0\) then integration by parts allows to get
We can then proceed similarly to the terms \(I_{2,1}\) and \(I_{2,2}\) in order to get that \(I_{2,2}\) is bounded. The next step will be to check that the function \(I_1\) is in \(\mathscr {C}^{\alpha }(0,\pi /2)\), for all \(\alpha \in (0,1)\). If we do the change of variable \(\varphi =\phi \theta \) we find
If \(\phi _1\) and \(\phi _2\) are in \((0,\pi /2)\), then
To estimate the function \(I_{1,1}\), let us consider the auxiliary function
First we may write for \(0<\phi _1\leqslant \phi _2 <\pi /2\),
Hence, it is enough to obtain an appropriate bound for \(\partial _sG_{1,\phi _1}\). Taking the derivative in s, applying (4.68) and some standard estimates used before we have
and so by Hölder inequality
for all \(\alpha \in (0,1)\). Let us now move to the estimate of the term \(I_{1,2}\). By (4.71) and some standard estimates used along the work we obtain, for any \(\alpha \in (0,1)\)
where in the last inequality we have used that \(\ln (\frac{1+\theta }{2})\simeq \frac{\theta -1}{2}\) if \(\theta \) is enough close to 1.
To estimate the term \(I_{1,3}, \) let us take the function
Now, \(|I_{2,3}|=|G_{2,\phi _2}(\phi _1)-G_{2,\phi _2}(\phi _2)|. \) As in the case of the function \(\zeta _1\) we will get the desired estimate through decomposing the integral in several terms. The first term to consider is
Using (4.59) and (4.68) one can see that this term is bounded by
For the next adding term, using (4.56) and (4.68) we obtain
For the next term we will use the regularityof \(r_0\) and (4.68), obtaining
Let us move to the last term in \(I_{1,3}\). By (4.68) and (4.62)
The two remaining terms in \(\zeta _2\) are left to the reader because they are very similar to the first one. It remains to estimate the term \(\zeta _3\) defined by (4.23). It can be split as follows,
Recall that we restrict to check the regularity for \(\phi \in (0,\pi /2)\) without loss of generality and later we extend it to \(\phi \in (0,\pi )\). Then, we will show that \(I_3\) belongs to \({\mathscr {C}}^\alpha (0,\frac{\pi }{2})\). We will skip here the details for the regularity of \(I_4\) since this term is less singular and the same procedure works, see the estimates for \(I_2\) previously done. To estimate \(I_3\) we proceed as before through the use of the change of variables \(\varphi =\phi \theta \),
Define the functions
In order to check that \(I_3\) belongs to \({\mathscr {C}}^\alpha (0,\frac{\pi }{2})\), it suffices to prove that each function \(H_{i,\phi }\) is in \({\mathscr {C}}^\alpha (0,\frac{\pi }{2})\) uniformly in \(\phi \in (0,\pi /2)\), for any \(i=1,\dots , 4\). Let us start with \(H_{1,\phi }\) showing that its derivative is bounded.
From straightforward calculus it is easy to check that for any \(0\le \phi \le s<\frac{\pi }{2}\),
Hence, we obtain
Using (4.43)–(4.46)–(4.47) allows to get
which is uniformly bounded on \(0\le \phi \le s<\frac{\pi }{2}\). We shall skip the details for \(H_{2,\phi }\) which can be analyzed following the same lines of the term \(T_{2,\phi }\) introduced in (4.51).
Let us now focus on the estimate of \(H_{3,\phi }\). Set
then using (4.46), we deduce
By ranging the expression of \({\mathscr {T}}\) as follows
and differentiating with respect to s we find
We will not give the full details for this estimate because the computations are long and tedious, but to get a more precise idea how this works we shall just explain the estimate of the first term in \(\partial _s {\mathscr {T}}\) given by
Note that the other terms can be treated similarly and we use similar estimates but with \(\sin \), \(\cos \) and \(r_0'\) instead of \(r_0\). In particular, here we use \(r_0\in {\mathscr {C}}^2\) in order to bound \(r_0''\). Define
Then, one has \(\partial _\theta g(\theta ,s)=r_0'(s\theta )\) and then
Since \(g(1,s)=0\), we can write by Taylor’s formula
and hence
Using (4.74), we achieve
Plugging this into the the definition of \({\mathscr {T}}_1\) and using the mean value theorem yields to the estimate
Now, interpolating between (4.72) and (4.73), we find that for any \(\alpha \in (0,1)\)
Using (4.47) we get
Combining this estimate with (4.75) and (4.47), we conclude that for any \(0\le s_1,s_2\le \phi \le \frac{\pi }{2}\)
for any \(\alpha \in (0,1)\). Let us finish working with \(H_{4,\phi }\). Using (4.54) and the standard inequality
one gets
As a consequence of (A.9), (4.70) and (4.78) one has
Interpolating between (4.76) and (4.79) we achieve
Finally, using (4.72) and (4.80) we obtain for any \(0\le s_1,s_2\le \phi \)
the convergence of the integral is guaranteed pro
vided that \(\alpha \in (0,1)\). This achieves the proof of \(\nu _\Omega \in {\mathscr {C}}^{1,\alpha }(0,\pi )\) for any \(\alpha \in (0,1)\).
(4) Since the function \(\nu _\Omega \) reaches its minimum at a point \(\phi _0\in [0,\pi ]\), we have that if this point belongs to the open set \((0,\pi )\) then necessary \(\nu _\Omega ^\prime (\phi _0)=0\). However when \(\phi _0\in \{0,\pi \}\) then from the point (3) of Proposition 4.1 we deduce also that the derivative is vanishing at \(\phi _0\). Using the mean value theorem, we obtain for any \(\phi \in [0,\pi ]\)
for some \({{\overline{\phi }}}\in (\phi _0,\phi )\). Since \(\nu _\Omega ^\prime \in {\mathscr {C}}^\alpha \) then
Notice that \(\Vert \nu _\Omega ^\prime \Vert _{{\mathscr {C}}^\alpha }\) is independent of \(\Omega \). Consequently
for some absolute constant C. In the particular case \(\Omega =\kappa \) we get from the definition (4.5) that \(\nu _\kappa (\phi _0)=0\) and therefore the preceding result becomes
\(\square \)
4.3 Eigenvalue problem
In Sect. 4.1 we have checked that the operator \({\mathcal {L}}_n^\Omega \) defined in (4.1) is of integral type. Then studying the kernel of this operator reduces to solving the integral equation
where the kernel \(K_n\) and the measure \(d\mu _\Omega \) are defined successively in (4.2) and (4.4). The parameter \(\Omega \) ranges over the interval \((-\infty ,\kappa )\). This latter condition is imposed to guarantee the positivity of the measure \(d\mu _\Omega \) through the positivity of \(\nu _\Omega \) according to Lemma 4.1. We point out that studying the kernel of \({\mathcal {L}}_n^\Omega \) amounts to finding the values of \(\Omega \) such that 1 is an eigenvalue of \({\mathcal {K}}_n^{{\Omega }}\). To investigate the spectral study of \({\mathcal {K}}_n^{{\Omega }}\) we need to introduce the Hilbert space \(L^2_{\mu _\Omega }\) of measurable functions \(f:[0,\pi ]\rightarrow {\mathbb {R}}\) such that
Notice that the space \(L^2_{\mu _\Omega }\) is equipped with the usual inner product:
Remarks 4.1
-
(1)
Since \(d\mu _\Omega \) is a nonnegative bounded Borel measure for any \(\Omega \in (-\infty ,\kappa )\), then the Hilbert space \(L^2_{\mu _\Omega }\) is separable.
-
(2)
For any \(\Omega \in (-\infty ,\kappa )\), the space \(L^2_{\mu _\Omega }\) is isomorphic to the space \(L^2_{\mu }\) where
$$\begin{aligned} d\mu (\varphi )=\sin (\varphi )\, r_0^2(\varphi )\, d\varphi . \end{aligned}$$This follows from Proposition 4.1-(2) which ensures that \(\nu _\Omega \) is nowhere vanishing. However this property fails for the critical value \(\Omega =\kappa \) because \(\nu _\kappa \) is vanishing at some points.
The next proposition deals with some basic properties of the operator \({\mathcal {K}}_n^{{\Omega }}\).
Proposition 4.2
Let \(\Omega \in (-\infty ,\kappa )\) and \(r_0\) satisfies the assumptions (H1) and (H2). Then, the following assertions hold true.
-
(1)
For any \(n\ge 1\), the operator \({\mathcal {K}}_n^{{\Omega }}:L^2_{\mu _\Omega }\rightarrow L^2_{\mu _\Omega }\) is Hilbert–Schmidt and self-adjoint.
-
(2)
For any \(n\ge 1\), the eigenvalues of \({\mathcal {K}}_n^{{\Omega }}\) form a countable family of real numbers. Let \(\lambda _n(\Omega )\) be the largest eigenvalue, then it is strictly positive and satisfies
for any function \(\varrho \) such that \(\displaystyle {\int _0^\pi \varrho ^2(\varphi )d\varphi =1.}\)
-
(3)
We have the following decay: for any \(\alpha \in [0,1)\) there exists \(C>0\) such that
-
(4)
The eigenvalue \(\lambda _n(\Omega )\) is simple and the associated nonzero eigenfunctions do not vanish in \((0,\pi )\).
-
(5)
For any \(\Omega \in (-\infty ,\kappa )\), the sequence \(n\in {{\,\mathrm{{\mathbb {N}}}\,}}^{\star }\mapsto \lambda _n(\Omega )\) is strictly decreasing.
-
(6)
For any \(n\ge 1\) the map \(\Omega \in (-\infty ,\kappa )\mapsto \lambda _n(\Omega )\) is differentiable and strictly increasing.
Proof
(1) In order to check that \({\mathcal {K}}_n^{{\Omega }}\) is a Hilbert–Schmidt operator, we need to verify that the kernel \(K_n\) satisfies the integrability condition
Indeed, by (4.2) and (3.3), one gets
for some constant \(C_n\) and R was defined in (3.2). Remark that
Moreover, according to Proposition 4.1 the function \(\nu _{\Omega }(\varphi )\) is not vanishing in the interval \([0,\pi ]\) provided that \(\Omega <\kappa \). Therefore we get
By (A.7) and the assumption (H2) we deduce that
It suffices now to use the inequality (4.10) to get
This concludes that the operator \({\mathcal {K}}_n^{{\Omega }}\) is bounded and is of Hilbert–Schmidt type. As a consequence from the general theory this operator is necessarily compact.
On the other hand, as we have mentioned before the kernel \(K_n\) is symmetric in view of the formula (4.6) and the symmetry of R defined in (3.2). Therefore we deduce that \({\mathcal {K}}_n^{{\Omega }}\) is a self-adjoint operator
(2) From the spectral theorem on self-adjoint compact operators, we know that the eigenvalues of \({\mathcal {K}}_n^{{\Omega }}\) form a countable family of real numbers. Define the real numbers
Since \({\mathcal {K}}\) is self-adjoint, we obtain \(\sigma ({\mathcal {K}}_n^{{\Omega }})\subset [m,M]\), with \(m\in \sigma ({\mathcal {K}}_n^{{\Omega }})\) and \(M\in \sigma ({\mathcal {K}}_n^{{\Omega }})\), where the set \(\sigma ({\mathcal {K}}_n^{{\Omega }})\) denotes the spectrum of \({\mathcal {K}}_n^{{\Omega }}\). Since \(\lambda _n(\Omega )\) is the largest eigenvalue, then
We shall prove that \(M>0\) and \(|m|\le M\). Indeed, for any \(h\in L^2_{\mu _\Omega }\), the positive function |h| belongs also to \( L^2_{\mu _\Omega }\) with the same norm and using the positivity of the kernel \(K_n\) we obtain
Using once again the positivity of the kernel one deduces that
Consequently, we obtain that \(M>0\). In order to prove that \(|m|\le M\), we shall proceed as follows. Using the positivity of the kernel, we achieve
This implies that M is nothing but the spectral radius of the operator \({\mathcal {K}}_n^{{\Omega }}\), that is,
From the Cauchy–Schwarz inequality, one deduces that
which implies that
For the lower bound, we shall work with the special function
with the normalized condition \(\Vert f\Vert _{\mu _{\Omega }}=1\) which is equivalent to
and
This gives the announced lower bound for the largest eigenvalue.
(3) From the expression of \(K_n\) given by (4.2) we easily get
Using the definition (4.5) of \(\kappa \) we infer
and we obtain
Applying Lemma 3.1 combined with the assumption (H2) yields for any \(0\le \alpha <\beta \le 1\)
By taking \(\beta <\frac{1}{2}\) we get the convergence of the integral and consequently we obtain the desired result,
(4) First, let us check that any nonzero eigenfunction associated to the largest eigenvalue \(\lambda _n(\Omega )\) should be with a constant sign. Indeed, let f be a nonzero normalized eigenfunction and assume that it changes the sign over a non negligible set. From the strict positivity of the kernel in the interval \((0,\pi )\), we deduce that
First, by the assumption on f we get
Second, from (4.85) we have that
Consequently,
achieving a contradiction. Hence, any nonzero eigenfunction of \(\lambda _n(\Omega )\) must have a constant sign. Now let us check that f is not vanishing in \((0,\pi )\). First we write
From (3.3) and Proposition 4.1 we get
The first assertion follows from the strict positivity of the associated hypergeometric function. Combined with the positivity of f we deduce that
Finally, we shall check that the subspace generated by the eigenfunctions associated to \(\lambda _n(\Omega )\) is one-dimensional. Assume that we have two independent eigenfunctions \(f_0\) and \(f_1\), which are necessarily with constant sign, then there exists \(a, b\in {{\,\mathrm{{\mathbb {R}}}\,}}\) such that the eigenfunction \(af_0+bf_1\) changes its sign. This is a contradiction.
(5) Using (4.2) combined with Lemma 3.1, we get that \(n\in {{\,\mathrm{{\mathbb {N}}}\,}}^{\star }\mapsto K_n(\phi ,\varphi )\) is strictly decreasing for any \(\varphi \ne \phi \in (0,\pi )\). Then, for any \(\Omega \in (-\infty ,\kappa )\) and for any nonnegative function f, we get
which implies in turn that
Since the largest eigenvalue \(\lambda _{n+1}(\Omega )\) is reached at some positive normalized function \(f_{n+1}\ge 0\), then
This provides the announced result.
(6) Fix \(\Omega _0\in (-\infty ,\kappa )\) and denote by \(f_n^\Omega \) the positive normalized eigenfunction associated to the eigenvalue \(\lambda _n(\Omega )\). Using the definition of the eigenfunction yields
The regularity follows from the general theory using the fact that this eigenvalue is simple. However we can in our special case give a direct proof for its differentiability in the following way. From the decomposition
we get according to the expression of \({\mathcal {K}}_n^{{\Omega }}\)
with
Therefore we obtain
As \({\mathcal {K}}_n^{{\Omega _0}}\) is self-adjoint on the Hilbert space \(L^2_{\mu _{\Omega _0}}\) then
Let us assume for a while that
Then we deduce that \(\Omega \mapsto \lambda _n(\Omega )\) is differentiable at \(\Omega _0\) with
Since
we find that \( \lambda _n^\prime (\Omega _0)>0\), which achieves the proof of the suitable result.
It remains to prove (4.90). First, for the numerator of the left hand side we first make the splitting
To estimate the second term \({\mathcal {I}}_2\) we use the identities (4.88) and (4.89) leading to
It follows that
Hence, applying Proposition 4.1-(2) combined with Cauchy-Schwarz inequality and the normalization assumption of the eigenfunctions in (4.87) we infer
From Remarks 4.1-(2), (4.86) and (4.87) we may write for \(\Omega \) close to \(\Omega _0\)
This obviously gives
Let us move to the term \({\mathcal {I}}_1\) introduced in (4.91). Then combining (4.89) with the fact that \({\mathcal {K}}_n^{{\Omega _0}}\) is self-adjoint on the Hilbert space \(L^2_{\mu _{\Omega _0}}\) allows to get
Applying Proposition 4.1-(2) we easily get that \(g\in L^2_{\mu _{\Omega _0}}\). Now we claim that
Before giving its proof, let us see how to conclude. It is easy to check from (4.94) and (4.95) that
Thus combining this result with (4.91) and (4.93) yields (4.90). It remains to check (4.95) which is a consequence of classical results on perturbation theory. One can use for instance [42] or [38, Chapter XII], where the analytic dependence of the eigenvalues and the associated eigenfunctions is analyzed. Let us briefly discuss the main arguments used to get the continuity of the eigenfunctions with respect to the parameter \(\Omega \). First we set
Then using (4.3) we finds
Then similarly to (4.88) we obtain the decomposition
By applying the lower bound of Proposition 4.1-(2) we get that \(A_m\) is bounded with
This shows that \(A(\Omega )\) is analytic for \(\Omega \) close enough to \(\Omega _0\). Now define the sets
Then it is known that the resolvent set \(A(\Omega )\) and \(\Gamma \) are open, see Theorem XII.7 in [38]. Now, since \(\lambda _n(\Omega _0)\) is an isolated simple eigenvalue and \(\Gamma \) is open then we may find \(\delta >0\) such that the oriented circle \(\gamma :=\big \{z, |z- \lambda _n(\Omega _0)|=\delta \big \}\) is contained in \(\rho (A(\Omega ))\) for all \(|\Omega -\Omega _0|\leqslant \delta \). Thus, for \(|\Omega -\Omega _0|\leqslant \delta \) the operator
is well-defined and it is analytic in view of (4.96). Notice that one may get the estimate
Then from classical results on spectral theory, see for instance Theorems XII.5-XII.6-XII.8 in [38], \(P_n(\Omega _0)\) is a projection on the one dimensional eigenspace associate to \(\lambda _n(\Omega _0)\). In addition, \(A_n(\Omega )\) admits only one eigenvalue inside the circle \(\Gamma \) which necessary coincides with \(\lambda _n(\Omega )\). Notice that this latter claim can be proved using the continuity in \(\Omega \) of the largest eigenvalue which can be checked from (4.85). Furthermore, \(P_n(\Omega )\) is still a one dimensional projection on the eigenspace associated to the \(\lambda _n(\Omega )\). As a consequence, if \(f_n^{\Omega _0}\) is a normalized eigenfunction of the operator \(A(\Omega _0)={\mathcal {K}}_n^{{\Omega _0}}\) associated to \(\lambda _n(\Omega _0)\), then \( P_n(\Omega )f_n^{\Omega _0} \) is an eigenfunction of \(A(\Omega )\) associated to \(\lambda _n(\Omega )\). Applying (4.97) yields
where we have used the fact that \(P_n(\Omega _0)f_n^{\Omega _0}=f_n^{\Omega _0}\). Now, by taking
we get a normalized eigenfunction in the sense of (4.87) and the family \(\Omega \mapsto f_n^\Omega \in L^2_{\mu _{\Omega _0}}\) is continuous at \(\Omega _0\), which ensures (4.95). This ends the proof of the desired result. \(\quad \square \)
Next we shall establish the following result.
Proposition 4.3
Let \(n\ge 1\) and \(r_0\) satisfies the assumptions (H1) and (H2). Set
Then the following holds true
-
(1)
The set \({\mathscr {S}}_n\) is formed by a single point denoted by \(\Omega _n\) .
-
(2)
The sequence \((\Omega _n)_{n\ge 1}\) is strictly increasing and satisfies
$$\begin{aligned} \lim _{n\rightarrow \infty }\Omega _n=\kappa . \end{aligned}$$
Proof
(1) To check that the set \({\mathscr {S}}_n\) is non empty we shall use the mean value theorem. From the upper bound in Proposition 4.2-(2) and (4.2) we find that
Thus by taking the limit as \(\Omega \rightarrow -\infty \) we deduce that
Next, we intend to show that
Using the lower bound of \(\lambda _n(\Omega )\) in Proposition 4.2-(2), we find by virtue of Fatou Lemma
for any nonnegative \(\varrho \) satisfying \(\displaystyle {\int _0^\pi \varrho ^2(\phi )d\phi =1}\). According to Proposition 4.1-(4), the function \(\nu _\kappa \) reaches its minimum at a point \(\phi _0\in [0,\pi ]\) and
There are two possibilities: \(\phi _0\in (0,\pi )\) or \(\phi _0\in \{0,\pi \}\). Let us start with the first case and we shall take \(\varrho \) as follows
with \(\beta <\frac{1}{2}\) and the constant \(c_\beta \) is chosen such that \(\varrho \) is normalized. Hence using the preceding estimates we get
Let \(\varepsilon >0\) such that \([\phi _0-\varepsilon ,\phi _0+\varepsilon ]\subset (0,\pi )\). According to (3.3) the function \(H_n\) is strictly positive in the domain \((0,\pi )^2\), hence there exists \(\delta >0\) such
Thus we obtain
By taking \(\frac{1+\alpha }{2}+\beta >1\), which is an admissible configuration, we find
Now let us move to the second possibility where \(\phi _0\in \{0,\pi \}\) and without any loss of generality we can only deal with the case \(\phi _0=0\). From (3.3) and using the inequality
we obtain
Combined with the assumption \({\mathbf{(H2)}}\), it implies
Plugging this into (4.101) we find
Let \(\varepsilon >0\) sufficiently small, then using Taylor expansion we get according to (3.2)
Thus
which gives after simplification
Making the change of variables \(\varphi =\phi \theta \) we obtain
This integral diverges provided that \(\alpha +2\beta >1\) and thus under this assumption
Hence we obtain (4.100). By the intermediate mean value, we achieve the existence of at least one solution for the equation
Consequently, using Proposition 4.2 we deduce by the mean value theorem that the set \({\mathscr {S}}_n\) contains only one element.
(2) Since \(\Omega _n\) satisfies the equation
According to Proposition 4.2-(5) the sequence \(k\mapsto \lambda _k(\Omega _n)\) is strictly decreasing. It implies in particular that
Hence by (4.100) one may apply the mean value theorem and find an element of the set \({\mathscr {S}}_{n+1}\) in the interval \((\Omega _n,\kappa )\). This means that \(\Omega _{n+1}>\Omega _n\) and thus this sequence is strictly increasing. It remains to prove that this sequence is converging to \(\kappa \). The convergence of this sequence to some element \({\overline{\Omega }}\le \kappa \) is clear. To prove that \({\overline{\Omega }}=\kappa \) we shall argue by contradiction by assuming that \({\overline{\Omega }}<\kappa \). By the construction of \(\Omega _n\) one has necessarily
Using the upper-bound estimate stated in Proposition 4.2-(2) combined with the point (3) we obtain for any \(\alpha \in (0,1)\)
By taking the limit as \(n\rightarrow +\infty \) we find
This contradicts (4.102) which achieves the proof. \(\quad \square \)
4.4 Eigenfunctions regularity
This section is devoted to the strong regularity of the eigenfunctions associated to the operator \({\mathcal {K}}_n^{{\Omega }}\) and constructed in Proposition 4.2. We have already seen that these eigenfunctions belong to a weak function space \(L^2_{\mu _\Omega }\). Here we shall show first their continuity and later their Hölder regularity.
4.4.1 Continuity
The main result of this section reads as follows.
Proposition 4.4
Let \(\Omega \in (-\infty ,\kappa )\), \(n\ge 1\), \(r_0\) satisfies the assumptions (H1) and (H2), and f be an eigenfunction for \({\mathcal {K}}_n^{{\Omega }}\) associated to a non-vanishing eigenvalue. Then f is continuous \(\hbox { over}\ [0,\pi ]\), and for \(n\ge 2\) it satisfies the boundary condition \(f(0)=f(\pi )=0\). However this boundary condition fails for \(n=1\) at least with the eigenfunctions associated to the largest eigenvalue \(\lambda _1(\Omega )\).
Proof
Let \(f\in L^2_{\mu _{\Omega }}\) be any non trivial eigenfunction of the operator \({\mathcal {K}}_n^{{\Omega }}\) defined in (4.82) and associated to an eigenvalue \(\lambda \ne 0\), then
Since \(f\in L^2_{\mu _\Omega }\), then the function \(g: \varphi \in [0,\pi ]\mapsto r_0^{\frac{3}{2}}(\varphi )f(\varphi )\)belongs to \( L^2((0,\pi );d\varphi )\). Therefore the equation (4.103) can be written in terms of g as follows
Coming back to the definition of \(H_n\) in (3.3) we obtain for some constant \(c_n\) the formula
Using (A.7) and the assumption (H2) yields
This implies, using Cauchy–Schwarz inequality and the fact that \(\nu _\Omega \) is bounded away from zero
It follows that g is bounded. Now inserting this estimate into (4.103) allows to get
Using once again estimat-1 and the assumption (H2) we deduce that
By symmetry we may restrict the analysis to \(\phi \in [0,\frac{\pi }{2}]\). Thus, splitting the integral given in (4.105) and using that
we obtain
It follows that
Using the change of variables \(\varphi =\phi \theta \) we get
Consequently we find
Inserting this estimate into (4.103) and using (4.104) yields
As before we can restrict \(\phi \in [0,\frac{\pi }{2}]\) and by using the fact
we deduce after splitting the integral
Making the change of variables \(\varphi =\phi \theta \) leads to
Consequently we get
This shows that f is bounded over \((0,\pi )\) and by the dominated convergence theorem one can show that f is in fact continuous on \([0,\pi ]\) and satisfies for \(n\ge 2\) the boundary condition
Last, we shall check that this boundary condition fails for \(n=1\) with the largest eigenvalue \(\lambda _1(\Omega )\). Indeed, according to (4.103) we have
However, from (3.3) we get
Combining this with the fact that f does not change the sign allows to get that \(f(0)\ne 0\). \(\quad \square \)
4.4.2 Hölder continuity
The main goal of this section is to prove the Hölder regularity of the eigenfunctions.
Proposition 4.5
Assume that \(r_0\) satisfies the conditions (H) and let \(\Omega \in (-\infty ,\kappa )\), then any solution h of the equation
with \(\lambda \ne 0\), belongs to \({\mathscr {C}}^{1,\alpha }(0,\pi )\), for any \(n\ge 2\). The functions involved in the above expression can be found in (3.3)–(4.2)–(4.3).
Proof
From the initial expression of the linearized operator (3.1) in Proposition 3.1 and combining it with Proposition 3.2, one has
with
It is clear that any solution h of (4.106) is equivalent to a solution of
From Proposition 4.1 we know that \(\nu _\Omega \in {\mathscr {C}}^{1,\alpha }(0,\pi )\) and does not vanish when \(\Omega \in (-\infty ,\kappa )\). Therefore to check the regularity \(h\in {\mathscr {C}}^{1,\alpha }(0,\pi )\) it is enough to establish that \({\mathcal {F}}_n(h)\in {\mathscr {C}}^{1,\alpha }(0,\pi )\), due to the fact that \({\mathscr {C}}^\alpha \) is an algebra. Since h is symmetric with respect to \(\phi =\frac{\pi }{2}\), then one can verify that \({\mathcal {F}}_n(h)\) preserves this symmetry and hence we shall only study the regularity in the interval \([0,\frac{\pi }{2}]\) and check that the left and right derivative at \(\pi /2\) coincide. Notice that Proposition 4.4 tells us that h is continuous in \([0,\pi ]\), for any \(n\ge 1\).
In order to prove such regularity, let us first check that
which is the key point in this proof. In order to do so, recall first from (2.18) that
On the other hand, define the function
which obviously verifies \(g_1(r_0(\phi ))={\widehat{R}}(\phi ,\varphi ,\eta )\). Such function has a minimum located at
Now we shall distinguish two cases: \(\cos \eta \in [0,1]\) and \(\cos \eta \in [-1,0]\). In the first case we get
From elementary trigonometric relations we deduce that
This implies in particular that, for \(\cos \eta \in [0,1]\)
As to the second case \(\cos \eta \in [-1,0]\), we simply notice that the critical point \(x_c\) is negative and therefore the second degree polynomial \(g_1\) is strictly increasing in \({{\,\mathrm{{\mathbb {R}}}\,}}_+\). This implies that
Therefore we get in both cases
By the symmetry property \({\widehat{R}}(\phi ,\varphi ,\eta )={\widehat{R}}(\varphi ,\phi ,\eta )\) we also get
Adding together (4.109)–(4.110)–(4.111), we achieve
It suffices now to combine this inequality with the assumption (H2) on \(r_0\) in order to get the desired estimate (4.108).
Let us now prove that \({\mathcal {F}}_n(h)\in {\mathscr {C}}^{1,\alpha }\) and for this aim we shall proceed in four steps.
\(\bullet \) Step 1: If \(h\in L^\infty \) then \({\mathcal {F}}_n(h)\in {\mathscr {C}}^\alpha (0,\pi )\).
Here we check that \({\mathcal {F}}_n(h)\in {\mathscr {C}}^\alpha (0,\pi )\) for any \(n\ge 1\). In order to avoid the singularity in the denominator coming from \(r_0\), we integrate by parts in the variable \(\eta \)
Introduce
and according to Chebyshev polynomials we know that
with \(U_{n}\) being a polynomial of degree n. Thus
Using the assumption (H2) combined with the estimate (4.108) for the denominator \({\widehat{R}}(\phi ,\varphi ,\eta )\), we achieve
Interpolating between the two inequalities
and
we deduce that for any \(\beta \in [0,1]\)
Then,
Let us now bound the derivative \(\partial _\phi K_1(\phi ,\varphi ,\eta )\). For this purpose, let us first show that
Indeed
Using the identity \(1-\cos (\eta )=2 \sin ^2(\eta /2)\) and (4.108), we get a constant C such that
achieving (4.115). Therefore, taking the derivative in \(\phi \) of \(K_1\) yields
Hence (4.114) allows to get or any \(\beta \in (0,1)\),
Here, we can use Proposition C.1 to the case where the operator \({\mathcal {K}}\) depends only on one variable by taking \(g_1(\theta ,\eta )=g_3(\theta ,\eta )=\sin ^{-\beta }(\eta /2)\). Then, we infer \({\mathcal {F}}_n(h)\in {\mathscr {C}}^\beta (0,\pi )\) for any \(\beta \in (0,1)\).
\(\bullet \) Step 2: For \(n\ge 2\), if h is bounded then \(h(0)=h(\pi )=0\).
Notice that this property was shown in Proposition 4.4 and we give here an alternative proof. Since \(\nu _\Omega \) is not vanishing then this amounts to checking that \({\mathcal {F}}_n(h)(0)=0\). By continuity, it is clear by Fubini that
which is vanishing if \(n\ge 2\). Hence, \(h(0)=0\), for any \(n\ge 2\). In the same way, we achieve \(h(\pi )=0\).
\(\bullet \) Step 3: If \(h\in {\mathscr {C}}^\alpha (0,\pi )\) and \(h(0)=h(\pi )=0\), then \({\mathcal {F}}_n(h)\in W^{1,\infty }(0,\pi )\).
We have shown before that \({\mathcal {F}}_n(h)\in {\mathscr {C}}^\beta (0,\pi )\) for any \(\beta \in (0,1)\). Then it is enough to check that \({\mathcal {F}}_n(h)^\prime \in L^\infty (0,\pi )\). For this aim, we write
Adding and subtracting some appropriate terms, we find
Let us bound each term separately. Using (4.108), (4.113) and (4.115) we achieve
We write in view of (4.114)
Therefore by imposing \(1-\alpha<\beta <1\) we get
which implies immediately that \(I_1\in L^\infty \). Now let us move to the boundedness of \(I_2\). From direct computations we get
Combining the assumption (H2) with \(r_0'\in W^{1,\infty }\) and the mean value theorem yields
Hence, using (4.108) and (4.113) we obtain
Interpolation inequalities imply
Therefore we get for any \(\phi \in (0,\pi /2), \varphi \in (0,\pi )\) and \(\eta \in (0,2\pi )\),
It follows that
which gives the boundedness of \(I_2\). It remains to bound the last term \(I_3\). Then integrating by parts we infer
To make the previous integration by parts rigorously, we should split the integral in \(\varphi \in (\varepsilon ,\phi -\varepsilon )\) and \(\varphi \in (\pi +\varepsilon ,\pi -\varepsilon )\) and later taking the limit as \(\varepsilon \rightarrow 0\).
Then, since \(h(0)=0\) and \(h\in {\mathscr {C}}^\alpha \) we find according to the assumptions (H), (4.108) and (4.113)
Applying (4.121) yields
That implies that \(h\in W^{1,\infty }(0,\pi /2)\).
Moreover, since \(r_0'(\pi /2)=0\) (this comes from the symmetry of \(r_0\) with respect to \(\pi /2\)) and using (4.116) we find that \(h'(\pi /2)=0\), which justifies why we can check the regularity only on \(\phi \in (0,\pi /2)\). Finally, we get the desired result, that is, \(h\in W^{1,\infty }(0,\pi )\).
\(\bullet \) Step 4: If \(h^\prime \in L^\infty (0,\pi )\) and \(h(0)=h(\pi )=0\), then \({\mathcal {F}}_n(h)^\prime \in {\mathscr {C}}^\beta (0,\pi )\) for any \(\beta \in (0,1)\).
Coming back to (4.117) and integrating by parts in the last integral we deduce
with
As in the previous step, integration by parts can be justified by splitting the integral in \(\varphi \in (\varepsilon ,\phi -\varepsilon )\) and \(\varepsilon \in (\phi +\varepsilon ,\pi -\varepsilon )\) and later taking limits as \(\varepsilon \rightarrow 0\).
We want to apply Proposition C.1 to each of those terms. First, for \(T_1\) we use (H) and (4.119) combined with (4.108), we arrive at
Since \(h'\in L^\infty \) and \(h(0)=h(\pi )=0\) then we can write \(h(\varphi )=\sin (\varphi ){\widehat{h}}(\varphi )\), with \({\widehat{h}}\in L^\infty (0,\pi )\).
Consequently,
Interpolating again, we find that for any \(\beta \in [0,1]\)
Let us mention that we have proven that
for any \(\beta \in (0,1)\), \(\phi \in (0,\pi /2), \varphi \in (0,\pi )\) and \(\eta \in (0,2\pi )\), which will be useful later.
Now we shall estimate the derivative of \(T_1\) with respect to \(\phi \). We start with
Using that \(r_0''\in L^\infty \), we find
Thus,
Using (4.108)–(4.119)–(4.125), we find
It follows that
Putting together this estimate with (4.123) we infer
for any \(\beta \in (0,1)\).
Concerning the estimate of the term \(T_2\), we first make appeal to (4.108) and (4.113) leading to
Applying (4.114) and (4.123), one finds
for any \(\beta \in (0,1)\). The next stage is devoted to the estimate of \(\partial _\phi T_2\) and one gets from direct computations
Using (4.115) and (4.108), it implies
Therefore (4.114) and (4.123) allows to get
for any \(\beta \in (0,1)\). Hence, by Proposition C.1, adapted to a one variable function, we achieve that \({\mathcal {F}}_n(h)'\in {\mathscr {C}}^\beta \), for any \(\beta \in (0,1)\), which achieves the proof of the announced result. \(\quad \square \)
4.5 Fredholm structure
In this section we shall be concerned with the Fredholm structure of the linearized operator \(\partial _f {\tilde{F}}(\Omega ,0)\) defined through (3.4) and (4.1). Our main result reads as follows.
Proposition 4.6
Let \(m\ge 2\), \(\alpha \in (0,1)\) and \(\Omega \in (-\infty ,\kappa )\), then \(\partial _f {\tilde{F}}(\Omega ,0):X_m^\alpha \rightarrow X_m^\alpha \) is a well-defined Fredholm operator with zero index. In addition, for \(\Omega =\Omega _m\), the kernel of \(\partial _f {\tilde{F}}(\Omega _m,0)\) is one-dimensional and its range is closed and of co-dimension one.
Recall that the spaces \(X_m^\alpha \) have been introduced in (2.15) and \(\Omega _m\) in Proposition 4.3.
Proof
We shall first prove the second part, assuming the first one. The structure of the linearized operator is detailed in (3.4) and one has for \(h(\phi ,\theta )=\sum _{n\ge 1} h_n(\phi )\cos (n\theta )\)
where
In view of (4.1) and (4.82), \({\mathcal {L}}_n^\Omega (h)=0\) can be written in the form
We define the dispersion set by
Hence \(\Omega \in {\mathcal {S}}\) if and only if there exists \(m\ge 1\) such that the equation
admits a nontrivial solution satisfying the regularity \(h_m\in {\mathscr {C}}^{1,\alpha }(0,\pi )\) and the boundary condition \(h_m(0)=h_m(\pi )=0\). By virtue of Propositions 4.4 and 4.5 the foregoing conditions are satisfied for any eigenvalue provided that \(m\ge 2\). On the other hand, we have shown in Proposition 4.2-(4) that for \(\Omega =\Omega _m\) the kernel of \({\mathcal {L}}_m\) is one-dimensional. Moreover, Proposition 4.2-(5) ensures that for any \(n>m\) we have \(\lambda _n(\Omega _m)<\lambda _m(\Omega _m)=1\). Since by construction \(\lambda _n(\Omega _m)\) is the largest eigenvalue of \({\mathcal {K}}_n^{\Omega _m}\), then 1 could not be an eigenvalue of this operator and the equation
admits only the trivial solution. Thus the kernel of the restricted operator \(\partial _f {\tilde{F}}(\Omega _m,0):X_m^\alpha \rightarrow X_m^\alpha \) is one-dimensional and is generated by the eigenfunction
We emphasize that this element belongs to the space \(X_m^\alpha \) because it belongs to the function space \({\mathscr {C}}^{1,\alpha }((0,\pi )\times {{{\,\mathrm{{\mathbb {T}}}\,}}})\) since \(\phi \mapsto h_m(\phi )\in {\mathscr {C}}^{1,\alpha }(0,\pi )\). That the range of \(\partial _f {\tilde{F}}(\Omega _m,0) \) is closed and of co-dimension one follows from the fact this operator is Fredholm of zero index.
Next, let us show that \(\partial _f {\tilde{F}}(\Omega ,0) \) is Fredholm of zero index. By virtue of the computations developed in Proposition 3.1 and the expression of (4.1), we assert that
with
Since \(\Omega \in (-\infty ,\kappa )\), the function \(\nu _\Omega \) is not vanishing. Moreover, by Proposition 4.1 one has that \(\nu _\Omega \in {\mathscr {C}}^{1,\beta }\), for any \(\beta \in (0,1)\).
Define the linear operator \(\nu _\Omega \text {Id}: X_m^\alpha \rightarrow X_m^\alpha \) by
We shall check that it defines an isomorphism. The continuity of this operator follows from the regularity \(\nu _\Omega \in {\mathscr {C}}^{1,\alpha }(0,\pi )\) combined with the fact that \( {\mathscr {C}}^{1,\alpha }((0,\pi )\times {{{\,\mathrm{{\mathbb {T}}}\,}}})\) is an algebra. The Dirichlet boundary condition, the m-fold symmetry and the absence of the frequency zero are immediate for the product \(\nu _\Omega h\), which finally belongs to \(X_m^\alpha \). Moreover, since \(\nu _\Omega \) is not vanishing, one has that \(\nu _\Omega \text {Id}\) is injective. In order to check that such an operator is an isomorphism, it is enough to check that it is surjective, as a consequence of the Banach theorem. Take \(k\in X_m^\alpha \), and we will find \(h\in X_m^\alpha \) such that \((\nu _\Omega \text {Id})(h)=k\). Indeed, h is given by
Using the regularity of \(\nu _\Omega \) and the fact that it is not vanishing, it is easy to check that its inverse \(\frac{1}{\nu _\Omega }\) still belongs to \({\mathscr {C}}^{1,\alpha }(0,\pi )\). Similar arguments as before allow to get \(h\in X_m^\alpha \). Hence \(\nu _\Omega \text {Id}\) is an isomorphism, and thus it is a Fredholm operator of zero index. From classical results on index theory, it is known that to get \(\partial _f {\tilde{F}}(\Omega ,0)\) is Fredholm of zero index, it is enough to establish that the perturbation \({G}: X_m^\alpha \rightarrow X_m^\alpha \) is compact. To do so, we prove that for any \(\beta \in (\alpha ,1)\) one has the smoothing effect
that we combine with the compact embedding \({\mathscr {C}}^{1,\beta }\big ((0,\pi )\times {{{\,\mathrm{{\mathbb {T}}}\,}}}\big )\hookrightarrow {\mathscr {C}}^{1,\alpha }\big ((0,\pi )\times {{{\,\mathrm{{\mathbb {T}}}\,}}}\big )\).
Take \(h\in X_m^\alpha \) and let us show that \(G(h)\in {\mathscr {C}}^{1,\beta }\big ((0,\pi )\times {{{\,\mathrm{{\mathbb {T}}}\,}}}\big )\), for any \(\beta \in (0,1)\). We shall first deal with a preliminary fact. Define the following function
By (2.17) we infer
According to the definition of the space \(X_m^\alpha \), the partial function \(\tau \mapsto h(\varphi ,\tau )\) is \(2\pi \)-periodic and with zero average, that is, \(\displaystyle {\int _{0}^{2\pi }h(\varphi ,\tau )d\tau =0}\). This allows to get that \(\eta \mapsto g_\theta (\varphi ,\eta )\) is also \(2\pi \)-periodic, and from elementary arguments we find
for any \(\varphi \in [0,\pi ]\) and \(\theta ,\eta \in [0,2\pi ]\). In addition, it is immediate that \(g_\theta \in {\mathscr {C}}^{1,\alpha }\big ((0,\pi )\times {{{\,\mathrm{{\mathbb {T}}}\,}}}\big )\) and
The same arguments as before show that the partial function \(\tau \mapsto \partial _\varphi h(\varphi ,\tau )\) is \(2\pi \)-periodic and with zero average. Moreover, \(\eta \mapsto \partial _\varphi g_\theta (\varphi ,\eta )\) is also \(2\pi \)-periodic and
for any \(\varphi \in [0,\pi ]\) and \(\theta ,\eta \in [0,2\pi ]\). Using the auxiliary function \(g_\theta \), one can integrate by parts in G(h) in the variable \(\eta \) obtaining
We can justify the integration by parts by splitting the integral in \(\eta \in (0,\theta -\varepsilon )\) and \(\eta \in (\theta +\varepsilon ,2\pi )\) and later taking limits as \(\varepsilon \rightarrow 0\). The boundary term in the above integration by parts is vanishing due to the periodicity in \(\eta \) of the involved functions. It follows from (4.108),
for any \(\phi ,\varphi \in (0,\pi )\) and \(\theta ,\eta \in (0,2\pi )\), and this estimate is crucial in the proof.
The boundedness of G(h) can be implemented by using (4.128) and (4.131). Indeed, we write
Therefore, we obtain
From the assumption \(\mathbf{(H2)}\) on \(r_0\) combined with (4.114) we get for any \(\beta \in (0,1)\), and then
Therefore
The next step is to check now that \(\partial _\phi G(h)\in {\mathscr {C}}^\beta \) by making appeal to Proposition C.1. From direct computations using (4.130) we find
Note that we can insert the derivative inside the integral, to make this rigorous, cut off the integral in \(\eta \) away from \(\theta \) and take a limit. Adding and subtracting in the numerator \(\partial _\varphi A(\phi ,\theta ,\varphi ,\eta )\), it can be written in the form
Integrating by parts in \(\varphi \) in the last term yields
The goal is to check the kernel assumptions for Proposition C.1 in order to prove that \({\mathscr {G}}_1\) and \({\mathscr {G}}_2\) belong to \({\mathscr {C}}^\beta \), for any \(\beta \in (0,1)\). For this aim, we define the kernels
and
Let us start with \(K_1\) and show that it satisfies the hypothesis of Proposition C.1. From straightforward calculus we obtain in view of the assumptions \({\mathbf{(H)}}\) and the mean value theorem
Using the inequality \(|ab|\le \frac{1}{2}(a^2+b^2)\) allows to get
Thus, applying (4.131) we deduce that
Then, putting together (4.128), \({\mathbf{(H2)}}\), (4.131) and (4.134) we find
As a consequence of (4.114), we immediately get
for any \(\beta \in (0,1)\). Let us compute the derivative with respect to \(\phi \) of \(K_1\),
From direct computations, we easily get
and
Then, it is clear from (4.131) that
In addition, one may check that
By using (4.128), \({\mathbf{(H2)}}\), (4.131), (4.136), (4.138) and (4.139), one achieves
Therefore, using some elementary inequalities allow to get
Applying (4.114) implies for any \(\beta \in (0,1)\)
Now, let us move to the estimate of the partial derivative \(\partial _\theta K_1\), given by
By definition of \(g_\theta \) in (4.127) and (4.128), one concludes in view of (2.17) and (4.131) that
Moreover, one gets
Using also the definition of A, we obtain
Then, with the help of (4.128), (4.131), (4.133) (4.140), (4.141) and (4.142), we can estimate \(\partial _\theta K_1\) as
Consequently we get
Therefore we obtain by virtue of (4.114)
for any \(\beta \in (0,1)\). Hence, all the hypothesis of Proposition C.1 are satisfied and therefore we deduce that \({\mathscr {G}}_1\) belongs to \({\mathscr {C}}^\beta \big ((0,\pi )\times {{{\,\mathrm{{\mathbb {T}}}\,}}}\big )\), for any \(\beta \in (0,1)\). The estimates of the kernel \(K_2\) we are quite similar to those of \(K_1\) modulo some slight adaptations. We shall not develop all the estimates which are straightforward and tedious. We will restrict this discussion to the analogous estimate to (4.135) and (4.143). First note that thanks to (4.128) and (4.129) one gets
This implies that
It follows from (4.131) and (4.114) that
which is the announced estimate. As to the estimate of \(\partial _\theta K_2\) we first write
Straightforward calculations using \({\mathbf{(H2)}}\) and (2.17) show that
Putting together (4.131) and (4.114) implies
Therefore we find
As to the second term of the right-hand side of (4.145), we get in view of \({\mathbf{(H2)}}\), (4.128) and (4.129)
It follows from (4.146) that
Concerning the last term of (4.145), we put together (4.142), (4.144), (4.131) and (4.114) that
Therefore collecting the preceding estimates allows to get the suitable estimate for \(\partial _\theta K_2\),
The estimate for \(\partial _\phi K_2\) can be done similarly in a straightforward way. Consequently the assumptions of Proposition C.1 hold true and one deduces that \({\mathscr {G}}_1\in {\mathscr {C}}^\beta \big ((0,\pi )\times {{{\,\mathrm{{\mathbb {T}}}\,}}}\big )\). Hence we obtain \(\partial _\phi G\in {\mathscr {C}}^\beta \big ((0,\pi )\times {{{\,\mathrm{{\mathbb {T}}}\,}}}\big )\), with the estimate
The next stage is to show that \(\partial _\theta G(h)\in {\mathscr {C}}^\beta \big ((0,\pi )\times {{{\,\mathrm{{\mathbb {T}}}\,}}}\big )\) following the same strategy as before. From (4.126), we get
Direct computations show that
It follows
On the other hand, integration by parts in \(\eta \) (this can be done by cutting off the integral in \(\eta \) away from \(\theta \) and taking a limit) yields
Thus we deduce by subtraction
Since \(h\in {\mathscr {C}}^{1,\alpha }\), then the mean value theorem implies
Moreover, by the \(2\pi \)-periodicity of h in \(\eta \) one obtains
Define the kernel
and let us check the hypothesis of Proposition C.1. First using (4.131), \({\mathbf{(H2)}}\) and (4.149) we obtain
Applying (4.114) yields
for any \(\beta \in (0,1)\). Let us estimate \(\partial _\phi K_3\) which is explicitly given by,
By virtue of (4.138) and (4.150), we achieve
for any \(\beta \in (0,1)\). It remains to establish the suitable estimates for \(\partial _\theta K_3\). First we have
Using (4.142) and (4.150) (where we exchange \(\beta \) by \(1-\beta \)) we get
For the second term of the right-hand side of \(\partial _\theta K_3\) we write in view of \({\mathbf{(H2)}}\)
Applying (4.146) yields
Concerning the last term of the right-hand side of \(\partial _\theta K_3\), it is similar to the foregoing one. Indeed, using (4.149) and \({\mathbf{(H2)}}\) we get
It suffices to use (4.146) to obtain
Therefore we get from the preceding estimates
Consequently, all the assumptions of Proposition C.1 are verified by the kernel \(K_3\) and thus we deduce that \(\partial _\theta G(h)\in {\mathscr {C}}^\beta \big ((0,\pi )\times {{{\,\mathrm{{\mathbb {T}}}\,}}}\big )\) for any \(\beta \in (0,1)\), with the estimate
Putting together this estimate with (4.147) and (4.132) yields
and this achieves the proof of the proposition. \(\quad \square \)
4.6 Transversality
We have shown in Proposition 4.6 that when \(\Omega \) belongs to the discrete set \(\{\Omega _m, m\ge 2\}\) then the linearized operator \(\partial _f {\tilde{F}}(\Omega ,0)\) is of Fredholm type with one-dimensional kernel. This property is not enough to bifurcate to nontrivial solutions for the nonlinear problem. A sufficient condition for that, according to Theorem B.1, is the the transversal assumption which amounts to checking
where \(f_m^\star \) is a generator of the kernel of \(\partial _f {\tilde{F}}(\Omega _m,0)\). Note that as a consequence of (4.1) and (4.2), for a function \(h:(\phi ,\theta )\mapsto \sum _{n\ge 1}h_n(\phi ) \cos (n\theta ) \in X_m^\alpha \), we get
with
where \({\mathcal {K}}_n^\Omega \) is defined in (4.82). Hence, the second mixed derivative takes the form,
Our main result of this section reads as follows.
Proposition 4.7
Let \(m\ge 2\), then the transversal condition holds true, that is,
where \(f_m^\star \) is a generator of the kernel of \(\partial _f {\tilde{F}}(\Omega _m,0)\).
Proof
Recall from the proof of Proposition 4.6 that the function \(f_m^\star \) has the form
and \(h_m^\star \) is a nonzero solution to the equation
It follows that
Assume that this element belongs to the range of \(\partial _{f} {\tilde{F}}(\Omega _m,0)\). Then we can find \(h_m\) such that
Dividing this equality by \(\nu _{\Omega _m}\) and taking the inner product with \(h_m^{\star }\), with respect to \(\langle \cdot ,\cdot \rangle _{\Omega _m}\) defined in (4.84) yields by the symmetry of \( {\mathcal {K}}_m^{\Omega _m}\)
Coming back to the definition of the inner product (4.84) and (4.4), we find
From the assumption \({\mathbf{(H)}}\) we know that \(r_0\) does not vanish in \((0,\pi )\). Then we get from the continuity of \(h_m^\star \) that this latter function should vanish everywhere in \((0,\pi )\), which is a contradiction. Hence, we deduce that \(f_m^\star \) does not belong to the range of \(\partial _f {\tilde{F}}(\Omega _m,0)\) and then the transversal condition is satisfied. \(\quad \square \)
5 Nonlinear Action
This section is devoted to the regularity study of the nonlinear functional \({\tilde{F}}\) defined in (2.13) that we recall for the convenience of the reader,
for any \((\phi ,\theta )\in (0,\pi )\times (0,2\pi )\) and where
the mean m is defined in (2.11) and
We would like in particular to analyze the symmetry/regularity persistence of the function spaces \(X_m^\alpha \) defined in (2.15) and (2.16) through the action of the nonlinear functional \({\tilde{F}}\).
5.1 Symmetry persistence
The main task here is to check the symmetry persistence of the function spaces \(X_m^\alpha \) defined in (2.15) through the nonlinear action of \({\tilde{F}}\). Notice that at this level, we do not raise the problem of whether or not this functional is well-defined and this target is postponed later in Sect. 5. First recall that
Proposition 5.1
Let \(\Omega \in {{\,\mathrm{{\mathbb {R}}}\,}}\), \(f\in X_m^\alpha \) with \(m\ge 1\) and assume that \(r_0\) satisfies the conditions \({\mathbf{(H)}}\). Then the following assertions hold true.
-
(1)
The equatorial symmetry:
$$\begin{aligned}{\tilde{F}}(\Omega ,f)\left( \pi -\phi ,\theta \right) ={\tilde{F}}(\Omega ,f)\left( \phi ,\theta \right) ,\quad \forall \, (\phi ,\theta )\in [0,\pi ]\times {{\,\mathrm{{\mathbb {R}}}\,}}. \end{aligned}$$ -
(2)
We get the algebraic structure,
$$\begin{aligned} {\tilde{F}}(\Omega ,f)(\phi ,\theta )=\sum _{n\ge 1} f_n(\phi )\cos (n\theta ), \end{aligned}$$for some functions \(f_n\) and for any \((\phi ,\theta )\in [0,\pi ]\times [0,2\pi ]\).
-
(3)
The m-fold symmetry: \({\tilde{F}}(\Omega ,f)(\phi ,\theta +\frac{2\pi }{m})={\tilde{F}}(\Omega ,f)(\phi ,\theta )\), for any \((\phi ,\theta )\in [0,\pi ]\times {{\,\mathrm{{\mathbb {R}}}\,}}\).
Proof
\({\mathbf{(1)}}\) From the expression of \({\tilde{F}}\) in (2.13), it is suffices to check the property for I(f). One can easily verify using the symmetry of the functions \(\cos \) and r combined with the change of variables \(\varphi \mapsto \pi -\varphi \)
\({\mathbf{(2)}}\) In order to get the desired structure, it suffices to check the following symmetry
To do that, we use the symmetry of r, that is \(r(\varphi ,-\theta )=r(\varphi ,\theta )\), combined with the change of variables \(\eta \mapsto -\eta \) allowing one to get
\({\mathbf{(3)}}\) First, since r belongs to \(X_m^\alpha \) then it satisfies \(r(\varphi , \theta +\frac{2\pi }{m})=r(\varphi ,\theta )\). Thus we get by the change of variables \(\eta \mapsto \eta +\frac{2\pi }{m}\)
Notice that we have used the fact that the Euclidean distance in \({\mathbb {C}}\) is invariant by the rotation action \(z\mapsto e^{i\frac{2\pi }{m}} z\). \(\quad \square \)
The next discussion is devoted to the symmetry effects of the surface of the vortices on the velocity structure. We shall show the following.
Lemma 5.1
If \(r_0\) satisfies \({\mathbf{(H)}}\) and \(f\in X_m^\alpha \), with \(m\ge 2\), then
As a consequence, the velocity field defined in (2.4) is vanishing at the vertical axis, that is,
for any \(z\in {{\,\mathrm{{\mathbb {R}}}\,}}\).
Proof
Set for any \(z\in {{\,\mathrm{{\mathbb {R}}}\,}}\),
Observe that from the periodicity in \(\eta \) we may write
Since \(f\in X_m^\alpha \), then \(r(\varphi ,-\eta )=r(\varphi ,\eta )\) and so \((\partial _\eta r)(\varphi ,-\eta )=-(\partial _\eta r)(\varphi ,\eta )\). Therefore making the change of variables \(\eta \mapsto -\eta \) allows to get \(I_1(z)=0\).
To check \(I_2(z)=0\) we shall use the m-fold symmetry of r. In fact by the change of variables \(\eta \mapsto \eta +\frac{2\pi }{m}\) and using the \(2\pi \)-periodicity in \(\eta \) and some elementary trigonometric identity, we find
Since \(m\ge 2\) and \(I_1(z)=0\) then we get \(I_2(z)=0\).
Coming back to (2.4) and following the change of variables giving (2.8) we easily get
which gives the announced result. \(\quad \square \)
5.2 Deformation of the Euclidean norm
The spherical change of coordinates used to recover both the velocity and the stream function from the surface geometry of the patch yields a deformation of the Green function. Notice that in the usual Cartesian coordinates the Green kernel is radial and thus it is isotropic with respect to all the variables. In the new coordinates we lose this property and the Green kernel becomes anisotropic and the north and south poles are degenerating points. To deal with these defects one needs refined treatments in the behavior of the kernel or also the adaptation of the function spaces which are of Dirichlet type. The following lemma is crucial to deal with the anisotropy of the kernel.
Lemma 5.2
Let \(m\ge 1, \alpha \in (0,1)\), \(r_0\) satisfies \({\mathbf{(H)}}\), \(f\in X_m^\alpha \) such that \(\Vert f\Vert _{X_m^\alpha }\le \varepsilon \) with \(\varepsilon \) small enough and set \(r=r_0+f\). Define for any \(\phi \in [0,\frac{\pi }{2}]\), \(\varphi \in [0,\pi ]\), \(\theta ,\eta \in [0,2\pi ]\) and \(s\in [0,1]\)
Then
with C an absolute constant. Remark that we have restricted \(\phi \) to \(\in [0,\pi /2]\) instead of \([0,\pi ]\) because of the symmetry of r with respect to \(\frac{\pi }{2}\).
Proof
Since \(f\in B_{X_m^\alpha }(\varepsilon )\), for some \(\varepsilon <1\), and \(r_0\) verifies (H2) then
In addition f satisfies (2.17) and in particular
It follows that,
By imposing \(\Vert f\Vert _{\text {Lip}}\le \varepsilon = \frac{C}{C_1}\), we infer
Consequently, we obtain
which gives the estimate (5.1). Let us now check the validity of (5.2). First, we remark that
Denote
and therefore we get the relation \(g_1(sr(\phi ,\theta ))=J_s(\phi ,\theta ,\varphi ,\eta )\). From variation arguments we infer that the function \(g_1\) reaches its global minimum at the point
Let us distinguish the cases \(\cos (\theta -\eta )\in [0,1]\) and \(\cos (\theta -\eta )\in [-1,0]\). In the first case, one has according to (5.3)
Using that \(\cos (\theta -\eta )\in [0,1]\), one gets
Moreover, since \(\phi \in [0,\frac{\pi }{2}]\) and \(\varphi \in [0,\pi ]\), we obtain
Hence
In the second case where \(\cos (\theta -\eta )\in [-1,0]\), the critical point is negative, \(x_c\le 0\), and one has from the variations of \(g_1\), the estimate (5.3) and (5.4)
Putting together (5.5) and (5.6), one deduces that
for any \(\phi \in [0,\pi /2]\), \(\varphi \in [0,\pi ]\) and \(\theta ,\eta \in [0,2\pi ]\).
Following the same ideas, we introduce the function
which satisfies \(g_2(r(\varphi ,\eta ))=J_s(\phi ,\theta ,\varphi ,\eta )\). Then as before we can check easily that the function \(g_2\) reaches its minimum at the point \( {\tilde{x}}_c=sr(\phi ,\theta )\cos (\theta -\eta ). \) Similarly we distinguishing between two cases \(\cos (\theta -\eta )\in [0,1]\) and \(\cos (\theta -\eta )\in [-1,0]\). For the first case, using (5.4), we have
Since \(\phi \in [0,\pi /2]\), we have that \(\sin (\phi )\ge \frac{2}{\pi }\phi \), and then
In the other case, i.e. \(\cos (\theta -\eta )\in [-1,0]\), one has that \({\tilde{x}}_c<0\) and then
By summing up (5.7)–(5.8)–(5.9) we achieve (5.2). \(\quad \square \)
5.3 Regularity persistence
In this section we shall investigate the regularity of the function \({\tilde{F}}\) introduced in (2.13). The main result reads as follows.
Proposition 5.2
Let \(m\ge 2, \alpha \in (0,1)\) and \(r_0\) satisfy \({\mathbf{(H)}}\). There exists \(\varepsilon \in (0,1)\) small enough such that the functional
is well-defined and of class \({\mathscr {C}}^1\). The function spaces \(X_m^\alpha \) are defined in (2.15) and (2.16).
Proof
First we shall split the functional \({\tilde{F}}\) into two pieces
with
Define also
Note that I(f) is defined (2.14) and it is nothing but the stream function \(\psi _0\) associated to the domain parametrized by
Thus
We point out that according to the general potential theory the stream function \(\psi _0\) belongs at least to the space \({\mathscr {C}}^{1,\alpha }({{\,\mathrm{{\mathbb {R}}}\,}}^3)\). The proof will be divided into three steps.
Step 1: \(f\mapsto {\mathscr {F}}_2(f)\) is \({\mathscr {C}}^1\). In this step, we check that \({\mathscr {F}}_2\) is well-defined and of class \({\mathscr {C}}^1\). Note that checking the regularity for \({\mathscr {F}}_2\) is equivalent to do it for \(F_2\). The first term of \(F_2\) is trivial to check. As to the second one, it is clear by Taylor’s formula using the boundary conditions and \({\mathbf{(H2)}}\) that the function \(\frac{f^2}{r_0}\) is bounded and vanishes at the points \(\phi =0,\pi \). For the regularity, we differentiate with respect to \(\phi \),
Using again Taylor’s formula and the assumptions \({\mathbf{(H)}}\) on \(r_0\) we deduce that the functions \((\phi ,\theta )\mapsto \frac{f(\phi ,\theta )}{\sin \phi }\) and \((\phi ,\theta )\mapsto \frac{\sin \phi }{r_0(\phi )}\) belongs to \({\mathscr {C}}^{\alpha }\). Thus using the algebra structure of this latter space we infer that \((\phi ,\theta )\mapsto \frac{f(\phi ,\theta )}{r_0(\phi )}\) belongs also to \({\mathscr {C}}^\alpha \). The same algebra structure allows to get \(\partial _\phi \left( \frac{f^2}{{ r_0}}\right) \in {\mathscr {C}}^\alpha \). Following the same argument we obtain \(\partial _\theta F_2\) belongs to \({\mathscr {C}}^\alpha \). Concerning the symmetry; it can be derived from Proposition 5.1 combined with the fact that frequency \(n=0\) is eliminated in the definition of \({\mathscr {F}}_2\) by subtracting the mean value in \(\theta \).
Now let us check the \({\mathscr {C}}^1\) dependence in f of \(F_2\). First we can check that its Frechet derivative takes the form
Using similar ideas as before, we can easily get that
This implies that \(f\mapsto \partial _f {\mathscr {F}}_2(f)\) is continuous and therefore \({\mathscr {F}}_2\) is of class \({\mathscr {C}}^1\).
Step 2: \(f\mapsto {\mathscr {F}}_1(f)\) is well-defined. This is more involved than \({\mathscr {F}}_2\). According to Proposition 5.1 the functional \({\mathscr {F}}_1\) is symmetric with respect to \(\phi =\frac{\pi }{2}\) and therefore it suffices to check the desired regularity in the range \(\phi \in (0,\pi /2)\) and check that the derivative is not discontinuity at \(\pi /2\). Let us emphasize that we need to check the regularity not for \(F_1\) but for \({\mathscr {F}}_1\) First, we shall check that \({\mathscr {F}}_1\) is bounded and satisfies the boundary condition \({\mathscr {F}}_1(0,\theta )=0\), for any \(\theta \in (0,2\pi )\). The remaining boundary condition \({\mathscr {F}}_1(\pi ,\theta )=0\) follows from the symmetry with respect to the equatorial. For this purpose, we write by virtue of Taylor’s formula
Making the substitution \(x_h=(r_0(\phi )+f(\phi ,\theta ))e^{i\theta }\) and using (5.10) we infer
We observe that the \(\cdot \) denotes the usual Euclidean inner product of \({{\,\mathrm{{\mathbb {R}}}\,}}^2\). Consequently, we obtain
Let us analyze the term \( {\mathscr {F}}_{1,1}\) and check its continuity and the Dirichlet boundary condition. First we observe from the assumption \({\mathbf{(H2)}}\) that 0 is a simple zero for \(r_0\) and we know that \(f(0,\theta )=0\), then one may easily obtain the bound
Furthermore, according to Lebesgue dominated convergence theorem we infer
and this convergence is uniform in \(\theta \in (0,2\pi )\). Notice that the same tool gives the continuity of \( {\mathscr {F}}_{1,1}\) in \([0;\pi /2]\times [0;2\pi ]\).
Now, applying Lemma 5.1 we get \(\nabla _h\psi _0\left( 0,0, 1\right) =0\), and therefore
This implies that \({\mathscr {F}}_1\) is continuous in \([0,\pi ]\times [0,2\pi ]\) and it satisfies the required Dirichlet boundary condition \({\mathscr {F}}_1(0,\theta )={\mathscr {F}}_1(\pi ,\theta )=0\).
The next step is to establish that \(\partial _\theta {\mathscr {F}}_1\) and \(\partial _\phi {\mathscr {F}}_1\) are \({\mathscr {C}}^\alpha \). We will relate such derivatives to the two-components velocity field \(U=\nabla _h^\perp \psi _0\). Differentiating (5.10) with respect to \(\theta \) leads to
where \(r(\phi ,\theta )=r_0(\phi )+f(\phi ,\theta )\) and recall that \(\cdot \) is the usual Euclidean inner product of \({{\,\mathrm{{\mathbb {R}}}\,}}^2\).
Concerning the regularity of the partial derivative in \(\phi \), we achieve
Define
and
Then from (5.14) we may write
Let us justify why we can restrict ourselves to prove the regularity for \(\phi \in [0,\pi /2]\) by symmetry. Indeed, since \(r(\pi -\phi ,\theta )=r(\phi ,\theta )\) and \(r_0(\pi -\phi )=r_0(\phi )\), then we obtain that
That implies that \(r_0'(\pi /2)=0\) and \(\partial _\phi r(\pi /2,\theta )=0\). By this way, it is easy to check that \({\mathscr {J}}_1(\pi /2,\theta )={\mathscr {J}}_2(\pi /2,\theta )=0\) yielding that there is not any jump at \(\pi /2\). Moreover, note that the last term belongs to \({\mathscr {C}}^\alpha \) for any \(\phi \in [0,\pi ]\) (here we do not need to resctrict ourselves to \(\phi \in [0,\pi /2]\)) and then, by symmetry, we can extend it to \((0,\pi )\). Indeed, as \((\phi ,\theta )\mapsto \big (r(\phi ,\theta )e^{i\theta },\cos (\phi )\big )\) belongs to \({\mathscr {C}}^{1,\alpha }\) and \(\partial _z\psi _0\in {\mathscr {C}}^{\alpha }({{\,\mathrm{{\mathbb {R}}}\,}}^3)\) then by composition we infer \((\phi ,\theta )\mapsto \partial _z\psi _0\big (r(\phi ,\theta )e^{i\theta },\cos (\phi )\big )\) is in \({\mathscr {C}}^{\alpha }((0,\pi )\times {{{\,\mathrm{{\mathbb {T}}}\,}}}\big )\). On the other hand, the function \(\frac{\sin }{r_0}\) belongs to \( {\mathscr {C}}^\alpha \) and thus by the algebra structure of \({\mathscr {C}}^\alpha \) we obtain the desired result.
Concerning the term \({\mathscr {J}}_{1}\), we use Taylor’s formula for the stream function \(\psi _0\) as in (5.11) finding that
We observe that the first term is singular and depends only on \(\phi \) and therefore it does not contribute in \({\mathscr {J}}_{1}-\langle {\mathscr {J}}_{1}\rangle _\theta \). Since \((\phi ,\theta )\mapsto \frac{r(\phi ,\theta )}{r_0(\phi )}\) belongs to \({\mathscr {C}}^\alpha \) then to get \({\mathscr {J}}_{1}-\langle {\mathscr {J}}_{1}\rangle _\theta \in {\mathscr {C}}^\alpha \) it suffices to prove that
On the other hand to obtain \({\mathscr {J}}_2\in {\mathscr {C}}^\alpha \) it is enough to get
From (5.13) we get that \(\partial _\theta F_1(f)\in {\mathscr {C}}^\alpha \) provided that (5.15) and (5.16) are satisfied together with
By virtue of (2.10) and the fact that \(U=\nabla _h^\perp \psi \), we find that
Next we intend to prove (5.15), (5.16) and (5.17).
\(\bullet \) Proof of (5.17). Using (5.18), we deduce that
Using the notation of Lemma 5.2 we find that
and therefore we may write
This can be split into two integral terms
Next, we shall prove that \({{\mathcal {I}}}_1\) is \({\mathscr {C}}^\alpha \). Notice that the second term \({\mathcal {I}}_2\) is easier to deal with than \({\mathcal {I}}_1\) because its kernel is more regular on the diagonal than that of \({\mathcal {I}}_1\). To get \({\mathcal {I}}_2\in {\mathscr {C}}^\alpha \) it suffices to use in a standard way Proposition C.1. We shall skip this part and focus our attention on the proof to the delicate part \({\mathcal {I}}_1\). For this aim let us define the kernel
We shall start with checking that \({\mathscr {K}}_1\) is bounded. For this goal we use Lemma 5.2 which implies
It is easy to check the inequality
By interpolating between
and
one finds that for any \(\beta \in [0,1]\) and \(\phi \le \pi /2\)
implying that
Therefore, we easily achieve that \({\mathcal {I}}_1\in L^\infty \). To establish that \({\mathcal {I}}_1\in {\mathscr {C}}^\alpha \), we proceed in a direct way using the definition. Before that we remark that to get the \({\mathscr {C}}^\alpha \) regularity in both variables \((\phi ,\theta )\) it is enough to check the \({\mathscr {C}}^\alpha \)-regularity separately in the partial variables. Thus we shall check that \(\phi \mapsto {\mathcal {I}}_1(\phi ,\theta )\) is \({\mathscr {C}}^\alpha (0,\pi )\) uniformly in \(\theta \in [0,2\pi ]\). Take \(\phi _1,\phi _2\in [0,\frac{\pi }{2}]\) with \(0<\phi _1<\phi _2\), then it is easy to check from some algebraic considerations that
Coming back to the definition of \(J_1\) seen in Lemma 5.2, we can check that
Since \(r\in \text {Lip}\) we infer by interpolation
and
Consequently we find
From straightforward calculus we observe that
and then we find
It follows that
Using (5.23), one finds
By virtue of (5.21), for any \(\beta \in (0,1)\) we obtain
Hence, we get in view of Lemma 5.2 and (2.17)
for any \(i=1,2\). This implies that
It is classical that
provided that \(0\le \alpha<\beta <1\). Similarly we prove under the same condition that
Putting together the preceding estimates yields for any \(0<\alpha<\beta <1\)
where the constant C is independent of \(\theta , \phi _1\) and \(\phi _2\). Let us move on the regularity of \({\mathcal {I}}_1\) in \(\theta \). Here we shall use the estimate later proved (5.34) for \(s=1\), that is, for \(\theta _1,\theta _2\in [0,2\pi ]\) we have
Note that
On the one hand, using (5.21) we easily get
for any \(\beta \in (0,1)\). On the other hand, we use (5.24) to estimate the second term:
By virtue of Lemma 5.2, we arrive at
for \(\gamma _1,\gamma _2,\gamma _3\in (0,1)\) and \(\gamma _2+\gamma _3<1\).
At the end, we can find that
Finally, this allows to get that \({\mathcal {I}}_1\) is \({\mathscr {C}}^\alpha ((0,\pi )\times {{{\,\mathrm{{\mathbb {T}}}\,}}})\).
\(\bullet \) Proof of (5.16). In fact we shall establish a more refined result:
uniformly with respect to \(s\in [0,1]\). This allows to get the results (5.16) and (5.15).
Coming back to (5.18) and using \(J_s\) introduced in Lemma 5.2 we find the expression
Moreover, by Lemma 5.1 we have
and then we can subtract this vanishing term obtaining
Notice that we eliminate .the variable \(\theta \) from the definition of \(J_0\) because it is independent of this parameter. Since \((\phi ,\theta )\mapsto \frac{r(\phi ,\theta )}{r_0(\phi )}\) is \({\mathscr {C}}^\alpha \), then to get the desired regularity it is enough to check it for the the integral term. Denote
and let us show first that
belongs to \(L^\infty \). It is plain that
Combined with the estimate (5.1), it yields
As we have mentioned before at different stages, the symmetry allows us to restrict the discussion to the interval \(\phi \in (0,\pi /2)\). Then using Lemma 5.2 and (2.17) we achieve that
Hence, the estimate of (2.17) allows to get
By interpolation we deduce for any \(\beta \in (0,1)\),
It follows that
Let us move to the \({\mathscr {C}}^\alpha \)-regularity of this latter function. This amounts to checking the partial regularity separately in \(\phi \) and \(\theta \). The strategy is the same for both of them and to alleviate the discussion, we shall establish the regularity in the variable \(\theta \), contrary to the preceding section where it was established for \({\mathcal {I}}_1\) in the direction of \(\phi \). The goal is to get a convenient estimate for the difference
where \(0\le \theta _1<\theta _2\le 2\pi \). Coming back to the definition of the kernel \({\mathscr {K}}_2\) in (5.26) one deduces through straightforward algebraic computations that
with
and
We shall estimate independently each one of those terms. Concerning the term \({\mathcal {I}}_3\) it can be estimated using (5.1)
Then by virtue of (5.27) and (2.17), we find
Combining this estimate with the interpolation inequality: for any \(\beta \in [0,1]\)
we infer
Thus
uniformly in \(\phi \in (0,\pi /2)\) and \(\theta _1,\theta _2\in (0,2\pi )\), provided that \(0<\beta <\alpha \le 1\).
Concerning the term \({\mathcal {I}}_4\), we first use the definition of \(J_s\) in Lemma 5.2 and one may check
Using the trigonometric identity
we get
From (2.17) combined with Taylor’s formula we find
Therefore we get by interpolation inequality
Hence
Combining (5.31) together with (5.30) implies
Define
Using once again (5.29) we get
Note that using the inequality
and interpolating, one achieves
for any \(\alpha \in (0,1)\). Putting everything together, one finds
Now using the assumption (2.17) and (5.2) we find
Inserting this inequality into (5.32) implies
It follows that
Thus we get
Applying (5.27) combined with (2.17) we arrive at
The right hand side terms \({\mathcal {J}}_1\) and \({\mathcal {J}}_2\) are treated similarly and we shall only focus on the first one. We find that
Using (5.2) we deduce, since \(\alpha \in (0,1)\), that
Thus
The same estimate holds true for \({\mathcal {J}}_2\). Therefore we find
Hence by interpolation inequality we get for all \(\gamma \in (0,1)\)
It follows that
uniformly in \(\phi \in (0,\pi )\) and \(\theta _1,\theta _2\in (0,2\pi )\), provided that \(0<\gamma <\min (1-\alpha , \alpha ^2)\).
Let us focus on the estimate of the term \({\mathcal {I}}_5\). First we make the decomposition
with
and
Let us start with the last term \({\mathcal {J}}_4\). Using Lemma 5.2 combined with (5.33) and (2.17) yields
From (5.2) we infer
Applying (5.27) leads to
Therefore
uniformly in \(\phi \in (0,\pi )\) and \(\theta _1,\theta _2\in (0,2\pi )\) provided that \(0<\beta<\alpha <1\). Next we shall deal with the term \({\mathcal {J}}_5\). Then by virtue of (5.31) combined with (2.17) we may write
It follows that
According to (5.36) and since \(\alpha \in (0,1)\) we find that
and similarly we obtain
Consequently, we get from (5.27)
By integration, we obtain
uniformly in \(\phi \in (0,\pi )\) and \(\theta _1\in (0,2\pi )\) provided that \(0<\gamma <\min (\alpha ^2,1-\alpha )\).
Following the same ideas as before and using (5.27) we get
It follows that
By integration, we deduce that
uniformly in \(\phi \in (0,\pi )\) and \(\theta _1\in (0,2\pi )\) provided that \(0<\alpha <1\). Putting together the preceding estimates allows to get
Let us finish estimating the term \({\mathcal {I}}_6\), which is quite similar to \({\mathcal {I}}_4\). Applying (5.34), Lemma 5.2 and the ideas done for \({\mathcal {I}}_4\) we obtain
Using again Lemma 5.2 we achieve
for some \(\gamma _1,\gamma _2,\gamma _3,\gamma _4\in (0,1)\) with \(\gamma _1+\gamma _2<1\) and \(\gamma _3+\gamma _4<1\). Notice that this is possible since \(1+2\alpha -\alpha ^2<2\) for any \(\alpha \in (0,1)\).
Therefore, we obtain
uniformly in \(\phi \in (0,\pi )\). This concludes the proof of the stability of the function spaces by \({\tilde{F}}\).
Step 3: \(F_1\) is \({\mathscr {C}}^1\). In this last step, we check that \(F_1\) is \({\mathscr {C}}^1\).
More precisely, we intend to prove the following
and
for some \(\gamma >0\), and where \(f_1,f_2\in B_{X_m^\alpha }(\varepsilon )\), for some \(\varepsilon <1\).
Notice that (5.37)–(5.38)–(5.39) will imply the \({\mathscr {C}}^1\)-regularity of \(F_1\). Denote \(r_i(\phi ,\theta )=r_0(\phi )+f_i(\phi ,\theta )\), for \(i=1,2\). We will check directly the estimates for the derivatives, i.e., (5.38)–(5.39) and leave the first estimate which is less delicate. From the expressions (5.13) and (5.14), it is enough to check the estimates for the terms: \(U\cdot e^{i\theta }\) and \(\frac{U}{r_0}\cdot ie^{i\theta }\).
As we can guess the computations are very long, tedious and share lot of similarities. For this reason we shall focus only on one significant term given by (5.19) to illustrate how the estimates work, and restrict the discussion to some terms of \({\mathcal {I}}_2\). Notice that the Frechet derivative of each of the previous terms will correspond again to a singular integral where the kernel has the same order of singularity and then the estimates work similarly, even if the computations are longer.
Then, let us show the idea for \({\mathcal {I}}_2\) and note that
where
and
Let us exhibit the main ideas of the term \({\mathcal {T}}_1\). The goal is to check
for some \(\gamma >0\). We observe that
Now we write for any \(\phi ,\varphi \in (0,\pi )\) and \(\theta ,\eta \in (0,2\pi )\)
By the \({\mathscr {C}}^{1,\alpha }\) regularity of r one has
In addition, we claim that
Indeed, and without any restriction to the generality we can impose that \(0\le \eta \le \theta \le 2\pi \). We shall discuss two cases: \(0\le {\theta -\eta }\le \pi \) and \(\pi \le {\theta -\eta }\le 2\pi \). In the first case, we simply write
with C a constant. As to the second case \(\pi \le {\theta -\eta }\le 2\pi \), by setting \({\widehat{\eta }}=\eta +2\pi \) we get
Since \(\eta \mapsto r(\varphi ,\eta )\) is \(2\pi \)-periodic then using the result of the first case yields
This achieves the proof of the (5.42). Consequently we find
From algebraic calculus we easily get
Therefore we deduce successively from (5.43)
and
By interpolation, we infer for any \(\gamma \in [0,1]\),
On the other hand, coming back to the definition of \(J_1\) we get
Thus, putting together this inequality with (5.44) and (5.41) yield
Now, we shall give an estimate of \({\mathcal {T}}_1(f_1)-{\mathcal {T}}_1(f_2)\) in \( L^\infty \). For this purpose, define the quantity
then one can easily check that
From this definition, it follows that
According to (5.2) we get
Combining this inequality with (5.45) and (5.46) leads to
Applying Lemma 5.2, we infer
Using the inequality \(\varphi ^2\ge \sin ^2(\varphi )\) for any \(\varphi \in {{\,\mathrm{{\mathbb {R}}}\,}}\), one achieves
The boundary conditions \(h(0,\eta )=h(\pi ,\eta )=0\) allow to cancel the singularity and one gets
Interpolating we find that for any \(\beta \in (0,1)\),
Thus, we have that \({\mathcal {I}}_7\) is integrable in the variable \((\varphi ,\eta )\) uniformly in \((\phi ,\theta )\), and then
The next purpose is to establish the partial \({\mathscr {C}}^\alpha \)-regularity in \(\phi \), and the partial regularity in \(\theta \) can be done similarly. We want to prove the following
For this goal we need to study the kernel \( |{\mathcal {I}}_7(\phi _1)-{\mathcal {I}}_7(\phi _2)|. \) To alleviate the notation we simply denote \({\mathcal {I}}_7(\phi , \theta ,\varphi ,\eta )\) by \({\mathcal {I}}_7(\phi )\) and \(J_1(f_i)(\phi _i,\theta ,\varphi ,\eta )\) by \(J_1(f_i)(\phi _i)\). Adding and subtracting some appropriate terms, one finds
with
and
The estimate of those terms are quite similar and we shall restrict the discussion to the term \({\mathcal {I}}_{10}\) which involves more computations. The analysis is straightforward and we will just give the basic ideas. First one should give a suitable estimate for the quantity
By using (5.41)–(5.44), one finds
Moreover,
and
In a similar way, we deduce first by triangular inequality
and second from (5.44)
Combining the preceding estimate we achieve
where we use the notation
Hence,
By using the definition of \(J_1\) in Lemma 5.2, we immediately get
that we combine with (5.43) in order to get
We shall analyze the term associated to \({\mathscr {E}}_{1,1}\) and the treatment of the other ones are quite similar. First we note
Making appeal to (5.27) and (2.17), we infer
By interpolation we obtain for any \(\gamma , \beta \in [0,1]\),
Combining the preceding inequalities gives for any \(\gamma _1, \gamma _2, \beta _1,\beta _2\in [0,1]\)
The majorant functions are integrable in the variable \((\varphi ,\eta )\) uniformly in \(\phi _1,\phi _2,\theta \) provided that
and under these constraints one can find admissible parameters. Consequently,
They, we are able to find that
for some \(\gamma \in (0,1)\).
Let us now move on \({\mathcal {T}}_2(f)h(\phi ,\theta )\) and show the main ideas. Define
and then
Let us analyze \({\mathcal {I}}_{13}\) and note that \({\mathcal {I}}_{14}\) is more involved (it includes more computations) but has same order of singularity. Moreover, recall the expression of \(\partial _f J_1\) in (5.40). Define also
Let us begin studying the \(L^\infty \) norm of \({\mathcal {T}}_{2,1}\). Indeed, by (5.40) and (5.43) we are able to find that
Then, using Lemma 5.2 we achieve
finding that \({\mathcal {T}}_{2,1}\) is bounded.
Since we showed the estimates in \(\phi \) of \({\mathcal {T}}_1\) (see (5.48)), let us work here with the variable \(\theta \). Indeed, our purpose will be showing
for any \(\theta _1,\theta _2\in [0,2\pi ]\) with \(\theta _1<\theta _2\). Since we have decomposed \({\mathcal {T}}_2(f_1)-{\mathcal {T}}_2(f_2)\) in two terms in (5.49), let us work with \({\mathcal {T}}_{2,1}\) and show
Here, we will use Proposition C.1 by fixing \(\phi \). The kernel of \({\mathcal {T}}_{2,1}\), i.e. \({\mathcal {I}}_{13}\), has been already bounded in (5.51). That gives us hypothesis (C.2) of such proposition and it remains to estimate \(\partial _\theta {\mathcal {I}}_{13}\) (see hypothesis (C.4)). By using the expression of \({\mathcal {I}}_{13}\) we get
For \({\mathcal {I}}_{13,1}\) we use (5.50) and Lemma 5.2 finding
That gives us hypothesis (C.4) for the first term \({\mathcal {I}}_{13,1}\). In order to work with \({\mathcal {I}}_{13,2}\), using (5.43) note that
and then \({\mathcal {I}}_{13,2}\) follows as
Similarly, we can work with \({\mathcal {I}}_{13,3}\). First note that
which, together with (5.50) implies
Putting everything together we achieves that \(\partial _\theta {\mathcal {I}}_{13}\) satisfies hypothesis (C.4) of Proposition C.1. Then, such proposition can be applied to find (5.53) concluding the proof. \(\quad \square \)
6 Main Result
In this section we shall provide a general statement that precise Theorem 1.1 and give its proof using all the previous results. Recall that the search of rotating solutions in the patch form to the equation (1.1), that is, solutions in the form
where D is a bounded simply-connected domain surrounded by a surface parametrized by
reduces to solving the following infinite-dimensional equation
with f in a small neighborhood of the origin in the Banach space \(X_m^\alpha \) and \({\tilde{F}}\) is introduced in (2.13). Notice that a solution is nontrivial means that the associated shape is not invariant by rotation along the vertical axis. Looking to the structure of the elements of space \(X_m^\alpha \) one can easily see that a nonzero element guarantees a nontrivial shape. Our result stated below asserts that solutions to this functional equation do exist and are organized in a countable family of one-dimensional curves bifurcating from the trivial solution at the largest eigenvalues of the linearized operator at the origin. More precisely, we have the following.
Theorem 6.1
Let \(m\ge 2\) be a fixed integer and \(r_0:[0,\pi ]\rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}\) satisfies the conditions:
-
(H1)
\(r_0\in {\mathscr {C}}^{2}([0,\pi ])\), with \(r_0(0)=r_0(\pi )=0\) and \(r_0(\phi )>0\) for \(\phi \in (0,\pi )\).
-
(H2)
There exists \(C>0\) such that
$$\begin{aligned} \forall \, \phi \in [0,\pi ],\quad C^{-1}\sin \phi \le r_0(\phi )\le C\sin (\phi ). \end{aligned}$$ -
(H3)
\(r_0\) is symmetric with respect to \(\phi =\frac{\pi }{2}\), i.e., \(r_0\left( \frac{\pi }{2}-\phi \right) =r_0\left( \frac{\pi }{2}+\phi \right) \), for any \(\phi \in [0,\frac{\pi }{2}]\).
Then there exist \(\delta >0\) and two one-dimensonal \({\mathscr {C}}^1\)-curves \(s\in (-\delta ,\delta )\mapsto f_m(s)\in X_m^\alpha \) and \( s\in (-\delta ,\delta )\mapsto \Omega _m(s)\in {{\,\mathrm{{\mathbb {R}}}\,}}, \) with
where \(\Omega _m\) is defined in Proposition 4.3, such that
Proof
The main material to prove this result is Crandall–Rabinowitz theorem, recalled in Theorem B.1. First the well-possednes and the regularity of \({\tilde{F}}:X_m^\alpha \rightarrow X_m^\alpha \) were discussed in Proposition 5.2. Thus it remains to check the suitable spectral properties of the linearized operator at the origin. The expression of this operator is detailed in Proposition 3.2 and it is a of Fredholm type of zero index according to Proposition 4.6. In addition for \(\Omega =\Omega _m\) the kernel is a one-dimensional vector space. Finally, the transversal condition is satisfied by virtue of Proposition 4.7. \(\quad \square \)
6.1 Special case: sphere and ellipsoid
In this section we aim to show the particular case of bifurcating from spherical or ellipsoidal shapes. The main particularity of these shapes is that their associated stream function is well-known in the literature, see [32]. More specifically, let \({\mathscr {E}}\) be an ellipsoid inside the region
The associated stream function given by
can be computed inside the ellipsoid as
In the case that \(a=b\) we have that the ellipsoid is invariant under rotations about the z-axis and then it defines a stationary patch, see Lemma 2.1. Moreover and without loss of generality we can take \(c=1\). Note that in this case
where
and
The sphere coincides with the case \(a=1\) having \(\alpha _1(1)=\alpha _2(1)=\frac{1}{6}\) and \(\alpha _3(1)=\frac{1}{2}\).
The above expression of the stream function together with Remark 3.2 gives us that
for any \(\phi \in [0,\pi ]\). Recall that \(H_n\) is defined in (3.3). Now, by virtue of Proposition 3.2 one has
where
Moreover, the function \(\nu _\Omega \) used in the spectral study and defined in (4.3) agrees with
which now is constant on \(\phi \). Also the constant \(\kappa \) in (4.5) equals now to \(2\alpha _1(a)\). Hence, the key point in Sect. 4.1 is the symmetrization of the above operator. For that reason, we have defined the signed measure \(d{\mu _\Omega }\) as
in (4.4) and the operator \({\mathcal {K}}_n^\Omega \) in (4.82). However, since in this case \(\nu _\Omega (\varphi )\) is constant on \(\varphi \), there is no need to introduce it in the measure with the goal of symmetryzing the operator. Following the ideas developed above, we deduce that the kernel study of the linearized operator agrees in this case with the following eigenvalue problem
Here, we define
which does not depend now on \(\Omega \) by definition of \({\mathcal {K}}_n^\Omega \). Note that both operators have similar properties. Hence \(\tilde{{\mathcal {K}}}_n\) sets the properties given in Proposition 4.2 taking the Lebesgue space \(L^2_{{\tilde{\mu }}_\Omega }\) with
Denote by \(\beta _{n,i}\) the eigenvalues of \(\tilde{{\mathcal {K}}}_n\) (for each n we have a family of eigenvalues). Then, we have necessary that
In Theorem 6.1, bifurcation occurs from \(\Omega _n^\star \) given by
with
Moreover, we know that \(\beta _n^\star \) is positive and then \(\Omega _n^\star <2\alpha _1(a)\). In particular, by Proposition 4.3, we have that \(\Omega _n^\star \) tends to \(\kappa =2\alpha _1(a)\). Furthermore, \(\Omega _n^\star \) increases in n and then we can bound it below by \(\Omega _1^\star \). Using the equation for \(\beta _1^\star \), that is
one finds that \(\beta _1^\star \le 2\alpha _1(a)\) and then \(\Omega _1^\star \) is positive. This implies that \(\Omega _n^\star \) is positive for any n. Then, in Theorem 6.1 bifurcation holds at some \(\Omega _n^\star \in (0,2\alpha _1(a))\). Let us remark that in the case of the sphere, meaning \(a=1\), one has \(2\alpha _1(a)=\frac{1}{3}\).
There is an interesting open problem concerning, first the spectral distribution of the eigenvalues \(\beta _{n,i}\) (whether or not they are finite, simple or multiple), and second if bifurcation occurs at the eigenvalues \(\Omega _n=2\alpha _1(a)-\beta _{n,i}\) (which is shown to happen only for the largest eigenvalue \(\beta _n^\star \)). Notice that the simplicity and the monotonicity of the eigenvalues is a delicate problem and could be related to the geometry of the revolution shape. Finally we observe that since \(\beta _{n,i}<\beta _n^\star \) then \(\Omega _n=2\alpha _1(a)-\beta _{n,i}>\frac{1}{3}-\beta _n^\star >0\).
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Acknowledgements
We would like to thank D. G. Dritschel for proposing this problem and for several discussions around it.
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C.G. has been partially supported by the MINECO-Feder (Spain) research Grant Number RTI2018-098850-B-I00, the Junta de Andalucía (Spain) Project FQM 954, the MECD (Spain) research grant FPU15/04094, and the European Research Council through Grant ERC-StG-852741 (CAPA), T. H. has been partially supported by the ANR grant ODA (ANR-18-CE40-0020-01), and J. M. has been partially supported by MTM2016-75390 (Mineco, Spain) and 2017-SGR-395 (Generalitat de Catalunya)
Appendices
Gauss Hypergeometric Function
We give a short discussion on Gauss hypergeometric functions and illustrate some of their basic properties. The formulas listed below were crucial in the computations of the linearized operator associated to the V-states equation and in the analysis of the main feature of its spectral properties. Recall that for any real numbers \(a,b\in {\mathbb {R}},\, c\in {\mathbb {R}}\backslash (-{\mathbb {N}})\) the hypergeometric function \(z\mapsto F(a,b;c;z)\) is defined on the open unit disc \({\mathbb {D}}\) by the power series
The Pochhammer’s symbol \((x)_n\) is defined by
and verifies
The series converges absolutely for all values of \(|z|<1\). For \(|z|=1\) we have the absolute convergence if \(\text {Re} (a+b-c)<0\) and it diverges if \(1\le \text {Re}(a+b-c)\). See [2] for more details.
We recall the integral representation of the hypergeometric function, see for instance [37, p. 47]. Assume that \( \text {Re}(c)> \text {Re}(b) > 0\), then
Notice that this representation shows that the hypergeometric function initially defined in the unit disc admits an analytic continuation to the complex plane cut along \([1,+\infty )\). Another useful identity is the following:
for \(\text {Re} (c)>\text {Re}(b)>0\). The function \(\Gamma : {\mathbb {C}}\backslash \{-{{\,\mathrm{{\mathbb {N}}}\,}}\} \rightarrow {\mathbb {C}}\) refers to the gamma function, which is the analytic continuation to the negative half plane of the usual gamma function defined on the positive half-plane \(\{\text {Re} z > 0\}\), and given by
and satisfies the relation \( \Gamma (z+1)=z\,\Gamma (z), \ \forall z\in {\mathbb {C}}\backslash (-{{\,\mathrm{{\mathbb {N}}}\,}}). \) From this we deduce the identities
provided that all the quantities in the right terms are well-defined.
We can differentiate the hypergeometric function obtaining
for \(k\in {{\,\mathrm{{\mathbb {N}}}\,}}\). Depending on the parameters, the hypergeometric function behaves differently at 1. When \(\text {Re}(c)>\text {Re}(b)>0\) and \(\text {Re} (c-a-b)>0 \), it can be shown that it is absolutely convergent on the closed unit disc and one finds the expression
see for example [37] for the proof. However, in the case \(a+b=c\), the hypergeometric function exhibits a logarithmic singularity as follows
see for instance [1] for more details. Next, we shall give a proof of the following classical result.
Lemma A.1
Let \(n\in {{\,\mathrm{{\mathbb {N}}}\,}}\), \(\beta \ge 0\) and \(A>1\), then
Proof
By a change of variables and using \(\cos (2\theta )=2\cos ^2(\theta )-1\), we arrive at
Since \(\frac{2}{1+A}<1\), we can use Taylor series in the following way,
Then,
At this stage we use the identity, see [43, p. 449],
for \(x>-1\) and \(y\in {{\,\mathrm{{\mathbb {R}}}\,}}\). That identity For \(x=2m\) and \(y=2n\), we obtain
We can use some properties of Gamma functions in order to find
which implies
\(\square \)
Now we propose to describe the boundary behavior of the Hypergeometric functions in some suitable cases that were very useful in the preceding sections.
Proposition A.1
The following assertions hold true.
-
(1)
Bound for F(a, a; 2a; x) : for \(a>1\), there exists \(C>0\) such that
$$\begin{aligned} \forall x\in [0,1),\quad F\left( a,a;2a; x\right) \le C\frac{|\ln (1-x)|}{x}\le C+C|\ln (1-x)|. \end{aligned}$$(A.7) -
(2)
Bound for \(F(a,a;2a-1,x):\) for \(a>2\), there exists \(C>0\) such that
$$\begin{aligned} \forall x\in [0,1),\quad F\left( a,a;2a-1; x\right) \le C\frac{1}{|1-x|}\cdot \end{aligned}$$(A.8) -
(3)
Bound for \(F(a,a;2a-2,x):\) for \(a>3\), there exists \(C>0\) such that
$$\begin{aligned} \forall x\in [0,1),\quad F\left( a,a;2a-2; x\right) \le C\frac{1}{|1-x|^2}\cdot \end{aligned}$$(A.9) -
(4)
For \(a>1\), there exists \(C>0\) such that
$$\begin{aligned} \forall x\in [0,1),\quad 0\le F(a,a;2a;x)-1\le C x\big (1+|\ln (1-x)|\big ). \end{aligned}$$(A.10) -
(5)
For \(a>2\), there exists \(C>0\) such that
$$\begin{aligned} \forall x\in [0,1),\quad | F(a,a;2a-1;x)-1|\le C \frac{x}{1-x}\cdot \end{aligned}$$ -
(6)
For \(a>1\), there exists \(C>0\) such that any \( \alpha \in [0,1]\)
$$\begin{aligned} \forall \, x_2\le x_1\in [0,1),\quad |F(a,a;2a;x_1)-F(a,a;2a;x_2)|\le C\frac{|x_1-x_2|^\alpha }{|1-x_1|^\alpha }\cdot \qquad \end{aligned}$$(A.11) -
(7)
For \(a>2\), there exists \(C>0\) such that any \( \alpha \in [0,1]\)
$$\begin{aligned}&\forall \, x_2\le x_1\in [0,1), \quad \! |F(a,a;2a-1;x_1)\!-\!F(a,a;2a-1;x_2)|\nonumber \\&\quad \le \! C\frac{|x_1-x_2|^\alpha }{|1-x_1|^{1+\alpha }}\cdot \end{aligned}$$(A.12)
Proof
The main tool is the integral representation of the Hypergeometric functions (A.2).
(1) From the integral representation (A.2), it is easy to get
Because \( t(1-t)\le 1-tx\) for any \(t,x\in [0,1]\), then we deduce
(2) As for (1), we find
Consequently, we infer from direction calculation
(3) We omit here the details of the proof by similarity with (1) and (2).
(4) First note from the integral representation that \(F(a,a;2a;x)>0\) provided that \(a>0\) and \(x\in [0,1)\). Moreover, it is strictly increasing function since from (A.4)
According to (A.1) one may check by construction that \(F(a,a;2a;0)=1\) and therefore
By the mean value theorem, we achieve
Combining this representation with (A.8), where we replace a by \(a+1\), we achieve
(5) By using similar arguments as the previous point, we obtain
Applying (A.9) by changing a with \(a+1\) allows to get
Then,
(6) Let \(t\in [0,1)\) and set \({g_t}(x)=(1-tx)^{-a}\). Take \(0\le x_2< x_1<1\), then direct computations, using in particular the mean value theorem, show that
Let \(\alpha \in [0,1]\) then by interpolation between the preceding inequalities we deduce that
It follows that
Since \(a>1\) and for any \(t,x_1\in [0,1)\),
then
(7) This is quite similar to the proof of the preceding one. Indeed,
\(\square \)
Bifurcation Theory
We shall briefly recall some basic facts around bifurcation theory which mainly focuses on the topological transitions of the phase portrait through the variation of some parameters. A particular case is to understand this transition in the equilibria set for the stationary problem \(F(\lambda ,x)=0\), where \(F:{{\,\mathrm{{\mathbb {R}}}\,}}\times X\rightarrow Y\) is a smooth function between Banach spaces X and Y. Assuming that one has a trivial solution, \(F(\lambda ,0)=0\) for any \(\lambda \in {{\,\mathrm{{\mathbb {R}}}\,}}\), we would like to explore the bifurcation diagram in the neighborhood of this elementary solution, and see whether multiple branches of solutions may bifurcate from a given point \((\lambda _0,0)\), called a bifurcation point. When the linearized operator around this point generates a Fredholm operator, then one may use Lyapunov–Schmidt reduction in order to reduce the infinite-dimensional problem to a finite-dimensional one, known as the bifurcation equation. For this latter problem we need some specific transversal conditions so that the Implicit Function Theorem can be applied. For more discussion in this subject, we refer to see [31, 33]. Notice that Theorem B.1 below is one of those interesting results that can cover various configurations and it is used in this paper to prove our main result. Before giving its precise statement, we need to recall some basic results on Fredholm operators.
Definition B.1
Let X and Y be two Banach spaces. A continuous linear mapping \(T:X\rightarrow Y\), is a Fredholm operator if it fulfills the following properties,
-
(1)
\(\text {dim Ker}\, T<\infty \),
-
(2)
\(\text {Im}\, T\) is closed in Y,
-
(3)
\(\text {codim Im}\, T<\infty \).
The integer \(\text {dim Ker}\, T-\text {codim Im}\, T\) is called the Fredholm index of T.
Next, we shall discuss the index persistence through compact perturbations, see [31, 33].
Proposition B.1
The index of a Fredholm operator remains unchanged under compact perturbations.
Now, we recall the classical Crandall-Rabinowitz Theorem whose proof can be found in [12].
Theorem B.1
(Crandall–Rabinowitz Theorem). Let X, Y be two Banach spaces, V be a neighborhood of 0 in X and \(F:{\mathbb {R}}\times V\rightarrow Y\) be a function with the properties,
-
(1)
\(F(\lambda ,0)=0\) for all \(\lambda \in {\mathbb {R}}\).
-
(2)
The partial derivatives \(\partial _\lambda F_{\lambda }\), \(\partial _fF\) and \(\partial _{\lambda }\partial _fF\) exist and are continuous.
-
(3)
The operator \(\partial _f F(0,0)\) is Fredholm of zero index and \(\text {Ker}(F_f(0,0))=\langle f_0\rangle \) is one-dimensional.
-
(4)
Transversality assumption: \(\partial _{\lambda }\partial _fF(0,0)f_0 \notin \text {Im}(\partial _fF(0,0))\).
If Z is any complement of \(\text {Ker}(\partial _fF(0,0))\) in X, then there is a neighborhood U of (0, 0) in \({\mathbb {R}}\times X\), an interval \((-a,a)\), and two continuous functions \(\Phi :(-a,a)\rightarrow {\mathbb {R}}\), \(\beta :(-a,a)\rightarrow Z\) such that \(\Phi (0)=\beta (0)=0\) and
Potential Theory
This last section is devoted to some results on the continuity of specific operators with singular kernels, taking the form
with \((x_1,x_2)\in [0,1]^2\) and the kernel \(K:[0,1]^2\times [0,1]^2\rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}\) is smooth out the diagonal.
Proposition C.1
Let \(K:[0,1]^2\times [0,1]^2\rightarrow {{\,\mathrm{{\mathbb {R}}}\,}}\) be smooth out the diagonal, satisfying
with \(\alpha \in (0,1)\) and \(g_1,g_2,g_3,g_4\in L^\infty ([0,1],L^1([0,1]))\). Then \({\mathcal {K}}:L^\infty ([0,1]\times [0,1])\rightarrow {\mathscr {C}}^\alpha ([0,1]\times [0,1])\) is well-defined and
with C an absolute constant.
Remark C.1
We give here this general proposition for a function of two variables, but let us remark that this can be also done for \({\mathcal {K}}\) depending only on one variable. Moreover, note that condition (C.4) (and also (C.5)) can be replaced by
for \(x_1<\tilde{x_1}\) and \(3|x_1-\tilde{x_1}|\le |y_1-x_1|\). The function g must satisfy
uniformly in \(x_1,\tilde{x_1},x_2\).
Proof
The \(L^\infty \) norm of \({\mathcal {K}}(f)\) can be estimated as
The convergence follows from the assumptions \(\alpha , \gamma \in (0,1)\). Hence,
For the Hölder regularity, take \(x_1, \tilde{x_1}\in [0,1]\) with \(x_1<\tilde{x_1}\). Define \(d=|x_1-\tilde{x_1}|\), \(B_{x_1}(r)=\left\{ y_1\in [0,1] : |y_1-x_1|<r\right\} \) and \(B^c_{x_1}(r)\) its complement set. Hence
Using (C.2), we arrive at
In order to work with \(I_2\), note that \(B_{x_1}(3d)\subset B_{\tilde{x_1}}(4d)\). Thus,
For the last term \(I_3\) we use the mean value theorem and (C.4) achieving
Note that if \(y_1\in B^c_{x_1}(3d)\), then
which implies
Putting together the preceding estimates yields
The same arguments enables to obtain
Then, we conclude that
\(\square \)
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García, C., Hmidi, T. & Mateu, J. Time Periodic Solutions for 3D Quasi-Geostrophic Model. Commun. Math. Phys. 390, 617–756 (2022). https://doi.org/10.1007/s00220-021-04290-w
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DOI: https://doi.org/10.1007/s00220-021-04290-w