Abstract
We show a global existence for the Cauchy problem with large initial data for the p-harmonic flow between two smooth, compact Riemannian manifolds. We devise new monotonicity type formulas of a local scaled energy and establish a partial regularity for the solution. The partial regularity obtained is almost optimal, comparing with that of the corresponding stationary case. The p-harmonic flow obtained also converges to a p-harmonic map along a certain time sequence tending to infinity.
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1 Introduction
Let \({\mathcal{M}}\) and \({\mathcal{N}}\) be smooth compact Riemannian manifolds of dimension m and n with metric g and h, respectively. We assume that, by Nash’s embedding theorem, \({\mathcal{N}}\) is isometrically embedded into \(\mathrm{I}\!\mathrm{R}^l\) \((l > n)\). For a smooth map u from \({\mathcal{M}}\) to \({\mathcal{N}} \subset \mathrm{I}\!\mathrm{R}^l\), we consider the p-energy
Here the unknown map \(u = \left( u^i \right) \), \(i = 1, \ldots , l\), is a vector-valued function, defined on \(\mathcal{M}\) with values into \({\mathcal{N}} \subset \mathrm{I}\!\mathrm{R}^l\). In a local coordinate \(x = (x_\alpha )\), \(\alpha = 1, \ldots , m\), on \({\mathcal{M}}\), the usual notation is used : \(g = \left( g_{\alpha \beta }\right) \), \(\left( g_{\alpha \beta }\right) ^{- 1} = \left( g^{\alpha \beta }\right) \), \(|g| = |\det \left( g_{\alpha \beta }\right) |\), and \(d {\mathcal{M}} = \sqrt{|g|} d x\) is a volume element with m-dimensional Lebesgue measure dx, and \(D_\alpha = \partial /\partial x_\alpha \), \(\alpha = 1, \ldots , m\), \(D u = \left( D_\alpha u^i\right) \) is the gradient of a map u, and \(|D u|^2 = \sum _{\alpha , \beta = 1}^m g^{\alpha \beta } D_\alpha u \cdot D_\beta u\) with an Euclidean inner product \(\cdot \) in \(\mathrm{I}\!\mathrm{R}^l\).
The p-harmonic map is a critical point of the p-energy and satisfies the Euler–Lagrange equation
where the p-Laplace operator is denoted by
and the second fundamental form A(u)(Du, Du) of \({\mathcal{N}} \subset \mathrm{I}\!\mathrm{R}^l\) is a vector field along the map \(u \in {\mathcal{N}}\) with values into the orthogonal complement of the tangent space of \({\mathcal{N}}\) at u (if necessary, the manifold \({\mathcal{N}}\) is assumed to be orientable).
An approach to look for p-harmonic maps is to exploit the gradient flow associated with the p-energy, called the p-harmonic flow, which are described by the evolutionary p-Laplacian system
where \(u = u (t, \, x)\) is defined on \({\mathcal{M}}_\infty = (0, \, \infty ) \times {\mathcal{M}}\) with values onto \(\mathrm{I}\!\mathrm{R}^l\), \(\partial _t u = \left( \partial _t u^i\right) \) is a partial derivative on time. In this paper we study a global existence and regularity of a solution to the Cauchy problem for the p-harmonic flow (1.4).
Let \(\mathrm{I}\!\mathrm{R}^l = \mathcal{T}_u {\mathcal{N}} \oplus (\mathcal{T}_u {\mathcal{N}})^\bot \) be the orthogonal decomposition of \(\mathrm{I}\!\mathrm{R}^l\) with respect to the tangent space \(\mathcal{T}_u {\mathcal{N}}\) at each \(u \in {\mathcal{N}}\). The corresponding orthonormal basis is \(\left( e_1 (u), \ldots , e_n (u)\right) \) of the tangent space \(\mathcal{T}_u {\mathcal{N}}\) and \(\left( e_{n + 1} (u), \ldots , e_l (u)\right) \) of its orthogonal complement \((\mathcal{T}_u {\mathcal{N}})^\bot \). Then we find an equivalent representation for the p-harmonic flow
In fact, there exists some vector-valued function \(\lambda = \left( \lambda ^j (u)\right) \), \(j = n + 1, \ldots , l\), such that
and, simply multiplying each of the orthonormal basis \(e_j (u)\), \(j = n + 1, \ldots , l\), by the second equation above, we have
where \(\partial _t u, \, D u \in \mathcal{T}_u {\mathcal{N}}\) because the map \(u = u (t, x)\) moves on \({\mathcal{N}}\), and thus, the usual Euclidean innner product in \(\mathrm{I}\!\mathrm{R}^l\) is taken, so that \(\partial _t u \cdot e_j (u) = 0\) and \(D u \cdot e_j (u) = 0\), for all \(j = n + 1, \ldots , l\). Here the last summation term in the equation above is nothing but the second fundamental form of \({\mathcal{N}}\) along the map u. Furthermore, the Euclidean inner product in \(\mathrm{I}\!\mathrm{R}^l\) of \(\partial _t u\) with the p-harmonic flow Eq. (1.4) leads the energy identity
integrated in \({\mathcal{M}}\) yielding, through integration by parts,
Thus, the p-energy E(u(t)) is decreasing along the solution u(t) of the p-harmonic flow and, in fact, the solution \(\left\{ u (t)\right\} \subset C^\infty ({\mathcal{M}}, {\mathcal{N}})\), \(0< t < \infty \), is the trajectory of negative direction gradient vector field of the p-energy
by the Euler–Lagrange equation (1.2), where \(\nabla E (u (t))\) is the G\(\hat{a}\)teaux derivative of \(E (\cdot )\) at \(u (t) \in C^\infty ({\mathcal{M}}, {\mathcal{N}})\). Therefore, a global in time solution to (1.4) for any initial data may converge to critical points of the p-energy, the p-harmonic maps, as time tends to \(\infty \). This heat flow method was originally realized by J. Eells and J. H. Sampson for the harmonic flow in the case \(p = 2\) under the condition that the sectional curvature of target manifold \({\mathcal{N}}\) is non-positive, in their pioneering work [15, 23]. This fundamental result in the harmonic flow case \(p = 2\) was also extends to hold similarly for the p-harmonic flow.
Theorem 1
[16, 31] Suppose that the sectional curvature of the target manifold \({\mathcal{N}}\) is non-positive, \({ Sect} ({\mathcal{N}}) \le 0\). Then, for any smooth initial map from \({\mathcal{M}}\) to \({\mathcal{N}}\), there exists a unique global in time weak solution of the Cauchy problem on \({\mathcal{M}}\) for p-harmonic flow (1.4). The solution u and its gradient are Hölder continuous in time-space. The solution and its gradient uniformly converge to a weak solution and its gradient, respectively, of the p-harmonic map, as time tends to \(\infty \), respectively, which are Hölder continuous.
We call the weak solution which is locally continuous on time-space together with its gradient the regular solution. The curvature restriction on the target manifold in general is necessary for the global existence of regular solution of the p-harmonic flow. In fact, without any curvature restriction on the target manifold, we have some example of a blowing up solution at a finite time (see [5] in the case \(p = m = 3\)). But, a global in time weak solution may be exist.
Theorem 2
[24] Let \(p = m \ge 2\) and the initial data be in the set of Sobolev maps \(W^{1, p} ({\mathcal{M}}, {\mathcal{N}})\) between two smooth, compact Riemannian manifolds \({\mathcal{M}}\) and \({\mathcal{N}}\). Then, there exists a global in time weak solution of Cauchy problem on \({\mathcal{M}}\) for the m-harmonic flow. The solution and its gradient are Hölder continuous on time-space, except for at most finitely many time slices.
In the case \(p = m = 2\), the global in time existence as above is also shown for the initial-boundary value problem of the two-dimensional harmonic flow. Moreover, the solution is smooth except for at most finitely many points [3, 38]. In the case \(p = m\), a nice Sobolev type inequality on time-space, referred as Ladyzhenskaya or Nash inequality, can be available and is crucial for regularity estimate in this case.
In the higher dimensional case \(m \ge 3\), M. Struwe et al. established the following fundamental result for global existence and regularity of the harmonic flow in the case \(p = 2\) in [8, 9, 39]
Theorem 3
[8, 9, 39] Let \(p = 2\). Let initial and boundary data \(u_0\) be smooth map from \({\mathcal{M}}\) into \({\mathcal{N}}\). Then, there exists a global in time weak solution u of the harmonic flow (1.4). The solution u satisfies the energy inequality: letting \({\mathcal{M}}_\infty = (0, \infty ) \times {\mathcal{M}}\),
There exists a relatively closed subset \(\Sigma \subset (0, \, \infty ] \times {\mathcal{M}}\) such that the solution u is smooth in the complement of \(\Sigma \), \({\mathcal{M}}_\infty {\setminus } \Sigma \); \(\Sigma \) is of at most finite m-dimensional Hausdorff measure with respect to the usual parabolic metric in \({\mathcal{M}}_\infty \), and furthermore, for any time \(t_0 > 0\) and some positive \(C_0= C ({\mathcal{M}}, {\mathcal{N}}, t_0, E (u_0))\), \(\mathcal{H}^{m - 2} (\{t_0\} \times \Sigma ) \le C_0\); As time suitably tends to \(\infty \), the solution converges to a weakly harmonic map \(u_\infty \) weakly in Sobolev space \(W^{1, \, 2} ({\mathcal{M}}, \, \mathrm{I}\!\mathrm{R}^l)\). There exists a closed set \(\Sigma _\infty \subset {\mathcal{M}}\) such that \(u_\infty \) is smooth on \({\mathcal{M}} {\setminus } \Sigma _\infty \); \(\Sigma _\infty \) is of at most finite \((m - 2)\)-dimensional Hausdorff measure: For some positive \(C_0^\prime = C^\prime ({\mathcal{M}}, {\mathcal{N}}, t_0, E (u_0))\), \(\mathcal{H}^{m - 2} (\Sigma _\infty ) \le C_0^\prime \).
There also exist blowing up solutions at a finite time (see [4, 7, 11, 22]).
If the target manifold is the standard unit sphere, the global in time existence of weak solution to the p-harmonic flow is also shown by use of the special structure of the target standard unit sphere [6, 25, 27, 32].
In differential geometry, the regularity has been studied under a smallness of image of a solution, instead of curvature condition [18], and the everywhere regularity of a small solution of harmonic flow is shown in [19, 20, 37]. Such regularity of a small solution of p-harmonic flow remained open (refer to [28]).
Theorem 3 implies the global in time existence of weak solution of the harmonic flow in the case \(p = 2\), which is partial regular in the sense of regularity outside exceptional closed set. It has remained open whether or not the corresponding result holds for the p-harmonic flow, since the important result, Theorem 2, was obtained for the case \(p = m\).
A compactness for regular p-harmonic flows with uniform boundedness of p-energy has been recently proved by the author in [33, 34] (see [39, Theorem 6.1; its proof, pp. 494–497] for the harmonic flow). The compactness result will be the key ingredient for the global in time existence of p-harmonic flow (refer to [9] for the harmonic flow case).
Theorem 4
(A compactness of regular p-harmonic flows with uniformly bounded p-energy) Let \(p > 2\). Suppose that a family \(\{u_k\}\) of regular p-harmonic flows on \(\mathrm{I}\!\mathrm{R}^m_\infty = (0, \, \infty ) \times \mathrm{I}\!\mathrm{R}^m\) satisfies the p-energy boundedness with uniform positive constant C
and converges to a limit map u in the sense
Then, the limit map u is a global weak solution on \(\mathrm{I}\!\mathrm{R}^m_\infty \) of the p-harmonic flow such that \(u \in {\mathcal{N}}\) almost everywhere in \(\mathrm{I}\!\mathrm{R}^m_\infty \), and the p-energy boundedness is valid, replacing \(u_k\) by u in (1.8). Moreover, the limit map u is partial regular in the sense : There exists a relatively closed set \(\Sigma \) in \(\mathrm{I}\!\mathrm{R}^m_\infty \) such that u and its gradient Du are locally in time-space continuous in the complement \(\mathrm{I}\!\mathrm{R}^m_\infty {\setminus } \Sigma \), and the size of \(\Sigma \) is also estimated by the Hausdorff measure : For any positive number \(\gamma _0\), \(2< \gamma _0 < p\), the set \(\Sigma \) is of at most locally zero m-dimensional Hausdorff measure with respect to the time-space metric \(|t|^{1/\gamma _0} + |x|\), and, furthermore, for any positive time \(\tau < \infty \), the \((m - \gamma _0)\)-dimensional Hausdorff measure of \(\{\tau \} \times \Sigma \) with respect to the usual Euclidean metric is locally zero.
In this paper we show the global existence and regularity of a weak solution of the Cauchy problem for the p-harmonic flow (1.4) with an initial data \(u_0\)
and a convergence of the solution of p-harmonic flow to a p-harmonic map along a time sequence tending to infinity. The Sobolev space on \({\mathcal{M}}\) is usually defined as
Definition 1
Definition 2
Let \(u_0 \in {\mathrm {W}}^{1, p} ({\mathcal{M}}, \, {\mathcal{N}})\). A map u is called a global weak solution of the Cauchy problem (1.12) if and only if u is a measurable vector-valued function defined on \({\mathcal{M}}_\infty := (0, \, \infty ) \times {\mathcal{M}}\) with values into \(\mathrm{I}\!\mathrm{R}^l\), satisfying the following four conditions :
-
(D1)
\(u \in {\mathrm {L}}^\infty (0, \infty ; \, {\mathrm {W}}^{1, p} ({\mathcal{M}}, \, \mathrm{I}\!\mathrm{R}^l))\), \(\partial _t u \in {\mathrm {L}}^2 ({\mathcal{M}}_\infty , \mathrm{I}\!\mathrm{R}^l);\)
-
(D2)
\(u \in {\mathcal{N}}\) almost everywhere in \({\mathcal{M}}_\infty ;\)
-
(D3)
u satisfies (1.4) in the sense of distributions, that is, for any smooth map \(\phi \) \(\in \) \(\mathrm {C}^\infty _0 ({\mathcal{M}}_\infty , \, \mathrm{I}\!\mathrm{R}^l)\),
$$\begin{aligned} \displaystyle { \int _{{\mathcal{M}}_\infty } \{ \partial _t u \, \cdot \, \phi + |D u|^{p - 2} D u \cdot D \phi - |D u|^{p- 2} \phi \cdot A (u) (D u, D u) \} \, d z = 0; } \end{aligned}$$ -
(D4)
u attains the initial data continuously in the Sobolev space
$$\begin{aligned} \left| u (t) - u_0 \right| _{{\mathrm {W}}^{1, p} ({\mathcal{M}}, \, \mathrm{I}\!\mathrm{R}^l)} \rightarrow 0 \quad \text{ as } t \rightarrow 0. \end{aligned}$$
Theorem 5
(A global existence and regularity for the p-harmonic flow) Let \(p > 2\). Let \(u_0 \in {\mathrm {W}}^{1, p} ({\mathcal{M}}, \, {\mathcal{N}})\). Then, there exists a global weak solution u of (1.12), satisfying the energy inequality
Moreover, the solution u is partial regular in the following sense : There exists a relatively closed set \(\Sigma \) in \({\mathcal{M}}_\infty = (0, \infty ) \times {\mathcal{M}}\) such that u and its gradient Du are locally in time-space continuous in the complement \({\mathcal{M}}_\infty {\setminus } \Sigma \), and the size of \(\Sigma \) is also estimated by the Hausdorff measure : For any positive number \(\gamma _0\), \(2< \gamma _0 < p\), the set \(\Sigma \) is of at most zero m-dimensional Hausdorff measure with respect to the time-space metric \(|t|^{1/\gamma _0} + |x|\), and, furthermore, for any positive time \(\tau < \infty \), the \((m - \gamma _0)\)-dimensional Hausdorff measure of \(\{\tau \} \times \Sigma \) with respect to the usual Euclidean metric is zero. As time suitably tends to \(\infty \), the solution converges to a weakly p-harmonic map \(u_\infty \) weakly in \(W^{1, \, p} ({\mathcal{M}}, \, \mathrm{I}\!\mathrm{R}^l)\). There exists a closed set \(\Sigma _\infty \subset {\mathcal{M}}\) such that \(u_\infty \) and its gradient \(D u_\infty \) are locally continuous on \({\mathcal{M}} {\setminus } \Sigma _\infty \); For any positive number \(\gamma _0\), \(2< \gamma _0 < p\), \(\Sigma _\infty \) is of at most zero \((m - \gamma _0)\)-dimensional Hausdorff measure.
Remark
Measuring by use of the time-space metric \(|t|^{1/p} + |x|\) on \({\mathcal{M}}_\infty \), the \((m + p - \gamma _0)\)-dimensional Hausdorff size of \(\Sigma \) is zero. The scale order in the estimate of singular set \(\Sigma \) is almost optimal, since the exponent \(\gamma _0\) can be as close to p as possible.
The contents of the paper are as follows :
-
1.
Introduction
-
2.
Penalty approximation
-
3.
Small energy regularity estimate
-
3.1
Preliminaries; 3.2 Local energy regularity estimate
-
4.
Passing to the limit
-
5.
Monotonicity estimate of a local scaled energy
-
6.
Appendix
In Sect. 2, we introduce the so-called penalty approximation for the p-harmonic flow. In Sect. 3, some preliminary estimates for the penalty approximating solutions are derived, those proofs are given in “Appendix”, and then, the small energy regularity estimate is shown to hold uniformly for the penalty approximating solutions, and is applied for their convergence to a weak solution of the p-harmonic flow in Sect. 4, based on the compactness result, Theorem 4. The monotonicity estimate, Lemmata 12 and 13, is demonstrated in Sect. 5.
2 Penalty approximation
In this section we set the approximation scheme for the p-harmonic flow. We will approximate the p-harmonic flow by the solutions of the gradient flow for the so-called penalized functional, introduced in [9] for the harmonic flow case \(p = 2\) (also refer to [29, 40]).
Since the manifold \({\mathcal{N}}\) is smooth and compact, there exists a tubular neighborhood \(\mathcal{O}_{2 \delta _{\mathcal{N}}}\) with width \(2 \delta _{\mathcal{N}}\) of \({\mathcal{N}}\) in \(\mathrm{I}\!\mathrm{R}^l\) such that any point \(u \in \mathcal{O}_{2 \delta _{\mathcal{N}}}\) has a unique nearest point \(\pi _{\mathcal{N}} (u) \in {\mathcal{N}}\) satisfying \( \text{ dist } \left( u, \, \pi _{\mathcal{N}} (u)\right) = \text{ dist } \left( u, \, {\mathcal{N}}\right) \) for the Euclidean distance \( \text{ dist } \left( \cdot , \cdot \right) \), where the projection \(\pi _{\mathcal{N}} \, : \, \mathcal{O}_{2 \delta _{\mathcal{N}}} \rightarrow {\mathcal{N}}\) is smooth, since the manifold \(\mathcal{N}\) is smooth. The distance function \( \text{ dist } (u, \, {\mathcal{N}})\) is Lipschitz continuous on \(u \in \mathcal{O}_{2 \delta _{\mathcal{N}}}\).
Let \(\chi \) be a smooth, non-decreasing real-valued function defined on \([0, \, \infty )\) such that \(\chi (s) = s\) for \(s \le (\delta _{\mathcal{N}})^2\) and \(\chi (s) = 2 (\delta _{\mathcal{N}})^2\) for \(s \ge 4 (\delta _{\mathcal{N}})^2\). Then, the function \(\chi \left( \text{ dist }^2 (u, \, {\mathcal{N}})\right) \) is smooth on \(u \in \mathrm{I}\!\mathrm{R}^l\). Its gradient at \(u \in \mathcal{O}_{2 \delta _{\mathcal{N}}}\) is computed as
parallel to the vector field \(u - \pi _{\mathcal{N}} (u)\) and orthogonal to \(\mathcal{T}_{\pi _{\mathcal{N}} (u)} {\mathcal{N}}\). We also have that, for any \(u \in {\mathcal{N}}\) and any tangent vector \(\tau \in \mathcal{T}_u {\mathcal{N}}\),
(see [2, Theorem 3.1, pp. 704–705], [1, Theorem 2.1]).
For positive parameters \(1 \le K \nearrow \infty \) and \(1 > \epsilon \searrow 0\), we consider the Cauchy problem in \({\mathcal{M}}_\infty \) with initial data \(u_0\) for the gradient flow, called the penalized equation,
associated with the penalized functional, defined by
where the positive constant \(C_0\) will be stipulated later, depending only on p, \({\mathcal{M}}\) and \({\mathcal{N}}\) (see Lemma 9 and its proof in “Appendix B”). The partial differential operator \(\Delta _{p, \, \epsilon }\) and its corresponding energy, called the regularized p-Laplace operator and the regularized p-energy, respectively, are defined as
We now state the global existence for (2.1). For the proof see “Appendix A”.
Lemma 6
(Existence for the penalty approximation) Let \(p > 2\) and let \(u_0 \in {\mathrm {W}}^{1, p} \left( {\mathcal{M}}, \, {\mathcal{N}}\right) \). For each positive numbers K and \(\epsilon \), there exists a weak solution \(u = u_{K, \, \epsilon }\) of the Cauchy problem for the penalized equation (2.1) such that \(u = u_{K, \, \epsilon }\) satisfies the energy inequality
and, that u, Du, \(\partial _t u\) and \(D^2 u\) are locally (Hölder) continuous on time and space (with some Hölder exponent) in \({\mathcal{M}}_\infty \) and u satisfies the penalized equation everywhere in \({\mathcal{M}}_\infty \).
3 Small energy regularity estimate
3.1 Preliminaries
In this section we show some regularity estimates for solutions \(u = u_{K, \, \epsilon }\) of the penalized equations (2.1). Those proofs are given in “Appendix”.
Lemma 7
(Energy inequality) Let \(u_0 \in {\mathrm {W}}^{1, \, p} \big ({\mathcal{M}}, \, {\mathcal{N}}\big )\) and \(u = u_{K, \, \epsilon }\) be a regular solution of (2.1). Then, it holds that
A solution of the penalized equation is uniformly bounded, that is used in the regularity estimate.
Lemma 8
(Boundedness) Let \(u = u_{K, \, \epsilon }\) be a regular solution of (2.1). Then it holds that \(\sup _{ {\mathcal{M}}_\infty } |u| \le H\), where the positive number H is so large that \(B (H) \supset \mathcal{O}_{ 2 \delta _{\mathcal{N}} } ({\mathcal{N}})\) in \(\mathrm{I}\!\mathrm{R}^l\).
We will put the setting for local estimates for the penalized Eq. (2.1). For this purpose we recall some standard geometrical settings. Let \(R_{\mathcal{M}} > 0\) be a lower bound for the injective radius of the exponential map on \({\mathcal{M}}\). Thus, for any positive number \(R < R_{\mathcal{M}}\) and any point \(x_0 \in {\mathcal{M}}\), the geodesic ball \(\mathcal{B} (R, \, x_0) \subset {\mathcal{M}}\) of radius R around \(x_0\) is well-defined and diffeomorphic to the Euclidean ball \(B (R, \, 0) \subset \mathrm{I}\!\mathrm{R}^m\), under the linear homeomorphism \(\mathcal{T}_{x_0} {\mathcal{M}} \cong \mathrm{I}\!\mathrm{R}^m\), through the exponential map
For any \(t \in (0, \, \infty )\), the map
is well-defined. Hereafter let \(x_0 \in {\mathcal{M}}\) be arbitrarily taken and fixed. We abbreviate as \(B (R_{\mathcal{M}}) = B (R_{\mathcal{M}}, \, 0)\). We denote \(u \left( t, \, \exp _{x_0} x\right) \) by \(u (t, \, x)\) for any \((t, \, x) \in \left( B (R_{\mathcal{M}})\right) _\infty : = (0, \, \infty ) \times B (R_{\mathcal{M}})\) and, furthermore, by translation, regard u as a map defined on \(\left( B (R_{\mathcal{M}})\right) _\infty \) with values into \(\mathrm{I}\!\mathrm{R}^l\).
Let us denote the penalized energy density for a map u by
We need the so-called Bochner type estimate for the penalized energy density. See “Appendix C” for the proof. Here the constant \(C_0\) in (2.1) is appropriately chosen.
Lemma 9
(Bochner type estimate) Let \(p > 2\) and \(u = u_{K, \, \epsilon }\) be a regular solution to (2.1). For brevity, put \(e (u) = e_{K, \, \epsilon } (u)\). Then, it holds in \((B_{R_{\mathcal{M}}})_\infty \) that
where
the summation convention over repeated indices is used and the positive constants \(C_i\) \((i = 1, 2, 3)\) depend on p, \(\mathcal{M}\) and \(\mathcal{N}\).
Let \(\lambda _0\) be a positive number, R be a positive number such that \(R < \min \{1, \, R_{\mathcal{M}}/2, \, T^{1/\lambda _0}\}\) and \((t_0, \, x_0)\) in the parabolic like envelope \(\mathcal{P} : = \big \{ (t, \, x) \, : \, T - R^{\lambda _0}< t \le T, \, |x|^{\lambda _0} < t - (T - R^{\lambda _0}) \big \}.\) In the following we use time-space local cylinder. For \(r, \, \tau > 0\), \(Q \left( \tau , \, r\right) (t_0, \, x_0) = \left( t_0 - \tau , \, t_0\right) \times B (r, \, x_0)\), where \(B (r, \, x_0)\) is an open ball in \(B_{R_{\mathcal{M}}}\) with center \(x_0\) and radius r. For brevity, we put \(u = u_{K, \, \epsilon }\), \(e (u) = e_{K, \, \epsilon } (u)\) in (3.3) and abbreviate the time-space Lebesgue measure \(d t \, d {\mathcal{M}}\) as dz.
Lemma 10
(Gradient boundedness on a small region) For some \((t_0, \, x_0) \in \mathcal{P}\), let \(\rho _0 : = \left( (t_0 - (T - R^{\lambda _0}))^{1/\lambda _0} - |x_0| \right) /4\). Suppose that, for \(\lambda _0 > 0\), \(r_0 > 0\), \(C_1 > 0\) and \(L > 0\),
Let \(q > 2\) be a positive number. Then there exists a positive number C depending only on q, p, \({\mathcal{M}}\) and \({\mathcal{N}}\), but, independent of L, such that
The detail of proof is presented in “Appendix D” (refer to [10, 12]).
3.2 Local regularity estimates
The partial regularity is based on the so-called small energy regularity estimate (refer to [39, Theorems 5.1, 5.3, 5.4; their proofs, pp. 491–494]). The small energy regularity estimate for the p-harmonic flow in the case \(p > 2\) has been recently established in [33, 34]. Our main task here is to demonstrate that the small energy regularity estimate holds uniformly for solutions of the penalized equations.
Theorem 11
(Small energy regularity) Let \(p > 2\). Let \(B_0\) and \(a_0\) be positive numbers satisfying the conditions
Let \(u = u_{K, \, \epsilon }\) be a regular solution of (2.1) on \((B (R_{\mathcal{M}}))_T = (0, \, T) \times B (R_{\mathcal{M}}, \, 0)\) for a positive \(T < \infty \), satisfying the energy bound
for a positive number \(C_1\) depending only on \({\mathcal{M}}\), p and \({\mathcal{N}}\). Then, there exists a small positive numeber \(R_0 < 1\), depending only on \({\mathcal{M}}\), \({\mathcal{N}}\), p, \(B_0\), \(a_0\) and \(C_1\), and the following holds true : Let \(\gamma _0\) be any positive number satisfying
If, for some small positive \(R < \min \{R_{\mathcal{M}}, \, R_0, \, T^{1/B_0}\}\),
then, there holds
where the positive constant \(C_2\) depends only on \(\gamma _0\), \(B_0\), \(a_0\), p, \({\mathcal{M}}\), \({\mathcal{N}}\) and \(C_1\).
The novelty here is a new monotonicity type estimate of a localized scaled energy, which may be of its own interest. Let us define our localized scaled energy in the following way: Let \(T \ge 0\) be given, and \((t_0, \, x_0)\) in the parabolic like envelope
The localized scaled energy is defined by
and \(\Lambda = \Lambda (r)\) is a function of a scale radius r, defined as
for any r, \(0< r < R_{\mathcal{M}}/2\), where we note that
The forward or backward in time Barenblatt like function, denoted by \(\mathcal{B}_-\) and \(\mathcal{B}_-\), respectively, are defined by
The localized function \(\mathcal{C}\) is defined and used as
We call \(E_+ (r)\) and \(E_- (r)\) the forward and backward localized scaled p-energy, respectively.
Our monotonicity type estimate of a scaled energy is the following. The proof is postponed by Sect. 5
Lemma 12
(Monotonicity estimate for the backward localized scaled p-energy) Let \(p > 2\) and \(q > 2\). For any regular solution u to (2.1) the following estimate holds for all positive numbers \(r, \rho \), \(r^{B_0} = \Lambda (r)^{2 - p} r^2 < \rho ^{B_0} = \Lambda (\rho )^{2 - p} \rho ^2 \le \min \{1, \, (R_{\mathcal{M}})^{B_0}, \, (t_0 - T)/2\}\),
where \(B_0\) as in (3.12), and the positive exponents \(\theta _0 \ge 2\) and \(\mu \) depend only on \(B_0\), p and \({\mathcal{N}}\), \({\mathcal{M}}\), p and \(B_0\), respectively, and the positive constant C depends only on the same ones as \(\mu \) and q.
Lemma 13
(Monotonicity estimate for the forward localized scaled p-energy) Let \(p > 2\) and \(q > 2\). For any regular solution u to (2.1) the following estimate holds for all positive numbers \(r, \rho \), \(r^{B_0} = \Lambda (r)^{2 - p} r^2 < \rho ^{B_0} = \Lambda (\rho )^{2 - p} \rho ^2 \le \min \{1, \, T - t_0 + (R_{\mathcal{M}})^{B_0}\}\)
where \(B_0\) as in (3.12), and the positive constants \(\theta _0 \ge 2\), \(\mu \) and C have the same dependence as those in Lemma 12.
From now on we show the validity of Theorem 11.
First of all we make parallel translation \(t^\prime = t - T\), \(x^\prime = x\) of the Eq. (1.4) and its solutions u on \((0, \, T) \times B (R_{\mathcal{M}})\) to those on \((- T, \, 0) \times B (R_{\mathcal{M}})\) with the same notation. The Eq. (1.4) is invariant under parallel transformation.
Under this setting the statement of Theorem 11 is rewritten as
Lemma 14
There exists a positive number \(R_0 < 1\), depending only on \(B_0\), p, \(\mathcal{M}\) and \(\mathcal{N}\), such that the following is valid : If
is satisfied for some small positive \(R \le R_0\) with
then, it holds that, for a positive constant \(C_2\) depending only on p, \(\mathcal{M}\), \({\mathcal{N}}\) and \(B_0\),
The proof of Lemma 14 consists of several steps, which are separately explained with those proofs. Our strategy of proof is based on a now classical argument similar to [9, 39], originally introduced by Schoen for the partial regularity of harmonic maps [36]. Here we carefully make local estimates under an intrinsic scaling to the evolutionary p-Laplace operator.
Hereafter in this section we put, for brevity,
Let positive numbers \(\lambda _0 > 2\) and \(a_0 < 1\) be determined later. According to \(\lambda _0\) and \(a_0\), we choose a positive number \(\epsilon \) such that
where we should choose \(a_0\) as
For t, \(- R^{\lambda _0} \le t \le 0\), we define a function f(t) as
where we notice by (3.20) that
Now we also define a function g(t) as
It is readily seen that, for any t, \(- R^{\lambda _0} \le t \le 0\),
Let t, \(- R^{\lambda _0} < t \le 0\), be arbitrarily taken and fixed. Then we can choose some time-space points \((t_0, \, x_0)\) such that \(t_0 \in (- R^{\lambda _0}, \, t]\) and \(x_0 \in B \big ( (t_0 + R^{\lambda _0})^{1/\lambda _0}, \, 0 \big )\), and
where we put
Here, if \(t_0 = - R^{\lambda _0}\) or \(|x_0| = (t_0 + R^{\lambda _0})^{1/\lambda _0}\), then \(f (t) = 0\) and \(g (t) = 0\).
Refined gradient boundedness on a small region By Lemma 10, we make the gradient bounded by a local scaled energy on a small region. We divide our consideration into two cases.
- Case 1.:
-
First we treat the case that \((\rho _0)^{a_0} \, (e (u (t_0, \, x_0)))^{\frac{1}{p}} \le 1\).
Then we have that
$$\begin{aligned}&\left( \frac{ (t_0 + R^{\lambda _0})^{\frac{1}{\lambda _0}} - |x_0| }{4} \right) ^{a_0} \, (e (u (t_0, \, x_0)))^{\frac{1}{p}} \le 1 \nonumber \\&\quad \Longleftrightarrow \displaystyle { \left( (t_0 + R^{\lambda _0})^{\frac{1}{\lambda _0}} - |x_0| \right) ^{a_0} \, (e (u (t_0, \, x_0)))^{\frac{1}{p}} \le 4^{a_0} } \nonumber \\&\quad \Longleftrightarrow \displaystyle { f (t) \le 4^{p \, a_0 \, A_0}.} \end{aligned}$$(3.27)$$\begin{aligned} \displaystyle { g (t) \le f (t) \le 4^{p \, a_0 \, A_0}. } \end{aligned}$$(3.28) - Case 2.:
-
Next we study the case that \((\rho _0)^{a_0} (e (u (t_0, \, x_0)))^{\frac{1}{p}} > 1\).
Then we have
$$\begin{aligned} \displaystyle { r_1 : = \left( \frac{1}{ (e (u (t_0, \, x_0)))^{\frac{1}{p}} } \right) ^{\frac{1}{a_0}} < \rho _0 \le 1. } \end{aligned}$$(3.29)
Let L be
It holds that
because
Under (3.31) we have
For the validity of (3.32), we observe from (3.22) and (3.25) that
Then we find that, for L in (3.30),
where we choose \(a_0\) as
Here we show the validity of (3.33), through (3.22) and (3.25). For any \(\tau \), \(t_0 - (\rho _0)^{\lambda _0} \le \tau \le t_0\), we find that
because it holds that for any \(\tau \), \(t_0 - (\rho _0)^{\lambda _0} \le \tau \le t_0\), and any \(x \in B (\rho _0, \, x_0)\)
where we note the definition \(\rho _0\) in (3.26) and use the simple algebraic inequality for any positive number a and b
From (3.36) we obtain that
where we use that for any \(\tau \), \(t_0 - (\rho _0)^{\lambda _0} \le \tau \le t_0\)
because by (3.37), for any \(\tau \), \(t_0 - (\rho _0)^{\lambda _0} \le \tau \le t_0\),
Thus, (3.33) is actually verified.
Under the choice of parameters \(\lambda _0 > 2\) and \(a_0\) in (3.21) and (3.35), we should have
and, (3.32) which verifies the condition (3.5) with letting \(r_0 = r_1/2\). Thus, we can apply Lemma 10 and take the \(L^\infty \)-estimate of gradient (3.6), yielding
where \(C > 0\) depends only on \(a_0\), p, \(\mathcal{M}\) and \({\mathcal{N}}\).
Multiplying the both sides of (3.40) by \(\left( L^{- p} e (u (t_0, \, x_0)) \right) ^{\frac{2}{p} - 1}\), we have
where by \(r_1\) in (3.29), L in (3.30) and (3.33) we compute as
Furthermore, we divide our estimations into two cases, depending on the size of \(r_1\).
The positive number \(q > 2\) is selected later. Recall that the positive number \(\epsilon \) is as in (3.20). Then \(q/\epsilon > 1\).
Case 2-1 : \(0 < r_1 \le (\rho _0)^{\frac{q}{\epsilon }}\) \(; \quad \) Case 2-2 : \((\rho _0)^{\frac{q}{\epsilon }}< r_1 < \rho _0\).
Case 2-1 \(0 < r_1 \le (\rho _0)^{\frac{q}{\epsilon }}\).
Lemma 15
Suppose that
Then there exists \(t_0^\prime \in \left[ t_0 - (r_1)^{\lambda _0}/4, \, t_0 \right] \) such that
where the positive constant C depends only on \(a_0\), m, p and \(\mathcal{N}\).
Proof
We will estimate both sides of (3.41).
By \(\rho _0\) in (3.42), \(r_1\) in (3.29) and L in (3.30), the left hand side of (3.41) is computed as
where by (3.42),
and the parameters \(a_0\) and \(\epsilon \) satisfy (3.38) and (3.20).
In the right hand side of (3.41), we take the supremum on time to have, by L and \(r_1\) in (3.30),
where by continuity of the gradient of solution, we choose some \(t_0^\prime \) such that
at which the supremum of the second line is attained. \(\square \)
Case 2-2 \((\rho _0)^{ \frac{q}{\epsilon } }< r_1 < \rho _0\).
Lemma 16
Suppose that
Then there exists \(t_0^\prime \in \left[ t_0 - (\rho _0)^{q \, \lambda _0/\epsilon }/4, \, t_0 \right] \) such that
where the positive constant C depends only on \(a_0\), p, \(\mathcal{M}\) and \(\mathcal{N}\).
Proof
First we take a look at the inequality (3.43) in Case 2-1. For \(r_1\), \(0 < r_1 \le (\rho _0)^{\frac{q}{\epsilon }}\) it holds that
where we use the definition of \(r_1\) in (3.29). In particular, (3.49) is valid for \(r_1 = (\rho _0)^{\frac{q}{\epsilon }}\) and the corresponding \(t_0^\prime \) as in (3.45) and (3.46)
Thus, for \(r_1\), \((\rho _0)^{ \frac{q}{\epsilon } }< r_1 < \rho _0\), we simply have
because of (3.20) and (3.38) again. \(\square \)
Now we derive an ordinary differential inequality for g(t), \(- R^{\lambda _0} \le t \le 0\).
Lemma 17
Let \(\lambda _0\), \(B_0\), \(a_0\) and \(\epsilon \) be positive parameters satisfying the conditions
Then the differential inequality holds for any positive \(R < 1\) and any t, \(- R^{\lambda _0} \le t \le 0\)
where the initial data \(g_0\) is
and the positive constant C depends only on \(\lambda _0\), p, \(\mathcal{M}\) and \(\mathcal{N}\).
Proof
Simply saying, our desired inequality (3.53) in Lemma 17 is obtained from combining the gradient \(L^\infty \)-estimate on a small region in Lemmata 15 and 16, and the monotonicity estimate of local scaled energy in Lemmata 12 and 13. Here we observe the admissible range of two parameters \(B_0\) in Lemmata 12 and 13, and \(\lambda _0\) in Lemmata 15 and 16, to choose as \(\lambda _0 = B_0\). By (3.12) and (3.39) we have
and thus, we can choose \(B_0\) and \(\lambda _0\) as in (3.51), because
The choice of \(a_0\) in (3.38) and \(\epsilon \) in (3.20) are as in (3.52).
By use of the monotonicity estimate in Lemmata 12 and 13. we estimate the local in space scaled integral of gradient in the right hand side of (3.43) in Lemma 15 and (3.48) in Lemma 16
Backward monotonicity estimate, Lemma 12 First we apply the backward monotonicity estimate, Lemma 12, for the local scaled energy in the right hand side of (3.43) in Lemma 15 and (3.48) in Lemma 16.
Let us choose the time-component \(t_0\) of the pole of Barenblatt function \(\mathcal{B}_-\) in (5.3) as follows: For each
let \(t_0\) be as \(t_1\)
Then, the local scaled integral in the right hand side of (3.43) and (3.48) is estimated as
because by (3.56) we have, for \(t : = t_0^\prime \),
Let \(\rho ^\prime \) be a positive number, chosen as
and then, the backward monotonicity estimate in Lemma 12 yields the upper-boundedness for (3.57) by
Forward monotonicity estimate, Lemma 13 Next, we use the forward monotonicty estmate in Lemma 13 for estimating the first scaled energy in (3.59).
By use of \(\mathcal{C}\), the first term of (3.59) is evaluated by the forward scaled energy
since by (3.58) we find that, for \(t : = t_1 - (\Lambda (\rho ^\prime ))^{2 - p} \, (\rho ^\prime )^2\),
and the function \(\mathcal{C}\) can be evaluated above as
because
and thus, for \(t : = t_1 -(\Lambda (\rho ^\prime ))^{2 - p} (\rho ^\prime )^2\),
Also the third term of (3.59) is bounded above by
because by the support of \(\mathcal{C}\) the region of \(L^\infty \) norm on space is actually
Then, by the forward monotonicity estimate in Lemma 13 (3.60) is bounded by
where again, we note that by (3.58)
The first scaled energy term above is estimated above as
Now we combine the estimations above, (3.43), (3.48), (3.57), (3.59), (3.60), (3.61), (3.62) and (3.63) to have
where the power exponent in the left hand side is positive by (3.52), and the second one in the right hand side is bounded as
where we recall \(t_1\) in (3.56) and \(\rho ^\prime \) in (3.58)
Differential inequality We gather (3.64) and (3.65) and then, multiply the resulting inequality by \(\left( \Big (t_0 + R^{\lambda _0}\Big )^{1/\lambda _0} - |x_0| \right) ^{a_0 \, A_0}\) to have
where we note by (3.51) that \(B_0 = \lambda _0\).
Moreover, we will modify some terms in (3.66) for our demand. By (3.24) the left hand side of (3.66) is estimated below by g(t).
In the third term in the right hand side of (3.66) the integrand is bounded by
since \(q > 2\) can be chosen to be large, comparing with \(a_0 p \theta _0\) and depending only on p and \(B_0\), in fact,
where \(\theta _0\) depends only on p and \(B_0\).
Finally, collecting (3.28) in Case 1, and (3.66) (3.67) in Case 2, we arrive at our desired estimation (3.53).
Here we observe that the principal integral quantity in (3.66) is rewritten as
In fact, by time-space continuity of Du, we have the estimation for sufficiently small positive \(\rho \)
which converges to 0, by taking the \(\limsup \) on \(\rho \) tending to 0 in the both side. \(\square \)
We are now in position to show the validity of Lemma 14. We solve the differential inequality (3.53) and (3.54), yielding the uniform gradient bound (3.19).
Proof of Lemma 14
The differential inequality (3.53) and (3.54) can be easily solved as
with the exponent
which is satisfied by \(\theta _0 > 1\) and choice in (3.52).
We simply obtain from (3.69)
under the choice of R such that
which is satisfied by
and so, let \(R_0\) be the positive number in the left hand side of the first inequality in (3.71). \(\square \)
4 Passing to the limit
In this section we present the proof of Theorem 5, based on Theorem 11. As before we abbreviate the time-space Lebesgue measure \(d t d {\mathcal{M}}\) as dz.
Let \(\{\epsilon _k\}\) and \(\{K_k\}\) be sequences such that \(\epsilon _k \searrow 0\) and \(K_k \nearrow \infty \) as \(k \rightarrow \infty \). Let \(u_{K_k, \, \epsilon _k}\), \(k = 1, 2, \ldots \), be a sequence of solutions of the Cauchy problem with initial data \(u_0\) for the penalized equations (2.1) with approximating numbers \(\epsilon = \epsilon _k\) and \(K = K_k\), obtained in Lemma 6. Hereafter we put \(u_k = u_{K_k, \, \epsilon _k}\) \(e_k (u_k) = e_{K_k, \, \epsilon _k} (u_{K_k, \, \epsilon _k})\), for brevity.
By the energy inequality (2.4), there exist a subsequence of \(\{u_k\}\), also denoted by the same notation, and the limit map u such that, as \(k \rightarrow \infty \),
where the strong convergence in (4.5) follows from (4.1) and (4.2) (see [6, Lemma 1.4, p. 28]). Thus, furthermore, for a subsequence \(\{u_k\}\) denoted by the same notation,
The use of convergence (4.3) and (4.2) in the energy inequality (3.1) for \(u_k\) also yields (1.14) for the limit map u.
We demonstrate that the limit map u is a partial regular weak solution of the p-harmonic flow, as in the statement of Theorem 5. The proof is divided to several steps and proceeded.
Let us define the regular set of the limit map u as
and thus, the singular set as the complement of \(\hbox {Reg}(u)\), \(\Sigma : = \hbox {Sing}(u) = {\mathcal{M}}_\infty {\setminus }\) \(\hbox {Reg}(u)\). By definition, \(\hbox {Reg}(u)\) is a relatively open set of \({\mathcal{M}}_\infty \) and \(\hbox {Sing}(u)\) is relatively closed in \({\mathcal{M}}_\infty \). Let \(R_0\) be a sufficient small positive number, determined in Theorem 11. For \(\tau \), \(0< \tau < \infty \), and R, \(0< R < \min \{R_0, \, \tau ^{1/B_0}\}\), we put two subsets in \(\mathcal{M}\) as
Then, let us define as
where \(\displaystyle {\mathop {\otimes }_{0< \tau < \infty }}\) means the direct product of sets on positive time \(\tau < \infty \).
Regularity of the limit map We will prove that \(\Sigma = \text{ Sing } (u) \subset \mathcal{S}\). For this purpose, we now show the regularity of limit map u in the complement of \(\mathcal S\). Let \((t_0, \, x_0)\) be in the complement of \(\mathcal S\). Thus, there exist a positive \(R < \min \{R_0, \, (t_0)^{1/B_0}\}\) and an infinite family \(\{u_k\}\) of regular solutions such that
Then we can apply Theorem 11 for each \(u_k\) above to obtain
where the positive constant C depends only on \(B_0\), p, \(\mathcal{M}\) and \({\mathcal{N}}\).
Put \(Q : = \Big (t_0 - (R/8)^{B_0}, \, t_0\Big ) \times B (R/8, \, x_0)\). From (4.9), there exists a subsequence of \(\{u_k\}\), denoted by the same notation, such that, as \(k \rightarrow \infty \),
Now we will show the uniform continuity of \(\{u_k\}\) in Q. For this purpose we will derive a local \(L^2\) estimate of derivative of the penalty term. For any smooth function \(\phi \) of compact support in Q, we multiply the Bochner type estimate (3.4) in Lemma 9 by \(\phi ^2 \, \sqrt{|g|}\) and integrate by parts in Q to have, letting \(K = K_k\), \(u = u_k\) and \(e (u) = e_k (u_k)\),
where we use the Cauchy inequality in the first inequality.
Let \((t_0, \, x_0) \subset Q\) be any point and \(r \le R/8\) be any positive number, and \(Q (r) = (t_0 - r^q, \, t_0) \times B (r, \, x_0)\) with \(q > 1\). In (4.11) we choose a smooth function \(\phi \) such that \(0 \le \phi \le 1\), \(\phi = 1\) in Q(r), \(\phi = 0\) outside Q(2r), and \(|D\phi | \le C/r\) and \(|\partial _t \phi | \le C/r^q\). Thus, from (4.9) and (4.11) we obtain
We also need the Poincaré inequality of parabolic type : Let \(u = u_k\). There exists a positive constant C, depending only on \(\mathcal{M}\) and p, such that, for any \(Q (r) \subset Q\),
where \({\bar{u}}_{Q (r)}\) is the integral mean of u in Q(r). For the proof refer to [28].
Substituting (4.9) and (4.12) into (4.13), we have, for any \((t_0, \, x_0) \subset Q\), any positive \(r \le R/8\), and \(Q (r) = (t_0 - r^q, \, t_0) \times B (r, \, x_0)\),
and thus, choosing \(q > 1\) in (4.14), we obtain from Campanato’s isomorphism theorem that \(\{u_k\}\) is uniformly Hölder continuous in Q with exponent \(\min \{1, \, q - 1, \, \frac{q}{2}\}\) on the metric \(|t|^{1/ q} + |x|\), uniformly on \(u_k\). Thus, we see that \(\{u_k\}\) is equicontinuous, and uniformly bounded in Q by Lemma 8. Therefore, by Arzela-Ascoli theorem we find for a subsequence of \(\{u_k\}\), denoted by the same notation, and the limit map u that, as \(k \rightarrow \infty \),
and that the limit map u is uniformly continuous in Q. From (4.9) and (4.15), we see that, as \(k \rightarrow \infty \),
Now we will show that the limit map u satisfies the p-harmonic flow equation in Q. From (4.9) and (4.11) we also see that \(\left\{ (K_k/2) \left. D_u \chi \big ({ \text{ dist }}^2 (u, \, {\mathcal{N}}\big ) \right| _{u = u_k} \right\} \) is bounded in \(L^2 (Q, \mathrm{I}\!\mathrm{R}^l)\) and then, there exists a vector-valued function \(\nu \in L^2 (Q, \mathrm{I}\!\mathrm{R}^l)\) such that, as \(k \rightarrow \infty \),
Let \(\mathcal{P}_{\mathcal{N}} \big (u (Q)\big )\) be a neighborhood of u(Q) in \(\mathcal{N}\). Let \(\tau (v)\) be any smooth tangent vector field of \({\mathcal{N}}\) on \(\mathcal{P}_{\mathcal{N}} \big (u (Q)\big )\), \(\tau (v) \in \mathcal{T}_v {\mathcal{N}}\) for any \(v \in \mathcal{P}_{\mathcal{N}} \big (u (Q)\big )\). By (4.15), we can choose a sufficiently large \(k_0\) such that, for any \(k \ge k_0\), \(u_k \in \mathcal{O}_{\delta _{\mathcal{N}}}\) in Q and \(\pi _{\mathcal{N}} (u_k) \in \mathcal{P}_{\mathcal{N}} \big (u (Q)\big ) \subset {\mathcal{N}}\) and \(\tau (\pi _{\mathcal{N}} (u_k)) \in \mathcal{T}_{\pi _{\mathcal{N}} (u_k)} {\mathcal{N}}\) in Q, where \(\mathcal{O}_{\delta _{\mathcal{N}}}\) is a tubular neighborhood in \(\mathrm{I}\!\mathrm{R}^l\) of \({\mathcal{N}}\) with width \(\delta _{\mathcal{N}}\), and \(\pi _{\mathcal{N}}\) is the nearest point projection to \(\mathcal{N}\) from the tubular neighborhood of \(\mathcal{N}\). Thus, we have that
because \(\left. D_u \text{ dist } (u, \, {\mathcal{N}}) \right| _{u = u_k (z)}\) is orthogonal to \(\mathcal{T}_{\pi _{\mathcal{N}} (u_k (z))} {\mathcal{N}}\) for any \(z \in Q\), and then,
By (4.15) and (4.17), we can take the limit as \(k \rightarrow \infty \) in (4.18) to have, for any smooth tangent vector field \(\tau (v)\) of \(\mathcal{N}\) on \(\mathcal{P}_{\mathcal{N}} \big (u (Q)\big ) \subset {\mathcal{N}}\), as \(k \rightarrow \infty \),
and thus, \(\nu (z)\) is a normal vector field along u(z) for any \(z \in Q\). In the weak form of (2.1), for any smooth map \(\phi \) with compact support in Q,
we pass to the limit as \(k \rightarrow \infty \) to find that the limit map u satisfies
where we use the convergence in the first line of (4.19) and, the strong convergence of gradients \(\{D u_k\}\), obtained from (2.1) with the convergence (4.1), (4.2) and (4.17) (see [6, Theorem 2.1 and its proof, pp. 31–33]). Therefore, we obtain that
We now observe that
Let \({\bar{z}} = ({\bar{t}}, \, {\bar{x}}) \in Q\) be arbitrarily taken and fixed. Let \(\gamma (v)\) be a smooth unit normal vector field of \({\mathcal{N}}\) in \(u (Q) \subset {\mathcal{N}}\) such that \(\gamma (v) \in (\mathcal{T}_v {\mathcal{N}})^\bot \), \(|\gamma (v)| = 1\) for any \(v \in u (Q)\) and \(\gamma (u ({\bar{z}})) = \nu ({\bar{z}}) / |\nu ({\bar{z}})|\). We take the composite map \(\gamma (u)\) of \(\gamma (\cdot )\) and the limit map u, and use a test function \(\sqrt{|g|} \gamma (u) \, \eta \) for any smooth real-valued function \(\eta \) with compact support in Q to have
where in the second line, we use that \(\partial _t u, D_\alpha u \in \mathcal{T}_u {\mathcal{N}}\), \(\alpha = 1, \ldots , m\), and \(\gamma (u) \in (\mathcal{T}_u {\mathcal{N}})^\bot \) in Q. The last line yields, at \(z = {\bar{z}}\),
Thus, (4.22) actually holds true.
Furthermore, there exists a positive constant C depending only on bounds of curvature of \({\mathcal{N}}\) and \((g^{\alpha \beta })\) such that
In fact, from (4.22) we obtain
By (4.23) and (4.10) we have that
where for the last statement of gradient continuity, we refer to [12, Theorem 1.1, p. 245; Sect. 4, p. 291; Sect. 1-(ii), pp. 217–218] (also see [26]).
As a consequence, we have that \((t_0, \, x_0)\) is a regular point and thus, \(\Sigma \subset \mathcal{S}\). Furthermore, from (4.21) and (4.22) it follows that the limit map u satisfies the p-harmonic flow Eq. (1.4) almost everywhere in Q.
Size estimate of the singular set We recall again that \(\Sigma = \text{ Sing } (u) \subset \mathcal{S}\). Let us estimate the size of \(\mathcal{S}\).
From the definition of limit supremum on k and (4.7), we see that, for every \(\tau \), \(0< \tau < \infty \), and R, \(0< R < \min \{R_0, \, \tau ^{1/B_0}\}\),
Here we have the estimation of size (see [17, Theorem 2.2; its proof, pp. 101–103], [21] for the proof) : It holds that, for every \(\tau \), \(0< \tau < \infty \), and R, \(0< R < \min \{R_0, \, \tau ^{1/B_0}\}\),
and so, by (4.25),
Thus, for any positive \(\tau < \infty \),
Then, the m-dimensional Hausdorff measure of \(\mathcal{S} \cap {\mathcal{M}}_\infty \) with respect to the time-space metric \(|t|^{1/\gamma _0} + |x|\) is locally zero : For any positive \(T < \infty \), letting \({\mathcal{M}}_T = (0, \, T) \times {\mathcal{M}}\),
Weak solution of the p-harmonic flow Now we set the two sequences of real-numbers as follows : Let \(\Lambda _0\) be a positive number. Let \(\epsilon \) be any small positive number and \(R_0 < 1\) be a sufficient small positive number, which are sent to zero, later. For positive constants \(M > 1\) and \(\theta < 1\), we define two geometrical progressions as
It is seen that \(\Lambda _l \nearrow \infty \) and \(R_l \searrow 0\) as \(l \rightarrow \infty \).
Let K be any time-space domain, \(K = (0, T) \times B (R_{\mathcal{M}}, \, x_0)\) for \(T > 0\) and a geodesic ball \(B (R_{\mathcal{M}}, \, x_0)\) in \(\mathcal{M}\). We set a family of sets \(\mathcal{S}_l\), \(l = 0, 1, 2, \ldots \), as
where \(\mathcal{S}\) is as in (4.8).
By the size of \(\mathcal{S}\) shown as before and the compactness of \(K \bigcap \mathcal{S}\), we can choose a covering of \(K \bigcap \mathcal{S}\) in the following way : There exist sequences of positive numbers \(\{r_{l \, i}\}\) and time-space points \(\{z_{l \, i}\}\), \(l = 0, 1, 2, \ldots \); \(i = 1, 2, \ldots , I (l)\) with finite integer I(l) depending on each l, such that, for each \(l = 0, 1, 2, \ldots \),
are a family of time-space cylinders and a covering of \(\mathcal{S}_l\) in the sense that
where \(\epsilon \) is firstly taken as small positive number.
Furthermore, by the compactness of \(K \bigcap \mathcal{S}\), we can take a covering of \(K \bigcap \mathcal{S}\) from \(\{P_{l \, i}^\prime \}\), obtained above, which consists of finitely many time-space cylinders \(P_{l \, i}^\prime \), \(l = 0, 1, 2, \ldots , L\) with finite integer L; \(i = 1, 2, \ldots , I (l)\), and has the properties
Let \(\eta \) be a smooth function on \({\mathcal{M}}_\infty \) such that \(0 \le \eta \le 1\), \(\eta = 1\) in \(P (1) (0) := (- 1, \, 1) \times B (1) (0)\) and the support of \(\eta \) is contained in \(P (2) (0) : = \big (- 2^{\gamma _0}, \, 2^{\gamma _0}\big ) \times B (2) (0)\), and \(|\partial _t \eta | + |D \eta | \le C\) with a positive number C depending only on m and \(\gamma _0\). For \(l = 0, 1, 2, \ldots \); \(i = 1, 2, \ldots , I (l)\), we denote by \(\eta _{\, l \, i}\) the scaled function
and then, \( \text{ the } \text{ support } \text{ of } \eta _{\, l \, i} \subset P_{l \, i}^{\prime \prime } : = P (10 \, r_{l \, i}) (z_{l \, i})\).
Let \(\mathcal{L} : = \{0, 1, 2, \ldots , L\}\), \(\mathcal{I} (l) : = \{1, 2, \ldots , I (l)\}\). Let \(\phi \) be any smooth map defined on \({\mathcal{M}}_\infty \) with values into \(\mathrm{I}\!\mathrm{R}^l\) with compact support in K. From (4.21) we obtain
We note that the number of overlaps of \(\left\{ P_{l \, i}^{\prime \prime } \right\} , l \in \mathcal{L} ;\) \(\, i \in \mathcal{I} (l)\), is at most finite and thus, there exists a subfamily \(\left\{ Q_{l \, i} \right\} \) of \(\left\{ P_{l \, i}^{\prime \prime } \right\} \) such that
for any \(z \in \bigcup _{l \in \mathcal{L}} \bigcup _{i \in \mathcal{I} (l)} \left( \text{ supp } \left( D \eta _{\, l \, i}\right) \bigcap P_{l \, i}^{\prime \prime } \bigcap \mathcal{S}_l \right) \). Thus, the last error term in (4.31) is estimated above by
of which the last integral is bounded by
with a positive constant C depending only on m, where we use that \(\gamma _0 > 1\) and that, by (4.27) and (4.28), for each \(l = 0, 1, 2, \ldots \),
Thus, it holds that
where we use (4.29) and the positive constant \(C^\prime \) depends only on m, \(\gamma _0\) and \(\sup _K |\phi |\). For summation on l, we choose the ratios \(M > 1\) and \(\theta < 1\) in (4.26) as
and compute
and thus,
provided that
Finally, we see from (4.29) and definition of \(\eta _{\, l \, i}\) that
and thus, we can take the limit as \(R_0 \searrow 0\) in (4.32) and (4.34), and use the Lebesgue convergence theorem with (4.35) in the second line of (4.31) to find that the limit map u satisfies the p-harmonic flow equation in the weak sense. \(\square \)
Convergence to the p-harmonic map at a time-infinity We will present the convergence of u to a p-harmonic map as time tends to infinity. By (1.14) we choose a sequence of time \(\{\tau _l\}\), \(\tau _l \nearrow \infty \), and a limit map \(u_\infty \) such that, as \(l \rightarrow \infty \),
where from (1.14) we obtain that, for some time-sequence \(\{t_l\}\), \(t_l \nearrow \infty \) as \(l \rightarrow \infty \),
Then, from the convergence (4.1), (4.2) and (4.3), there exists a subsequence of \(\{u_k (\tau _k)\}\), satisfying the same convergence as in (4.36), (4.37) and (4.38) with \(u (\tau _l)\) replaced by \(u_k (\tau _k)\), as \(k \nearrow \infty \).
Let us define the regular set of \(u_\infty \) as
and the singular set \(\hbox {Sing}(u_\infty )\) as the complement of \(\hbox {Reg}(u_\infty )\), \(\Sigma _\infty : = \hbox {Sing}(u_\infty ) = {\mathcal{M}} {\setminus }\) \(\hbox {Reg}(u_\infty )\). By definition, \(\hbox {Reg}(u_\infty )\) is relatively open in \(\mathcal{M}\) and \(\hbox {Sing}(u_\infty )\) is relatively closed in \({\mathcal{M}}\).
Let us put, for \(0< R < R_0\),
Then, similarly as in Size estimate of the singular set before, we have that, for any positive \(R < R_0\),
We will show that \(\Sigma _\infty \subset \mathcal{S}_\infty \). Let \(x_0\) be in the complement of \(\mathcal{S}_\infty \) and then, there exist a positive \(R < R_0\), a subsequence of \(\{u_k (\tau _k)\}\), denoted by the same notation as before, such that
Then, by Theorem 11, we have
where the positive constant C depends only on p, \(\mathcal{M}\) and \(\mathcal{N}\). Based on (4.42), we can proceed the same limit process as in (4.11)–(4.24) to find that \(u_\infty \) is regular in \(B (R/8, \, x_0)\) and thus, \(x_0 \in \text{ Reg } (u_\infty ) = {\mathcal{M}} {\setminus } \Sigma _\infty \). Therefore, the complement of \(\mathcal{S}_\infty \) is contained in that of \(\Sigma _\infty \), \( \text{ Reg } (u_\infty )\), and thus, \(\Sigma _\infty \subset \mathcal{S}_\infty \). By use of the size estimate of \(\mathcal{S}_\infty \) in (4.41), we can adopt the similar argument as in (4.26)–(4.35), where time-space regions used are replaced by the corresponding space regions, and thus, find that \(u_\infty \) is a weak solution of the p-harmonic map. \(\square \)
5 Monotonicity estimate of a local scaled energy
We now prove the monotonicity type estimate.
We make parallel translation on time of the Eq. (2.1) and its solutions u on \((T, \, \infty ) \times B (R_{\mathcal{M}})\) to those on \((0, \, \infty ) \times B (R_{\mathcal{M}})\) with the same notation.
Hereafter we assume that the metric \(g = \big (g_{\alpha \beta }\big )\) is the identity matrix. In the general case with \(\big (g_{\alpha \beta }\big )\), the lower order terms containing the derivatives of \(g_{\alpha \beta }\) only appear and controlled well as in the following estimations.
Let \((t_0, \, x_0)\) in the parabolic like envelope \(\displaystyle { \left\{ (t, \, x) \, : \, \min \{1, \, (R_{\mathcal{M}})^{B_0}\} > t \ge |x|^{B_0} \right\} , }\) \(\displaystyle { B_0 > 2. }\)
First we prove the backward monotonicity estimate, Lemma 12. Our localized scaled penalized energy is defined as
with weight
Hereafter, for brevity, we use the notation as above.
Lemma 18
Let \(p > 2\) and \(q > 2\). For any regular solution u to (2.1) the following estimate is valid for any positive number \(r < \rho \le \min \{1, \, R_{\mathcal{M}}, \, \left( t_0/2\right) ^{1/B_0}\}\)
where
and the positive exponents \(\theta _0 \ge 2\) and \(\mu \) depend only on \(B_0\), p and \(\mathcal{N}\), m, p and \(B_0\), respectively, and the positive constant C depends only on the same ones as \(\mu \) and q.
The proof is proceeded similarly as in [33, Lemmta 5 and 6]. Here we will study how to control well the approximating term, the derivative of penalty term.
Proof of Lemma 18
As before, let
and let r be any positive number in the range \(0 < r \le \min \left\{ 1, \, R_{\mathcal{M}}, \, \left( t_0/2\right) ^{1/B_0}\right\} \). First we make a scaling transformation intrinsic to the evolutionary p-Laplace operator
and, under the scaling transformation
Then the scaled solution v is a solution of the scaled equation on \(\{s = - 1\} \times \{y \in \mathrm{I}\!\mathrm{R}^m \, : \, x_0 + r y \in B(R_{\mathcal{M}})\}\)
and we put the notation
The scaled penalized energy is rewritten as
where the integral in (5.7) is well-defined by \(\text{ supp } (\mathcal{C})\) and \(\text{ supp } (\mathcal{B})\) and we simply compute as
Our main task in monotonicity estimate is to derive appropriate values of parameter such that
Step 1 : differentiation of E(r) on r. We compute differentiation of E(r) on r.
since
Estimations of II and III. The term II is nonnegative.
The term III is estimated by Young’s inequality as
where \(\mathcal{C} (s, \, y) : = \Big ( (t_0 + r^{B_0} \, s)^{1/B_0} - |x_0 + r \, y| \Big )_+\) is a Lipschitz function and the derivative of \(\mathcal C\) on r is computed as
and thus, on the support \(\{y \in \mathrm{I}\!\mathrm{R}^m \, : \, |y| < 1\}\) of \(\mathcal{B} (-1, \, y)\)
because of the conditions
The 1st term of (5.10) is scaled back and bounded below by
where we use \(\Lambda ^{2 - p} \, r^2 = r^{B_0}\),
and the notation
Each term of I is separately estimated in the following.
Estimation of I. By integration by parts, we have
where the generator of dilation is computed as
Estimation of \(I_1\).
where \({\bar{v}}\) is a weighted integral mean as in (5.14) below, and
because of Gauss’s divergence theorem and the compactness of support of \(\mathcal{B}\) and \(\mathcal{C}\).
Each term \(I_{11}\) and \(I_{12}\) is separately estimated in the following.
Estimation of \(I_{11}\). \(I_{11}\) is computed as
Now we will estimate each of three terms in (5.13).
For estimation of \(I_{111}\) we use the Poincaré type inequality with weight of Barenblatt like function [35, Theorem 5.3.4, p. 134]. Let \({\bar{v}}\) be a weighted integral mean
Lemma 19
(Poincaré inequality)
\(I_{111}\) of (5.13). \(I_{111}\) is estimated by Cauchy’s inequality for small \(c > 0\) as
where by definition of \(\Lambda \), \(1 + r \, \Lambda ^{- 1} \, \Lambda ^\prime = (p - B_0) (p -2)^{- 1}.\) The 1st time-derivative term is absorbed into that of (5.24) below, later. By the Poincaré inequality (5.15) and Young’s inequality with \(\delta > 0\), the 2nd term is bounded below by
where the last term is obtained from the derivative of \(\mathcal C\) on y, scaling back, a boundedness of the map u with a bound H depending only on \({\mathcal{N}}\) in Lemma 8
\(I_{112}\) of (5.13). By Cauchy’s inequality,
where the 1st one is the same as the 2nd term in (5.16) and bounded below for \(\delta > 0\) as in (5.17) and, the 2nd one of (5.19), together with the 1st one of (5.17), is estimated below by
where we make a scaling back and compute as
\(I_{113}\) of (5.13). By the boundedness (5.18) of derivative of \(\mathcal C\) and Cauchy’s inequality,
of which the 1st term is estimated, similarly as in (5.20), below by
and the 2nd term is bounded below as in (5.17).
Estimation of \(I_{12}\). By Cauchy’s inequality with \(\delta > 0\), we estimate as
where we use a boundedness of u with \(H > 0\) depending only on \({\mathcal{N}}\) in Lemma 8 to have
Estimation of \(I_2\). As before by Cauchy’s inequality
of which the 1st term is estimated below by (5.20).
Estimation of \(I_3\). \(I_3\) is treated as
Moreover each term of (5.23) is arranged as
Now each term in (5.24) is separately estimated.
\(I_{31}\) of (5.24). \(I_{31} \ge 0\) by the positivity of the coefficient. In fact, by definition of \(\Lambda \) and \(s = - 1\)
\(I_{32}\) of (5.24). By Cauchy’s inequality for small \(c > 0\),
The time-derivative term is absorbed into \(I_{31}\). By Young’s inequality the 2nd term is estimated below for \(\delta > 0\) by
of which the 1st term is bounded below by (5.20).
\(I_{33}\) of (5.24). Clearly, \(I_{33} \ge 0\).
\(I_{34}\) of (5.24).
By Cauchy’s inequality for small \(c > 0\),
where in the 1st inequality \((2 - p) \, r \, \Lambda ^{- 1} \, \Lambda ^\prime + 2 = B_0\) as before. The 1st term of (5.26) is absorbed into \(I_{31}\). The 2nd term of (5.26) is estimated below by (5.20).
\(I_{35}\) of (5.24). By Young’s inequality and the estimation (5.18) of derivative of \(\mathcal{C}\),
where the 1st term can be absorbed into \(I_{31}\) and the 2nd and 3rd terms are bounded below by
because \(2 (q - 1) (p - 1)/p > q - 2\) \(\Longleftarrow \) \(q > 2\).
Resulting estimation. Combining all of the estimations above we have
with \(\Lambda = r^{(B_0 - 2)/(2 - p)}\), according to (5.20), (5.21) and (5.28), and
The term J is clearly nonnegative. From (5.29) integrated on the interval \(\left( r, \, \rho \right) \) we derive
Step 2 : a uniform bound. We will make a bound of each term \(U_i\), \(i = 1, 2, 3\), in the right hand side of (5.30).
\(U_1\) of (5.30). The 1st integrals on r in the 2nd line of (5.30) are computed as
\(U_3\) of (5.30). \(- U_3\) is computed as
where by definition of \(\Lambda \)
and, in the last term we make a changing of variable
Here the exponent of power of \(\left( t_0 - t\right) \) in (5.32) is estimated as
and then,
and thus, the right hand side of (5.32) is bounded above by
\(U_2\) of (5.30). \(U_2\) is given by the approximation term, the derivative of penalty term in (2.1) and our task is to control \(U_2\) well in the appropriate way. \(U_2\) is evaluated by use of the Bochner type estimate for the penalty term
with positive constants \(C_0^\prime \) and C depending only on p, \({\mathcal{M}}\) and \({\mathcal{N}}\). The derivation of (5.33) is done similarly as in “Appendix C”, under the scaling settings (5.5) and (5.6), by using (6.8) below.
Let \(\bar{r}\) be as \(r \le {\bar{r}} < \rho \) and chosen later. In the following we will replace r by \(\bar{r}\), \(r \le {\bar{r}} < \rho \), and proceed to the similar estimations as for \(U_2\).
Multiplying a test function \(\mathcal{B} \, \mathcal{C}^q\) in (5.33) and then, integrating the resulting inequality on y in \(\{s = -1\} \times \mathrm{I}\!\mathrm{R}^m\) and on r in a interval \(({\bar{r}}, \, \rho )\), the estimation for \(U_2\) is done as
where by Cauchy’s inequality with a small \(c > 0\), the integrand term in the 3rd line is obtained from
Each term in the right hand side of (5.34) is separately treated in the following.
\(U_{22}\) in (5.34) The 3rd and 4th lines in the right hand side of (5.34) , \(U_{22}\), is scaled back and
where in the 2nd line we compute as
and, the integral on y in the 3rd line is bounded by a constant as before, since
The integral on r in the 1st and 2nd lines is transformed into that on time by changing a variable \(t = t_0 - \Lambda (r)^{2 - p} r^2 = t_0 - r^{B_0}\)
where the power exponents of scale radius are computed as
\(U_{21}\) in (5.34) \(- U_{21}\) is computed as
Each term in (5.36) is separately estimated in the following.
\(U_{212}\) in (5.36) \(U_{212}\) is estimated by using
and thus, by scaling back and a changing of variable \(t = t_0 - \Lambda (r)^{2 - p} r^2 = t_0 - r^{B_0}\),
where \(\displaystyle { \mathcal{B} = \left. \mathcal{B} (s, \, y)\right| _{s = - 1} = \left( 1 - |y|^\frac{p}{p - 1}\right) _+^{\frac{p - 1}{p - 2}} }\) and the exponents are computed as
\(U_{211}\) in (5.36) \(U_{211}\) is transformed into an integral on time by scaling back.
where we put
By changing a variable \(t = t_0 - \Lambda (r)^{2 - p} r^2 = t_0 - r^{B_0}\), we have
and an elementary computation
Thus, we have
where we put
We will estimate each term in (5.39).
Now, we set \(\bar{r}\) as
Then, by integration by parts in the integral on t, we have
Clearly, \(U_{2112} \ge 0\). By integration by parts in the integral on y, we also have
The last integral is estimated below as
where we use Young’s inequality with an exponent \(\alpha _0 > 1\) and compute as
and we choose \(\alpha _0 > 1\) as
By (5.41) and (5.42) substituted into (5.39), we have
yielding, with (5.36) and (5.37) for \(U_{212}\),
By definition of P(t) in (5.39) and \(\bar{r}\) in (5.40), P(t) is the local scaled integral of the penalty term, because by changing a variable \(x = x_0 + (t_0 - t)^{1/B_0} y\)
and it holds that
Collecting the estimations for \(U_1\), \(U_2\) and \(U_3\) above in (5.30), we have, for \({\bar{r}}\) in (5.40),
Let us put, for \(\alpha _0 > 1\) in (5.43),
From (5.46), our desired monotonicity estimate is shown to hold true in the range of scale radius \([{\bar{r}}, \, \rho ]\). Also (5.45) is the monotonicity estimate in the range \([r, \, {\bar{r}}]\) of the local scaled integral of the penalty term. Therefore, it remains to estimate the local scaled p-energy in the range of scale radius \([r, \, {\bar{r}}]\).
Step 3 : Monotonicity of the scaled p-energy. We now show a monotonicity estimate for the scaled p-energy without the penalty term. Under the same notation as before we denote the scaled p-energy by
and compute the differentiation of F(r) on a scale radius r
Clearly, \(H_2 \ge 0\). \(H_3\) is similarly estimated as III of (5.9) in (5.10), and
where we use an integration by parts and the dilation derivative (5.12).
Estimation of \(H_{11}\) We have
In the bracket of the right hand side, the 1st time-derivative term, the 3rd term and the 4th term are the same ones as in \(I_{111}\), \(I_{112}\) and \(I_{113}\) of \(I_{11}\) in (5.13), respectively. These terms are estimated as for \(I_{111}\), \(I_{112}\) and \(I_{113}\).
The 2nd term containing the derivative of penalty term is estimated in the following.
The 1st term is the same as \((- 1) \times \)2nd one in (5.16) and thus, estimated above by \((- 1) \times \)the right hand side of (5.17). The 2nd term is estimated in the following. Multiplying a test function \(\mathcal{B} \, \mathcal{C}^q\) by (5.33), we have, by Cauchy’s inequality with a small \(c > 0\),
where \(\displaystyle { \mathcal{B}^\prime = |y|^{\frac{2}{p - 1}} \left( 1 - \, |y|^{\frac{p}{p - 1}} \right) _+^{\frac{3 - p}{p - 2}} }\), and
The inequality (5.51) is integrated on y and then, estimated by integration by parts as
Estimations of \(H_{12}\) and \(H_{13}\) \(H_{12}\) is the same as \(I_2\) in (5.11) and thus, estimated as in (5.22) and (5.20).
\(H_{13}\) is the same as \(I_3\) in (5.11) except the derivative term of the penalty term and thus, is estimated similarly as for \(I_{3i}\), \(i = 1, \ldots , 5\), and the estimation (5.52) for the derivative of the penalty term in \(H_{11}\).
Gathering the estimations above and scaling back, we have, for \(\delta > 0\),
where in the 2nd line we estimate as
and, integrated on r in \((r, \, \rho )\), yielding
The 1st term in the right hand side is the same as \(U_1\) in (5.30) and estimated as in (5.31). The term in the 2nd and 3rd lines is, by changing a variable \(t = t_0 - \Lambda ^{2 - p} r^2 = t_0 - r^{B_0}\), computed as
where the power exponent of scale radius is evaluated as
Finally, we collect the estimations (5.45), (5.46) in Step 2, and (5.53), (5.54) in Step 3 to complete the proof of (5.4). \(\square \)
Now we show the validity of the forward monotonicity estimate, Lemma 13.
As before by parallel transformation let the Eq. (2.1) and its solutions u be defined on \((0, \, \infty ) \times \mathrm{I}\!\mathrm{R}^m\) with the same notation.
Let \((t_0, \, x_0)\) in the parabolic like envelope \(\displaystyle { \left\{ (t, \, x) \, : \, \min \{1, \, (R_{\mathcal{M}})^{B_0}\} > t \ge |x|^{B_0} \right\} }\), \(\displaystyle { B_0 > 2 }\).
The forward localized scaled penalized energy is
with weight
The notation as above is used.
Lemma 20
Let \(p > 2\) and \(q > 2\). For any regular solution u to (2.1) the following estimate is valid for any positive number \(r < \rho \le \min \{1, \, \big ((R_{\mathcal{M}})^{B_0} - t_0\big )^{1/B_0}\}\)
where
and the positive exponents \(\theta _0 \ge 2\) and \(\mu \) depend only on \(B_0\), p and \({\mathcal{N}}\), m, p and \(B_0\), respectively, and the positive constant C depends only on the same ones as \(\mu \) and q.
Proof of Lemma 20
As before we put
and let r any positive number in the range \(0 < r \le \min \{1, \, \big ((R_{\mathcal{M}})^{B_0} - t_0\big )^{1/B_0}\}\). We make a scaling transformation intrinsic to the evolutionary p-Laplace operator
and, under the scaling transformation it holds that
The scaled solution v is a solution of the scaled equation on \(\{s = 1\} \times \mathrm{I}\!\mathrm{R}^m\)
Hereafter we use the same notation as in (5.6).
Similarly as the backward case, the scaled energy is rewritten as
where the integral in (5.61) is well-defined by \(\text{ supp } (\mathcal{C})\) and \(\text{ supp } (\mathcal{B})\).
The computation and estimation are similar as in those for the backward monotonicity estimate. In the following we indicate only the part of estimations, different from the backward monotonicity. In the following the integral region on y is changed to \(\{s = 1\} \times \mathrm{I}\!\mathrm{R}^m\).
Similarly as in (5.9) in the backward case, we make differentiation of E(r) on r
Estimation of II and III. By Young’s inequality and \(0< \bar{\epsilon } = \Lambda ^{- 2} \epsilon \le 1\) the term II is bounded with \(\delta > 0\) by
For estimation of III the derivative of \(\mathcal{C}\) on r is computed as
and thus, on the support \(\{y \in \mathrm{I}\!\mathrm{R}^m \, : \, |y| < 1\}\) of \(\mathcal{B} (1, \, y)\)
because of the conditions
Thus, exactly as (5.10) in the backward case, we have
The estimation of I is exactly same as (5.11) in the backward case. The terms corresponding to \(I_1\) are bounded above by \((-1) \times \)the terms (5.20), (5.21) and some controllable integral terms containing \(\mathcal{B}\), \(\mathcal{C}\) and their derivatives, where the integral region is replaced by \(\{s = 1\} \times \mathrm{I}\!\mathrm{R}^m\). The term corresponding to \(I_2\) is estimated above by \((- 1) \times \)the right hand side of (5.22) with the integral region replaced by \(\{s = 1\} \times \mathrm{I}\!\mathrm{R}^m\).
\(I_3\) is computed exactly as (5.23) and (5.24) with the integral region replaced by \(\{s = 1\} \times \mathrm{I}\!\mathrm{R}^m\). In the term corresponding to \(I_{31}\) we note by \(s = 1\) that
The term corresponding to \(I_{33}\) is estimated above by
The other terms corresponding to \(I_{3i}\), \(i = 2, 4, 5\), are bounded above by \((- 1) \times \)the terms of the right hand side of (5.25), (5.26), (5.27) and (5.28).
Combining all of the estimations above we have
where \(\Lambda = r^{(B_0 - 2)/(2 - p)}\), but
The term J is clearly nonpositive. From (5.63) integrated on the interval \(\left( r, \, \rho \right) \) we derive
The terms in the right hand side of (5.64) correspond to those in (5.30). Note that \(U_1\), \(U_2\) and \(U_3\) are just \((- 1) \times \) the corresponding terms in (5.30). \(U_1\) and \(U_3\) can be estimated exactly similarly as the corresponding terms in (5.30).
We have to care the estimation of \(U_2\). Under the scaling setting (5.59) and (5.60) in the forward case now, we also have the Bochner type estimate (5.33) for the penalty term. We can proceed to the estimations, similarly as in (5.34), to obtain
The estimation for \(U_{22}\) is the same as in (5.35) in the backward case.
\(U_{21}\) is also computed as in (5.36) in the backward case
The estimation for \(|U_{212}|\) is done in the same way as in (5.37) in the backward case. The estimation for \(U_{211}\) is performed in the following. By changing a variable \(t = t_0 + \Lambda (r)^{2 - p} r^2 = t_0 + r^{B_0}\), we have
and a computation
Thus, we have
where we put
We will estimate each term in (5.65).
Now, we set \(\bar{r}\) as
Then, by integration by parts in the integral on t, we have
By integration by parts in the integral on y, we also have
The 1st term is nonnegative, since \(m - \frac{p (B_0 - 2)}{p - 2} > m - p \ge 0\), and the last integral is estimated below as
where \(\alpha _0 > 1\) is as in (5.43) in the backward case. Therefore we have
By definition of P(t) in (5.65) and \(\bar{r}\) in (5.66), P(t) is the local scaled integral of the penalty term, because by changing a variable \(x = x_0 + (t - t_0)^{1/B_0} y\)
and it holds that
Collecting the estimations for \(U_1\), \(U_2\) and \(U_2\) above in (5.64), we have, for \({\bar{r}}\) in (5.66),
Let \(\theta _0\) be as in (5.43) and (5.47) in the backward case. By (5.70), our desired monotonicity estimate holds true in the range of scale radius \([{\bar{r}}, \, \rho ]\), and (5.69) is the monotonicity estimate in the range \([r, \, {\bar{r}}]\) of the local scaled integral of the penalty term. Therefore, it remains to estimate the local scaled p-energy in the range of scale radius \([r, \, {\bar{r}}]\). The monotonicity estimate of the local scaled p-energy in the range of scale radius \([r, \, {\bar{r}}]\) is estimated exactly as Step 3 in the backward case. In fact, letting as in (5.48)
we arrive at the estimate corresponding to (5.53)
where the last term is controlled as in (5.54). \(\square \)
References
Ambrosio, L., Mantegazza, C.: Curvature and distance function from a manifold. J. Geom. Anal. 8(5), 723–748 (1998) (Dedicated to the memory of Fred Almgren)
Ambrosio, L., Soner, H.M.: A level set approach to the evolution of surfaces of any codimension. J. Differ. Geom. 43, 693–737 (1996)
Chang, K.-C.: Heat flow and boundary value problem for harmonic maps. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 6(5), 363–395 (1989)
Chang, K.-C., Ding, W.-Y., Ye, R.: Finite-time blow up of the heat flow of harmonic maps from surfaces. J. Differ. Geom. 36(2), 507–515 (1992)
Chen, C.-N., Cheung, L.F., Choi, Y.S., Law, C.K.: On the blow-up of heat flow for conformal \(3\)-harmonic maps. Trans. AMS 354(12), 5087–5110 (2002)
Chen, Y.-M., Hong, M.-C., Hungerbuhler, N.: Heat flow of \(p\)-harmonic maps with values into spheres. Math. Z. 215, 25–35 (1994)
Chen, Y.-M., Ding, W.-Y.: Blow-up and global existence for heat flows of harmonic maps. Invent. Math. 99(3), 567–578 (1990)
Chen, Y.-M., Lin, F.H.: Evolution of harmonic maps with Dirichlet boundary conditions. Commun. Anal. Geom. 1(3–4), 327–346 (1993)
Chen, Y.-M., Struwe, M.: Existence and partial regularity results for the heat flow for harmonic maps. Math. Z. 201, 83–103 (1989)
Choe, H.J.: Hölder continuity of solutions of certain degenerate parabolic systems. Nonlinear Anal. 8(3), 235–243 (1992)
Coron, J.M., Ghidaglia, J.M.: Explosion en temps fini pour le flot des applications harmoniques. C. R. Acad. Sci. Paris Ser. I(308), 339–344 (1989)
DiBenedetto, E.: Degenerate Parabolic Equations. Springer, New York (1993)
Duzaar, F., Fuchs, M.: On removable singularities of \(p\)-harmonic maps. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7(5), 385–405 (1990)
Duzaar, F., Mingione, G.: The \(p\)-harmonic approximation and the regularity of \(p\)-harmonic maps. Calc. Var. Partial Differential Equations 20, 235–256 (2004)
Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–169 (1964)
Fardoun, A., Regbaoui, R.: Heat flow for \(p\)-harmonic maps between compact Riemannian manifolds. Indiana Univ. Math. J. 51(6), 1305–1320 (2002)
Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton (1983)
Giaquinta, M., Hildebrandt, S.: A priori estimates for harmonic mappings. J. Reine Angew. Math. 336, 123–164 (1982)
Giaquinta, M., Struwe, M.: On the partial regularity of weak solutions of nonlinear parabolic systems. Math. Z. 179, 437–451 (1982)
Giaquinta, M., Struwe, M.: An optimal regularity result for a class of quasilinear parabolic systems. Manuscr. Math. 36, 223–240 (1981)
Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific, Singapore (2005)
Grotowski, J.F.: Finite time blow-up for the harmonic map heat flow. Calc. Var. Partial Differ. Equ. 1(2), 231–236 (1993)
Hamilton, R.: Harmonic Maps of Manifolds with Boundary \(m\)-Harmonic Flow. Lecture Notes in Mathematics, vol. 471, pp. 593–631. Springer, Berlin (1975)
Hungerbühler, N.: \(m\)-harmonic flow. Ann. Scuola Norm. Sup. Pisa CI. Sci. 24, 593–631 (1997)
Hungerbühler, N.: Global weak solutions of the \(p\)-harmonic flow into homogeneous spaces. Indiana Univ. Math. J. 45(1), 275–288 (1996)
Karim, C., Misawa, M.: Gradient Hölder regularity for nonlinear parabolic systems of \(p\)-Laplacian type. Differ. Integral Equ. 29(3–4), 201–228 (2016)
Leone, C., Misawa, M., Verde, A.: A global existence result for the heat flow of higher dimensional \(H\)-systems. J. Math. Pures Appl. (9) 97(3), 282–294 (2012)
Leone, C., Misawa, M., Verde, A.: The regularity for nonlinear parabolic systems of \(p\)-Laplacian type with critical growth. J. Differ. Equ. 256, 2807–2845 (2014)
Misawa, M.: Approximation of \(p\)-harmonic maps by the penalized equation. Nonlinear Anal. 47, 1069–1080 (2011)
Misawa, M.: Local Hölder regularity of gradients for evolutional \(p\)-Laplacian systems. Ann. Mat. Pura Appl. (IV) 181, 389–405 (2002)
Misawa, M.: Existence and regularity results for the gradient flow for \(p\)-harmonic maps. Electron J. Differ. Equ. 36, 1–17 (1998)
Misawa, M.: On the \(p\)-harmonic flow into spheres in the singular case. Nonlinear Anal. Ser. A Theory Methods 50(4), 485–494 (2002)
Misawa, M.: Local regularity and compactness for the \(p\)-harmonic map heat flows. Adv. Calc. Var. (2017). https://doi.org/10.1515/acv-2016-0064
Misawa, M.: Regularity for the evolution of \(p\)-harmonic maps. J. Differ. Equ. 264, 1716–1749 (2018)
Saloff-Coste, L.: Aspects of Sobolev-Type Inequalities. Lecture Note Series, vol. 289. London Mathematical Society
Schoen, R.: Analytic aspects of the harmonic map problem. In: Chern, S.S. (ed.) Seminar on Nonlinear Partial Differential Equations. MSRI Publications, vol. 2, pp. 321–358. Springer, New-York (1984)
Struwe, M.: On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems. Manuscr. Math. 35, 125–145 (1981)
Struwe, M.: On the evolution of harmonic maps of Riemannian surfaces. Comment. Math. Helv. 60(4), 558–581 (1985)
Struwe, M.: On the evolution of harmonic maps in higher dimensions. J. Differ. Geom. 28, 485–502 (1988)
Wang, C.: Limits of solutions to the generalized Ginzburg–Landau functional. Commun. Partial Differ. Equ. 27(5–6), 877–906 (2002)
Acknowledgements
I would like to record here my sincere thanks to the referee for kindly reading this long paper and giving some corrections.
Funding
The work by M. Masashi was partially supported by the Grant-in-Aid for Scientific Research (C) No. 15K04962 and No. 18K03375 at Japan Society for the Promotion of Science.
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6. Appendix
6. Appendix
1.1 Appendix A A global existence and regularity of a weak solution of (2.1).
Proof od Lemma 6
We use the Galerkin method and the monotonicity trick for p-Laplace operator to solve the Cauchy problem (2.1). The proof is standard and we can refer to [6, Theorem 1.5 and its proof, pp. 29–31].
Regularity of a weak solution. Let \(u = u_{K, \, \epsilon }\) be a weak solution of (2.1). The lower-order term is bounded by the definition of \(\chi \) as
and thus, we can apply the Hölder regularity result for the evolutionary p-Laplace operator in [12, Theorem \(1.1^\prime \), p. 256] (also see [26]) to find that the solution u and its gradient are locally Hölder continuous on \({\mathcal{M}}_\infty \). We also have that the second derivative is integrable : \(\big (\epsilon + |D u|^2\big )^{\frac{p - 2}{2}} |D^2 u|^2\) is locally integrable in \({\mathcal{M}}_\infty \) and that the gradient Du is locally bounded in \({\mathcal{M}}_\infty \) (see [12, Proposition 3.1, p. 223; Theorem 5.1 , p. 238]. Then, expanding the principal part of the p-Laplace operator, the solution u is also satisfies the linear parabolic systems with Hölder continuous coefficients and lower-order terms almost everywhere. Thus, it follows from the Schauder regularity theory that u, Du, \(D^2 u\) and \(\partial _t u\) are locally Hölder continuous in \({\mathcal{M}}_\infty \). \(\square \)
1.2 Appendix B Energy inequality and maximum principle
We present the proof of Lemmata 7 and 8.
Proof of Lemma 7
The energy inequality (3.1) is shown to be valid in the proof of Lemma 6. However, as a priori estimates for regular solutions of (2.1), we naturally multiply (2.1) by \(\partial _t u \, \sqrt{|g|}\) and integrate by parts on space variable in \({\mathcal{M}}_T\) for any \(T > 0\). \(\square \)
Proof of Lemma 8
We multiply (2.1) by \(\sqrt{|g|} u \big (|u|^2 - H^2\big )_+\) and integrate in \({\mathcal{M}}_\infty \), where \(\big (f\big )_+\) is the positive part of a function f. Since the support of \(\chi ^\prime \) is in \(\mathcal{O}_{2 \delta _{\mathcal{N}}} ({\mathcal{N}}) \subset B (H)\), \(\chi ^\prime (\mathrm{dist}^2 (u, \, {\mathcal{N}}))\) is zero in \(\mathrm{I}\!\mathrm{R}^l {\setminus } B (H, \, 0)\). Also \(u_0 \in {\mathcal{N}} \subset B (H)\). Hence, we have
and thus, \(|u (t)| \le H\) in \({\mathcal{M}}\) and any \(t \ge 0\). \(\square \)
1.3 Appendix C Proof of the Bochner estimate.
Proof of Lemma 9
In the proof, for brevity, let the regularized p-energy density be
In the general case in \({\mathcal{M}}\), the terms containing the spatial derivative of \(g_{\alpha \beta }\) only appear and are bounded by \(C \, (\epsilon + |D u|^2)^{p - 1}\). In fact, in (6.5) below, by a direct computation, we have the terms with derivatives of the metric
which are bounded by such terms as
with a positive constant C depending only on \(\mathcal{M}\), p and m. Here the 1st term is controllable lower-order one. By Cauchy’s inequality with \(c > 0\), the 2nd term are estimated above as
of which the 1st term with a small \(c >0\) in the right hand side is abosrbed into the squared 2nd derivative term of the solution in (6.5) below, and the 2nd term is a controllable one. The controllable terms \(C \, f\) above are multiplied by \((p f)^{1 - \frac{2}{p}}\) in (6.6) below, and thus, becomes \(C \, f^{2 (1 - \frac{1}{p})}\).
Hereafter, we assume that the metric \(g = \left( g_{\alpha \beta }\right) \) is the identity matrix.
Since u, Du and \(D^2 u\) are continuous in \(\mathrm{I}\!\mathrm{R}^m_\infty \), it holds in the distribution sense that
Hereafter the summation convention over repeated indices is used. Since \(\chi \big ({ \text{ dist }}^2 (u, \, {\mathcal{N}})\big ) = 2 (\delta _{\mathcal{N}})^2\) for \(u \in \mathrm{I}\!\mathrm{R}_l {\setminus } \mathcal{O}_{2 \delta _{\mathcal{N}}}\), \(D \Big (D_u \chi \big ({ \text{ dist }}^2 (u, \, {\mathcal{N}})\big )\Big ) = 0\) if \( \text{ dist } (u, \, {\mathcal{N}}) > 2 \delta _{\mathcal{N}}\) and then, we have (3.4) by (2.1). We treat the case that \( \text{ dist } (u, \, {\mathcal{N}}) \le 2 \delta _{\mathcal{N}}\). Noting that \(\chi \big ({ \text{ dist }}^2 (u, \, {\mathcal{N}})\big )\) is smooth, by a direct calculation we have
where the arguments in \(\chi ^\prime \) are omitted. By (6.3) and (6.4) with (2.1), we have
Furthermore, multiplying (6.5) by \((p \, f)^{1 - \frac{2}{p}}\), we obtain
where we use the fact that
By differentiation of the penalty term
and (6.4), we have
where in particular, multiplying (2.1) by the derivative of penalty term we compute
By the support of \(\chi ^{\prime \prime }\), we have
and thus, the 2nd terms in the 2nd line of (6.6) and the 3rd line of (6.7), and the 3rd term in the 3rd line of (6.8) are estimated above by
with a positive constant C depending only on p and \(\chi \), because
By Schwarz’s and Cauchy’s inequalities, the terms in the 3rd lines of (6.6) and in the 4th line of (6.7), (6.8) are bounded by
where by a positive constant \(C ({\mathcal{N}})\) depending on a bound for the curvature of \({\mathcal{N}}\), we have the boundedness for any \(u \in {\mathcal{N}}\)
of which the validity will be shown later.
The terms in the 5th line of (6.7) are bounded by
Gathering (6.9), (6.10) and (6.12) in (6.6) and (6.7), respectively, we obtain
and thus, from (6.13), the desired inequality (3.4) is obtained, if the constant \(C_0\) is so large that
We present the proof of (6.11). We follow the argument as in [2, Theorems 3.1 and 3.2, their proofs, pp. 704–707] (also refer to [1, Theorem 2.2]). \(\square \)
Lemma 21
There exists a positive constant C depending only on a bound of curvatures of \(\mathcal{N}\) such that, for any \(u \in \mathcal{O}_{2 \delta _{\mathcal{N}}}\) and \(q \in \mathrm{I}\!\mathrm{R}^l \cong \mathcal{T}_u \mathrm{I}\!\mathrm{R}^l\),
Proof
For any \(u \in \mathcal{O}_{2 \delta _{\mathcal{N}}}\) such that \(u \notin {\mathcal{N}}\), we make parallel transformation with the direction \(\big (u - \pi _{\mathcal{N}} (u)\big )/\big |u - \pi _{\mathcal{N}} (u)\big |\) and follow the following argument. Therefore, we treat the case that \(u \in {\mathcal{N}}\) and thus, \(\pi _{\mathcal{N}} (u) = u\). For any \(v \in \mathcal{O}_{2 \delta _{\mathcal{N}}}\) let us put
We know that the squared distance function \(\eta (v)\) is smooth on \(v \in \mathcal{O}_{2 \delta _{\mathcal{N}}}\). Let \(q \in \mathrm{I}\!\mathrm{R}^l\) be any vector in \(\mathrm{I}\!\mathrm{R}^l\) and be fixed. Under the orthogonal decomposition of \(\mathrm{I}\!\mathrm{R}^l\) with respect to the tangent space \(\mathcal{T}_u {\mathcal{N}}\) at \(u \in {\mathcal{N}}\), \(\mathrm{I}\!\mathrm{R}^l = \mathcal{T}_u {\mathcal{N}} \oplus (\mathcal{T}_u {\mathcal{N}})^\bot \), we set as
where p is the unit normal vector along the normal component of q and f(t) is the distance to \(\mathcal{N}\) measured along p. Then, we compute as
and, also, for any \(v \in \mathcal{O}_{2 \delta _{\mathcal{N}}}\), \(v \notin {\mathcal{N}}\),
Thus, letting, for any \(t \in (0, \, 2 \delta _{\mathcal{N}}]\),
we have
where by (6.17) we have, as \(t \searrow 0\),
and the 1st convergence is valid because \(D_{v^i} D_{v^j} \eta (u) = \left. D_{v^i} D_{v^j} \eta (v)\right| _{v = u}\) is the orthogonal projection on \((\mathcal{T}_u {\mathcal{N}})^\bot \) (see [2, Theorem 3.1, p.704]). Therefore, from l’Hospital’s theorem, we obtain
where we use that \(\displaystyle { p \cdot D_v d (u + t p) = \big |p\big |^2 = 1 }\) and \(\displaystyle { D_v (p \cdot D_u d (u + t p)) = 0. }\) Thus, we have
where in the last inequality the positive constant C depends only on a bound of curvatures of \(\mathcal{N}\) (see [2, Remark 3.3; Theorem 3.5, its proof, pp. 707–709]). \(\square \)
1.4 Appendix D Proof of the gradient boundedness.
Here we demonstrate the proof of Lemma 10, relying on Moser’s iteration method as usual. Such estimate has been originally done for the evolutionary p-Laplacian system with controllable growth lower-order terms, by DiBenedetto, developing the intrinsic scaling transformation to the evolutionary p-Laplace operator (refer to [10, 12]).
However, the emphasis here is to make localization by use of the cut-off function \(\mathcal C\).
Proof of Lemma 10
In the following we use the same notation as in Lemma 10.
By use of a scaling transformation intrinsic to the evolutionary p-Laplace operator
we now rewrite (3.4) in Lemma 9 by the scaled solution v on \(Q (1, \, 1) : = Q (1, \, 1) (0)\),
satisfying in \(Q (1, \, 1)\)
We put the notation
As in “Appendix C”, we assume that the metric \((g_{\alpha \beta })\) is the identity matrix and, in the general case in \({\mathcal{M}}\), the terms with the derivative of \(g_{\alpha \beta } (x_0 + r y)\) appear and are bounded by \(C \, f (v)^{2 (1 - \frac{1}{p})}\), as in (6.1) and (6.2), where we note that \(D_y g^{\alpha \beta } (x_0 + r y) = r D g^{\alpha \beta } (x)\) and \(r \le 1\). We proceed to the same computation as (6.6) and (6.7), where the quantities appeared are transformed to the corresponding ones, respectively, defined by the scaled solution v as above. Now we will look at the transformed estimation for the scaled solution v.
By the support of \(\chi ^{\prime \prime }\), we have
and thus, the corresponding terms for the scaled solution v to 2nd terms in the 2nd line of (6.6) and the 3rd line of (6.7), and the 3rd term in the 3rd line of (6.8) are estimated above by
with a positive constant C depending only on p and \(\chi \), because, by the definition of \(r_0\),
By Schwarz’s and Cauchy’s inequality, the terms in the 3rd line of (6.6) and in the 4th line of (6.7), (6.8) are estimated as
where by the definition of \(r_0\) as before, and a positive constant \(C ({\mathcal{N}})\) depending on a bound for the curvature of \({\mathcal{N}}\), we compute as
The terms in 5th line of (6.7) are bounded by
Gathering all of the estimations above yields
Finally we make Moser’s iteration estimate by (6.24) and scaling back to have (3.6). Now, taking care of localization by the cut off function \(\mathcal C\), we proceed to the estimations.
Let \(B (\rho ) = B (\rho , \, 0)\) be a ball in \(\mathrm{I}\!\mathrm{R}^m\) with radius \(\rho \le \min \{1, \, R_{\mathcal{M}}/2, \, T^{1/\lambda _0}\}\) and center of origin. Let \(0< r < \rho \). We use local parabolic cyllinders \(Q (r^2, \, r) = (- r^2, 0) \times B (r)\) and \(Q (\rho ^2, \, \rho ) = (- \rho ^2, 0) \times B (\rho )\), Let \(\eta \) be a smooth real-valued function on \(\mathrm{I}\!\mathrm{R}^m\) such that \(0 \le \eta \le 1\), the support of \(\eta \) is contained in \(B (\rho )\) and \(\eta = 1\) on B(r). Let \(\sigma = \sigma (t)\) be a smooth real-valued function on \(\mathrm{I}\!\mathrm{R}\) such that \(0 \le \sigma \le 1\), \(\sigma = 1\) on \([- r^2, \, \infty )\) and \(\sigma = 0\) on \((- \infty , \, - \rho ^2]\). We denote by the original notation the scaled function under (6.19). Put
and also write as \(z = (s, \, y) \in Q (1, 1)\) and \(d z = d {\mathcal{M}} d s\).
Put \(w = e (v)\) in the Bochner type estimate (6.24). Let \(\alpha \) be nonnegative number and use the test function \(w^\alpha \eta ^2 \sigma \, \mathcal{C}^q \, \sqrt{|g|}\) in the weak form of (6.24). After a routine computation we have the so-called reverse Poincaré inequality
where we compute as
Applying the Sobolev embedding \(W^{1, 2}_0 (B (\rho )) \rightarrow L^{2 m/(m - 2)} (B (\rho ))\) we have
which is combined with (6.25) and yields
By Hölder’s inequality and (6.26) we compute as
where we use a simple inequality valid for \(\alpha \ge 0\)
and also estimate the derivative of \(\mathcal{C}\) as
because, by the range Q(1, 1) of \((s, \, y)\) and the condition (3.5) of \(r_0\),
and so, we have the estimations
We arrange some terms in an appropriate way to have
Here let \(\{\rho _k\}\) be a sequence of radii, defined as
and \(\{\alpha _k\}\) be a sequence of exponents
We choose \(r = \rho _{k + 1}\), \(\rho = \rho _k\) and \(\alpha = \alpha _k\) in (6.27) and make routine computation to have
which is computed by sequences (6.28) and (6.29) as
An iterative application of (6.31) yields, as \(k \rightarrow \infty \),
where we use the relation of exponents
and the limit as \(k \rightarrow \infty \)
Finally, scaling back in (6.32) yields the desired estimate (3.6). \(\square \)
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Misawa, M. Global existence and partial regularity for the p-harmonic flow. Calc. Var. 58, 54 (2019). https://doi.org/10.1007/s00526-019-1500-9
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DOI: https://doi.org/10.1007/s00526-019-1500-9