Abstract
In this paper, we investigate a nonlinear differential inclusion with Dirichlet boundary conditions containing a weak Laplace operator of fractional orders (defined via the spectral decomposition of the Laplace operator \(-{\varDelta }\) under Dirichlet boundary conditions). Using variational methods, we characterize solutions of such a problem. Our approach is based on tools from convex analysis (properties of a Legendre–Fenchel transform).
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1 Introduction
In recent years, a fractional Laplacian (more precisely, the fractional powers of the Laplacian) attracted attention of many scientists due to its applications in various areas. It appears including in probability (cf. [1, 6, 7, 13]), economics and finance (cf. [1, 14]), mechanics (cf. [5, 7]), optimal control theory (cf. [9, 20]), fluid mechanics and hydrodynamics (cf. [8, 10,11,12, 28,29,30]). There exist many definitions of such operators (e.g., Fourier transform [15, 17], hypersingular integral [15], Riesz potential operator [24], Bochner’s definition [27], spectral decomposition (cf. [4, 14, 18]).
In this paper, we are concerned with the study of solutions to the following differential inclusion
where \(\alpha _i>0\), \(i=0,\dots ,k\) (\(k\in {\mathbb {N}}\cup \{0\}\)), \(0\le \beta _0<\beta _1<\dots <\beta _k\), \({\varOmega }\subset {\mathbb {R}}^N\) is an open and bounded set, \(F:{\varOmega }\times {\mathbb {R}}\rightarrow [0,\infty )\), \(\partial _uF\) denotes a subgradient of F with respect to u, \([(-{\varDelta }_\omega )]^{\gamma }\) denotes a weak fractional Laplace operator of order \(\gamma >0\) with zero Dirichlet boundary values on \(\partial {\varOmega }\) (the term “weak” is explained in Sect. 2). This work is based on the spectral definition of the mentioned operator (we shall call it a weak fractional Dirichlet–Laplace operator). More precisely, the definition of the Dirichlet Laplacian comes from the functional calculus for unbounded self-adjoint operators in a Hilbert space [16, 21] and is based on the spectral integral representation theorem for such operators [23, 25].
Let us note that if \(F(t,\cdot )\) is differentiable on \({\mathbb {R}}\), then (1) reduces to the following boundary value problem
Motivated by the paper [3], we characterize solutions of the problem (1). They are minimizers of a some integral functional J related to (1) due to the Legendre–Fenchel transform of the function F and the Fenchel–Young inequality (cf. Theorem 3). Using a standard method of the calculus of variations, we prove the existence and uniqueness of minimizers of J on an appropriate space of functions, being the space of solutions of (1) (cf. Theorem 4). Finally, obtained results are illustrated by some examples. In particular, we show that if F is adequately chosen, then the minimizer of J is a solution of (2). To the best of our knowledge, the problem of a type (1) was not investigated by other authors.
2 Preliminaries
In this section, we give some preliminary notions and results that will be used in the remaining part of this paper.
2.1 Weak Fractional Dirichlet–Laplace Operator
In a whole paper we assume that \({\varOmega }\subset {\mathbb {R}}^N\) is an open and bounded set.
Let \(T_0=-{\varDelta }:C^\infty _c({\varOmega },{\mathbb {R}})\subset L^2({\varOmega },{\mathbb {R}})\rightarrow L^2({\varOmega },{\mathbb {R}})\) be the classical Dirichlet–Laplace operator.
Definition 1
(cf. [21]) We say that \(u:{\varOmega }\rightarrow {\mathbb {R}}\) has a weak (minus) Dirichlet–Laplacian if \(u\in H^1_0({\varOmega },{\mathbb {R}})\) and there exists a function \(g\in L^2({\varOmega },{\mathbb {R}})\) such that
for any \(v\in H^1_0({\varOmega },{\mathbb {R}})\). The function g will be called the weak Dirichlet–Laplacian and denoted by \((-{\varDelta })_\omega u\).
The operator \((-{\varDelta })_\omega \) is called the weak Dirichlet–Laplace operator and the set
is named the domain of the operator \((-{\varDelta })_\omega \).
Using the Friedrich’s extension procedure, we obtain (cf. [21, Theorem 3.1], [23])
Theorem 1
The operator
is the self-adjoint extension of the operator \(T_0\) and
where \(-{\varDelta }:H^1_0({\varOmega },{\mathbb {R}})\cap H^2({\varOmega },{\mathbb {R}})\subset L^2({\varOmega },{\mathbb {R}})\rightarrow L^2({\varOmega },{\mathbb {R}})\) is the (strong) Dirichlet–Laplace operator.
Remark 1
In [2, Section 8.2] \((-{\varDelta })_\omega \) is called the Laplace–Dirichlet operator (without “weak”) and denoted by \(-{\varDelta }\).
It is known (cf. [21]) that the spectrum \(\sigma ((-{\varDelta })_\omega )\) of \((-{\varDelta })_\omega \) contains only the eigenvalues of \((-{\varDelta })_\omega \) that can be written in a non-decreasing sequence, repeating each eigenvalue according to its multiplicity \(0<\lambda _1\le \lambda _2\le \dots \le \lambda _j\rightarrow \infty \). Moreover, a system \(\{e_j\}\) of eigenfunctions of the operator \((-{\varDelta })_\omega \), corresponding to \(\lambda _j\), is a hilbertian basis in \(L^2({\varOmega },{\mathbb {R}})\).
Remark 2
In paper [21], the author proved that if \({\varOmega }\) is an open and bounded set of class \(C^{1,1}\) or a convex polygon in \({\mathbb {R}}^2\), then the weak and strong Dirichlet–Laplace operators coincide.
Now, let \(\beta >0\). By the weak fractional Dirichlet–Laplace operator of order \(\beta \) we mean the operator \([(-{\varDelta })_{\omega }]^\beta :D([(-{\varDelta })_{\omega }]^\beta )\subset L^2({\varOmega },{\mathbb {R}})\rightarrow L^2({\varOmega },{\mathbb {R}})\) defined in the following way (cf. [21, Section 3]):
where
(here E is the spectral measure given by \((-{\varDelta })_\omega \) and the convergence of the series is meant in \(L^2({\varOmega },{\mathbb {R}})\)).
It is well known that the operator \([(-{\varDelta })_{\omega }]^\beta \) is self-adjoint, its spectrum \(\sigma ([(-{\varDelta })_{\omega }]^\beta )\) contains only proper values \((\lambda _j)^\beta \), \(j\in {\mathbb {N}}\). Moreover, eigenspaces, corresponding to \((\lambda _j)^\beta \)-s and eigenspaces for \([(-{\varDelta })_{\omega }]\), corresponding to \(\lambda _j\)-s are the same.
In the space \(D([(-{\varDelta })_{\omega }]^\beta )\), we define two scalar products:
which generate norms:
respectively. Norms (3) and (4) are equivalent due to the following Poincaré inequality in \(D([(-{\varDelta })_{\omega }]^\beta )\) (cf. [21, inequality (3.2)]):
where
(here \(\lambda _1>0\) is the first (the smallest) eigenvalue of the operator \((-{\varDelta })_\omega \)).
Now, let us consider the function
where \(0\le \beta _0<\beta _1<\dots <\beta _k\), \(k\in {\mathbb {N}}\cup \{0\}\), \(\alpha _i>0\), \(i=0,\dots ,k\). Then, we can define the operator \(w((-{\varDelta })_{\omega }):D(w((-{\varDelta })_{\omega }))\subset L^2({\varOmega },{\mathbb {R}})\rightarrow L^2({\varOmega },{\mathbb {R}})\) as follows:
The operator \(w((-{\varDelta })_{\omega })\) is self-adjoint and \(D(w((-{\varDelta })_{\omega }))=D(((-{\varDelta })_{\omega })^{\beta _k})\) (cf. [21]). Moreover, in \(D([(-{\varDelta })_{\omega }]^{\beta _k})\) one can introduce a new scalar product of the form
which generates a norm
Norms (4) and (8) (the norm (8) will be used in the main part of this article) are equivalent in \(D([(-{\varDelta })_{\omega }]^{\beta _k})\) (cf. [21, Lemma 3.6]) and
where \(C_1, C_2>0\) are constants from [21, Lemma 3.6]. Hence and from (5) we obtain the following Poincaré inequality in \(D([(-{\varDelta })_{\omega }]^{\beta _k})\):
Moreover, the space \(D(w((-{\varDelta })_{\omega }))=D(((-{\varDelta })_{\omega })^{\beta _k})\) with the scalar product (7) is complete (cf. [21, Lemma 3.6]).
2.2 Basic Facts from Convex Analysis
In this part we recall some basic definitions and facts concerning the convex analysis. More details can be found in [19].
Let \(f : {\mathbb {R}}^n\rightarrow {\mathbb {R}}\). We shall say that a vector \(y\in {\mathbb {R}}^n\) is a subgradient of f at \(x\in {\mathbb {R}}^n\) if
The set of all subgradients of f at x is called a subdifferential and is denoted by \(\partial f(x)\). If \(\partial f(x)\ne \emptyset \) then the function f is called subdifferentiable at x. If f is a convex function, then the subdifferential is a nonempty, convex and compact set. Moreover, if f is Gateaux differentiable at x, then \(\partial f(x)=\{\nabla f(x)\}\).
The function \(f^*:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\cup \{\infty \}\) defined by
is called the Legendre–Fenchel transform of the function f.
In the main part of this paper we use the following facts:
Theorem 2
(Fenchel–Young inequality) Let \(f : {\mathbb {R}}^n\rightarrow {\mathbb {R}}\). Any points \(x,y\in {\mathbb {R}}^n\) satisfy the inequality
Equality holds if and only if \(y\in \partial f(x)\).
In conclusion of this section we give an another useful fact (cf. [22, Part II, Lemma 4.3.1]).
Lemma 1
Let \({\varOmega }\subset {\mathbb {R}}^d\) be open, \(f:{\varOmega }\times {\mathbb {R}}^d\rightarrow {\mathbb {R}}\), with \(f(\cdot ,v)\) measurable for all \(v\in {\mathbb {R}}^d\), \(f(x,\cdot )\) continuous for almost all \(x\in {\varOmega }\), and
where \(a\in L^1({\varOmega },{\mathbb {R}})\), \(b\in {\mathbb {R}}\), \(p\ge 1\). Then, the functional \({\varPhi }:L^p({\varOmega },{\mathbb {R}}^d)\rightarrow {\mathbb {R}}\cup \{\infty \}\),
is a sequentially lower semicontinuous on \(L^p({\varOmega },{\mathbb {R}}^d)\).
Remark 3
The above lemma can be also proved in case of the function f defined on \({\varOmega }\times {\mathbb {R}}^n\), where \(n\ne d\).
3 Differential Inclusion with Fractional Dirichlet–Laplace Operators
In this section we prove the main result of this paper, namely a theorem on the existence of minimizers of a some functional associated with the problem (1).
By a solution of inclusion (1) we mean a function \(u\in D(w((-{\varDelta })_\omega ))\) satisfying (1) a.e. on \({\varOmega }\).
Let us assume that F satisfies the following conditions:
- (F1)::
-
\(F(\cdot ,{u})\) is measurable on \({\varOmega }\) for all \(u\in {\mathbb {R}}\) and \(F(x,\cdot )\) is convex on \({\mathbb {R}}\) for a.e. \(x\in {\varOmega }\).
- (F2)::
-
\(F(x,\cdot )\) is coercive on \({\mathbb {R}}\) for a.e. \(x\in {\varOmega }\), i.e.
$$\begin{aligned} \underset{|u|\rightarrow \infty }{\lim }\tfrac{F(x,u)}{|u|}=\infty ,\quad x\in {\varOmega }\,\,\hbox {a.e.} \end{aligned}$$(12) - (F3)::
-
there exist a function \(v\in L^1({\varOmega },{\mathbb {R}})\) and a constant \(c>0\) such that
$$\begin{aligned} F(x,u)\le v(x)+c|u|^2,\quad x\in {\varOmega }\,\,\hbox {a.e.}, u\in {\mathbb {R}}. \end{aligned}$$(13)
Remark 4
Of course, since \(F(x,\cdot )\) is convex, therefore it is continuous on \({\mathbb {R}}\) for a.e. \(x\in {\varOmega }\) (cf. [26, Corollary 10.1.1]).
Let us consider the function \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) given by
where \(c>0\) and \(v\in {\mathbb {R}}\) are fixed. Then, it easy to check that
In the next result we shall use the following
Lemma 2
Let \(G:{\varOmega }\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) be a Carathéodory function and the function \(H:{\varOmega }\rightarrow {\mathbb {R}}\cup \{\infty \}\) is defined by
Then H is measurable on \({\varOmega }\).
Proof
From the fact that \(G(\cdot ,w)\) is continuous on \({\mathbb {R}}\) it follows that
where \({\mathbb {Q}}\) denotes the set of rational numbers. Since the supremum of the countable set of measurable functions is a measurable set, therefore H is measurable on \({\varOmega }\).\(\square \) \(\square \)
Now, we formulate and prove the following result.
Proposition 1
Let us assume that conditions (F1)–(F3) are satisfied. Then, the Legendre–Fenchel transform of the function F with respect to u \(F^*:{\varOmega }\times {\mathbb {R}}\rightarrow (-\infty ,\infty ]\) given by
has the following properties:
-
(a)
\(F^*(x,z)\) is finite for a.e. \(x\in {\varOmega }\) and all \(z\in {\mathbb {R}}\),
-
(b)
\(F^*(\cdot ,z)\) is measurable on \({\varOmega }\) for all \(z\in {\mathbb {R}}\) and \(F^*(x,\cdot )\) is continuous on \({\mathbb {R}}\) for a.e. \(x\in {\varOmega }\),
-
(c)
for a.e. \(x\in {\varOmega }\) and all \(z\in {\mathbb {R}}\)
$$\begin{aligned} F^*(x,z)\ge -v(x)+\frac{1}{4c}|z|^2, \end{aligned}$$(16)where the constant c and the function v are from assumption (F3).
Proof
proof of the property (a): Let us fix any \(z\in {\mathbb {R}}\) and \(x\in S\subset {\varOmega }\), where \(\mu ({\varOmega }{\setminus } S)=0\) (\(\mu \) denotes the Lebesque measure on \({\mathbb {R}}^N\)). From (F2) it follows that for any \(R>0\) there exists \(r>0\) such that for \(|u|>r\) we have \(F(x,u)\ge R|u|\). In particular, putting \(R=|z|\), we get
Consequently,
Hence and from continuity of the function \(F(\cdot ,u)\) we obtain
This means that \(F^*\) is finite for a.e. \(x\in {\varOmega }\) and all \(z\in {\mathbb {R}}\).
proof of the property (b): Measurability of \(F^*(\cdot ,z)\) on \({\varOmega }\) for any fixed \(z\in {\mathbb {R}}\) follows from Lemma 2. From the definition of the Legendre–Fenchel transform it follows that \(F^*(x,\cdot )\) is convex on \({\mathbb {R}}\) for a.e. \(x\in {\varOmega }\). Thus and from the proved condition (a) of this proposition we obtain continuity of \(F^*\) with respect to z for a.e. \(x\in {\varOmega }\) (cf. [26, Corollary 10.1.1]).
proof of the property (c): The condition (16) follows from assumption (F3), property (11) and equality (15). \(\square \)
Now, let us consider the functional \(J:D(w((-{\varDelta })_\omega ))\longrightarrow {\mathbb {R}}\cup \{\infty \}\) given by:
Let us note that since all terms in the above integral are measurable on \({\varOmega }\) (the first and third terms are even summable on \({\varOmega }\)), therefore J is well defined.
In the proof of the main result we use the following fact.
Lemma 3
If assumptions (F1)–(F3) are satisfied, then the functional
\(I:D(w((-{\varDelta })_{\omega }))\rightarrow {\mathbb {R}}\cup \{\infty \}\) given by
is sequentially weakly lower semicontinuous on \(D(w((-{\varDelta })_{\omega }))\).
Proof
Let us consider the functional \(I_1:L^2({\varOmega },{\mathbb {R}})\rightarrow {\mathbb {R}}\cup \{\infty \}\) given by
From Proposition 1, Lemma 1 and Remark 3 it follows that \(I_1\) is a sequentially lower semicontinuous functional on \(L^2({\varOmega },{\mathbb {R}})\).
Let \((u_n)_{n\in {\mathbb {N}}}\subset D(w((-{\varDelta })_{\omega }))\) be a sequence convergent to a some element u in the space \(D(w((-{\varDelta })_{\omega }))\). Then, the sequence \((w((-{\varDelta })\omega )u_n)_{n\in {\mathbb {N}}}\subset L^2({\varOmega },{\mathbb {R}})\) is convergent to \(w((-{\varDelta })\omega )u\) in \(L^2({\varOmega },{\mathbb {R}})\). From the proved part of this Lemma it follows that the functional I is sequentially lower semicontinuous on \(D(w((-{\varDelta })_{\omega }))\). Since I is convex on \(D(w((-{\varDelta })_{\omega }))\) (\(F^*(x,\cdot )\) is convex on \({\mathbb {R}}\)), therefore it is sequential weak lower semicontinuous on \(D(w((-{\varDelta })_{\omega }))\).
The proof is completed. \(\square \)
The next theorem plays a key role in the study of the existence of solutions to problem (1).
Theorem 3
Assume that conditions (F1)–(F3) are satisfied. A function \(u_*\in D(w((-{\varDelta })_{\omega }))\) is a solution of the inclusion (1) if and only if \(u_*\) is a minimizer of J on \(D(w((-{\varDelta })_{\omega }))\) and \(J(u_*)=0\).
Proof
Let \(u_*\in D(w((-{\varDelta })_{\omega }))\) be a solution of the problem (1). From Theorem 2 it follows that
Consequently, \(J(u_*)=0\). On the other hand, for every \(u\in D(w((-{\varDelta })_{\omega }))\), using once again Theorem 2, we have
This means that \(J(u)\ge 0\) for every \(u\in D(w((-{\varDelta })_{\omega }))\), so \(u_*\) is a minimizer of J on \(D(w((-{\varDelta })_{\omega }))\).
Now, we assume that \(u_*\) is a minimizer of J on \(D(w((-{\varDelta })_{\omega }))\) and \(J(u_*)=0\). This means in particular that
On the other hand, from Theorem 2 it follows that
Consequently,
The second part of Theorem 2 guaranties that \(u_*\in D(w((-{\varDelta })_{\omega }))\) is a solution of the inclusion (1).
The proof is completed. \(\square \)
Remark 5
From the above theorem it follows that if \(u_*\in D(w((-{\varDelta })_{\omega }))\) is a solution of the problem (1) then it is a minimizer of J. Consequently, only minimizers of J can be solutions of (1) (they are in effect solutions of (1) if additionally \(J(u_*)=0\), where \(u_*\) denotes a minimizer of J).
Now, we formulate and prove the main result of this paper, namely a theorem on the existence of a minimizer of the functional J on \(D(w((-{\varDelta })_{\omega }))\). We have
Theorem 4
Assume that conditions (F1)–(F3) are satisfied and
where \(\lambda _1\) is the first eigenvalue of \((-{\varDelta })_\omega \) and c is a constant from the assumption (F3). Then there exists a minimizer \(u_*\in D(w((-{\varDelta })_{\omega }))\) of the functional J. Moreover, if \(F(x,\cdot )\) is strictly convex on \({\mathbb {R}}\) for a.e. \(x\in {\varOmega }\), then the minimizer is unique.
Proof
Since the space \(D(w((-{\varDelta })_{\omega }))\) is reflexive (as the Hilbert space), therefore it is sufficient to show that the functional J is coercive and sequentially weakly lower semicontinuous on \(D(w((-{\varDelta })_{\omega }))\).
Coercivity: Let \((u_l)_{l\in {\mathbb {N}}}\subset D(w((-{\varDelta })_{\omega }))\) be any sequence such that
From the proof of Theorem 3 it follows that \(J(u_l)\ge 0\). Moreover, using nonnegativity of F, conditions (16), (20), the Hölder inequality and the Poincaré inequality (10) we obtain
where \(C=-\int \limits _{\varOmega }v(x)\hbox {d}x\). This means that J is coercive.
Sequential weak lower semicontinuity of J : Assume that \((u_l)_{l\in {\mathbb {N}}}\subset D(w((-{\varDelta })_{\omega }))\) is a sequence such that
and let
where
First, let us note that since F is a Carathéodory function, therefore \((F(\cdot ,u_{l}(\cdot )))_{l\in {\mathbb {N}}}\) is the sequence of measurable functions. Moreover all terms of this sequence are nonnegative. Consequently, using Fatou’s Lemma, we conclude
[21, Proposition 3.10] implies
So, there exists a subsequent \((u_{l_j})_{j\in {\mathbb {N}}}\) such that
Hence and from the fact that F is continuous with respect to the second variable we get
Supposing contrary and repeating the above argumentation we assert that
Consequently, the inequality (21) can be written as
This means that \(J_1\) is sequentially weakly lower semicontinuous on \(D(w((-{\varDelta })_{\omega }))\). The functional \(J_2\) has also such a property due to Lemma 3.
Now, we show that \(J_3\) is sequentially weakly continuous on \(D(w((-{\varDelta })_{\omega }))\). Indeed, using once again [21, Proposition 3.10] we assert that
This fact and convergence (22) imply
Finally, we conclude that J is sequentially weakly lower semicontinuous on \(D(w((-{\varDelta })_{\omega }))\).
Uniqueness of a solution: It is clear that since F and \(F^*\) are convex with respect to the second variable, therefore J is convex on \(D(w((-{\varDelta })_{\omega }))\). Let us suppose that \(u_*\) and \(v_*\) are two different minimizers of J on \(D(w((-{\varDelta })_{\omega }))\) and assume that \(F(x,\cdot )\) is strictly convex on \({\mathbb {R}}\) for a.e. \(x\in {\varOmega }\). This implies that \(J_1\), so also J, are strictly convex on \(D(w((-{\varDelta })_{\omega }))\). This means in particular that
This contradicts the assumption that \(u_*\) (\(v_*\)) is a minimizer of J on \(D(w((-{\varDelta })_{\omega }))\).
The proof is completed. \(\square \)
Remark 6
Let us note that only the order \(\beta _k\) and the value \(\alpha _k\) impact the existence of a minimizer of J. This is a consequence of the fact that domains \(D(((-{\varDelta })_\omega )^{\beta _k})\) and \(D(w((-{\varDelta })_{\omega }))\) coincide.
4 Illustrative Examples
In this section we present two simple theoretical examples which illustrate results obtained in Sect. 3.
Example 1
Let us consider the following problem:
where \({\varOmega }=(0,\pi )\times (0,\pi )\subset {\mathbb {R}}^2\) and \(F:{\varOmega }\times {\mathbb {R}}\rightarrow [0,\infty )\) is given by
It is easy to check that
Moreover,
We check that
and
Consequently,
Of course assumptions (F1)–(F3) are satisfied. In particular,
so the condition (13) holds with \(c=1\). Moreover, it is well known (cf. [2, Proposition 8.5.3]) that the first eigenvalue of \((-{\varDelta })_\omega \) equals to 2, so the assumption (20) is also satisfied. Consequently, using Theorem 4, we assert that there exists a minimizer (not necessarily unique) of the functional J given by (17) which can be the only possible solution of the problem (23) (cf. Remark 5).
Example 2
Let us consider the following boundary value problem:
where \(\alpha _i>0\), \(i=0,\dots ,k\) (\(k\in {\mathbb {N}}\cup \{0\}\)), \(0\le \beta _0<\beta _1<\dots <\beta _k\), \({\varOmega }\subset {\mathbb {R}}^N\), is any open and bounded set, \(F:{\varOmega }\times {\mathbb {R}}\rightarrow [0,\infty )\) is given by
where \(1<p<2\) and \(a\in L^\infty ({\varOmega },{\mathbb {R}}_+)\).
We shall show that the above problem has the only trivial solution. Indeed, first let us note that
and the partial derivative of F with respect to u is given by
Moreover, assumptions (F1)–(F3) hold. In particular, for any \(c>0\) there exists a sufficiently large constant \(R>0\) (dependent on the constants c, p and the function a) such that
so, the condition (13) is satisfied for any \(c>0\).
Consequently, since all assumptions of Theorem 4 are satisfied and F is strictly convex with respect to u for a.e. \(x\in {\varOmega }\), therefore the functional J given by
has a unique minimizer \(u_*\in D(w((-{\varDelta })_{\omega }))\).
On the other hand it is clear that \(u_*=0\) is a solution of the problem (24), so it must minimize J on \(D(w((-{\varDelta })_{\omega }))\). This means that \(u_*=0\) is the only solution of (24).
Remark 7
If \(p=2\) in the above example then the condition (13) holds for \(c=\frac{\Vert a\Vert _{L^\infty }}{2}\). Consequently, the linear problem of the form
has the only trivial solution provided that
Here it is worth to note that replacing the functional J given by (25) (for \(p=q=2\)) with the following one
we can prove (similarly as in Sect. 3) counterparts of Theorems 3 and 4 for the functional \(J_1\) (in particular, under assumption (26) one can obtain the existence of a unique minimizer of \(J_1\) on \(D(w((-{\varDelta })_{\omega }))\)) and the following, more general linear problem
where \(f\in L^2({\varOmega },{\mathbb {R}})\). Additionally, if a is a constant function, then the problem (27) has a unique solution. Indeed, since \(J_1\) has a unique minimizer \(u_*\) on \(D(w((-{\varDelta })_{\omega }))\), therefore
where \(\delta J_1(u_*,h)\) denotes the first variation of \(J_1\) at \(u_*\) in the direction h.
So,
for any \(h\in D(w((-{\varDelta })_{\omega }))\).
Since \(a<\frac{\sqrt{M_{\beta _k}}\alpha _k}{2}\), therefore the constant a is not the eigenvalue of the operator \(w((-{\varDelta })_{\omega })\). Consequently, the kernel of the self-adjoint operator \(L{:}D(w((-{\varDelta })_{\omega }))\rightarrow L^2({\varOmega },{\mathbb {R}})\), defined as
is trivial, so the image of L satisfies the equality \(R(L)=L^2({\varOmega },{\mathbb {R}})\). Hence and from (29) we conclude
This means that \(u_*\) is a unique solution of (27).
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Communicated by Rosihan M. Ali.
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Kamocki, R. On a Differential Inclusion Involving Dirichlet–Laplace Operators of Fractional Orders. Bull. Malays. Math. Sci. Soc. 43, 4089–4106 (2020). https://doi.org/10.1007/s40840-020-00910-1
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DOI: https://doi.org/10.1007/s40840-020-00910-1
Keywords
- Fractional Dirichlet–Laplace operator
- Differential inclusion
- Legendre–Fenchel transform
- Variational methods