On a Differential Inclusion Involving Dirichlet–Laplace Operators of Fractional Orders

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 -Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-{\varDelta }$$\end{document} 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).


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 (x, u(x)), x ∈ Ω a.e., (1) where α i > 0, i = 0, . . . , k (k ∈ N ∪ {0}), 0 ≤ β 0 < β 1 < · · · < β k , Ω ⊂ R N is an open and bounded set, F : Ω × R → [0, ∞), ∂ u F denotes a subgradient of F with respect to u, [(−Δ ω )] γ denotes a weak fractional Laplace operator of order γ > 0 with zero Dirichlet boundary values on ∂Ω (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, ·) is differentiable on 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.

Preliminaries
In this section, we give some preliminary notions and results that will be used in the remaining part of this paper.

Weak Fractional Dirichlet-Laplace Operator
In a whole paper we assume that Ω ⊂ R N is an open and bounded set.
be the classical Dirichlet-Laplace operator.
Definition 1 (cf. [21]) We say that u : Ω → R has a weak (minus) Dirichlet-Laplacian if u ∈ H 1 0 (Ω, R) and there exists a function g ∈ L 2 (Ω, R) such that The function g will be called the weak Dirichlet-Laplacian and denoted by (−Δ) ω u.
The operator (−Δ) ω is called the weak Dirichlet-Laplace operator and the set is the self-adjoint extension of the operator T 0 and Remark 1 In [2, Section 8.2] (−Δ) ω is called the Laplace-Dirichlet operator (without "weak") and denoted by −Δ.

Remark 2
In paper [21], the author proved that if Ω is an open and bounded set of class C 1,1 or a convex polygon in R 2 , then the weak and strong Dirichlet-Laplace operators coincide.
where a j -s is such that (here E is the spectral measure given by (−Δ) ω and the convergence of the series is meant in L 2 (Ω, R)).
It is well known that the operator contains only proper values (λ j ) β , j ∈ N. Moreover, eigenspaces, corresponding to (λ j ) β -s and eigenspaces for [(−Δ) ω ], corresponding to λ j -s are the same.

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 : R n → R. We shall say that a vector y ∈ R n is a subgradient of The set of all subgradients of f at x is called a subdifferential and is denoted by ∂ f (x).
If ∂ f (x) = ∅ 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, 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 : R n → R. Any points x, y ∈ R n satisfy the inequality

Equality holds if and only if y ∈ ∂ f (x).
In conclusion of this section we give an another useful fact (cf. [22, Part II, Lemma 4.3.1]).
is a sequentially lower semicontinuous on L p (Ω, R d ).

Remark 3
The above lemma can be also proved in case of the function f defined on Ω × R n , where n = d.
Let us assume that F satisfies the following conditions: (F3): there exist a function v ∈ L 1 (Ω, R) and a constant c > 0 such that Let us consider the function f : R → R given by where c > 0 and v ∈ R are fixed. Then, it easy to check that In the next result we shall use the following Lemma 2 Let G : Ω × R → R be a Carathéodory function and the function H : Then H is measurable on Ω.
Proof From the fact that G(·, w) is continuous on R it follows that where 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 Ω.
Now, we formulate and prove the following result.  F(x, u)] , x ∈ Ω a.e., z ∈ R has the following properties: (a) F * (x, z) is finite for a.e. x ∈ Ω and all z ∈ R, (b) F * (·, z) is measurable on Ω for all z ∈ R and F * (x, ·) is continuous on R for a.e.
x ∈ Ω, (c) for a.e. x ∈ Ω and all z ∈ R where the constant c and the function v are from assumption (F3).
Proof proof of the property (a): Let us fix any z ∈ R and x ∈ S ⊂ Ω, where μ(Ω\S) = 0 (μ denotes the Lebesque measure on 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) ≥ R|u|. In particular, putting R = |z|, we get Consequently, Hence and from continuity of the function F(·, u) we obtain This means that F * is finite for a.e. x ∈ Ω and all z ∈ R.
Let us note that since all terms in the above integral are measurable on Ω (the first and third terms are even summable on Ω), therefore J is well defined.
In the proof of the main result we use the following fact.
Proof Let us consider the functional I 1 : L 2 (Ω, R) → R ∪ {∞} given by From Proposition 1, Lemma 1 and Remark 3 it follows that I 1 is a sequentially lower semicontinuous functional on L 2 (Ω, R).
. From the proved part of this Lemma it follows that the functional I is sequentially lower semicontinuous on D(w((−Δ) ω )). Since I is convex on D(w((−Δ) ω )) (F * (x, ·) is convex on R), therefore it is sequential weak lower semicontinuous on D(w((−Δ) ω )).
The proof is completed.
The next theorem plays a key role in the study of the existence of solutions to problem (1).
where λ 1 is the first eigenvalue of (−Δ) ω and c is a constant from the assumption (F3). Then there exists a minimizer u * ∈ D(w((−Δ) ω )) of the functional J . Moreover, if F(x, ·) is strictly convex on R for a.e. x ∈ Ω, then the minimizer is unique.
Proof Since the space D(w((−Δ) ω )) 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((−Δ) ω )).
Coercivity: Let (u l ) l∈N ⊂ D(w((−Δ) ω )) be any sequence such that From the proof of Theorem 3 it follows that J (u l ) ≥ 0. Moreover, using nonnegativity of F, conditions (16), (20), the Hölder inequality and the Poincaré inequality (10) we obtain This means that J is coercive.

Sequential weak lower semicontinuity of
is a sequence such that u l l→∞ u * weakly in D(w((−Δ) ω )) and let First, let us note that since F is a Carathéodory function, therefore (F(·, u l (·))) l∈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∈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((−Δ) ω )). 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((−Δ) ω )). 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((−Δ) ω )).
This contradicts the assumption that u * (v * ) is a minimizer of J on D(w((−Δ) ω )).
The proof is completed.

Remark 6
Let us note that only the order β k and the value α k impact the existence of a minimizer of J . This is a consequence of the fact that domains D(((−Δ) ω ) β k ) and D(w((−Δ) ω )) coincide.

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 Ω = (0, π) × (0, π) ⊂ R 2 and F : Ω × R → [0, ∞) is given by It is easy to check that Moreover, We check that 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 (−Δ) ω 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 is any open and bounded set, F : Ω × R → [0, ∞) is given by where 1 < p < 2 and a ∈ L ∞ (Ω, R + ).
We shall show that the above problem has the only trivial solution. Indeed, first let us note that |z| q , 1 p + 1 q = 1 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 ∈ Ω, therefore the functional J given by has a unique minimizer u * ∈ D(w((−Δ) ω )).
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((−Δ) ω )). 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 = a L ∞ 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((−Δ) ω ))) and the following, more general linear problem where f ∈ L 2 (Ω, 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((−Δ) ω )), therefore where δ J 1 (u * , h) denotes the first variation of J 1 at u * in the direction h. So, for any h ∈ D(w((−Δ) ω )).
Since a < √ M β k α k 2 , therefore the constant a is not the eigenvalue of the operator w((−Δ) ω ). Consequently, the kernel of the self-adjoint operator L:D(w((−Δ) ω )) → L 2 (Ω, R), defined as This means that u * is a unique solution of (27).
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