Theory, Methods, and Applications of Fractional CalculusView this Special Issue
Stability of Nonlinear Dirichlet BVPs Governed by Fractional Laplacian
We consider a class of partial differential equations with the fractional Laplacian and the homogeneous Dirichlet boundary data. Some sufficient condition under which the solutions of the equations considered depend continuously on parameters is stated. The application of the results to some optimal control problem is presented. The methods applied in the paper make use of the variational structure of the problem.
Consider the following fractional partial differential equation with some variable distributed parameters of the form where , is a bounded domain with a Lipschitzian boundary , and with . We shall assume that the distributed parameter belongs to the space for some suitably chosen and .
The equation under consideration is the generalization of the nonlinear Poisson equation involving the Brownian diffusion expressed by the local Laplace operator fully analyzed in [1–3]. We extend our considerations to cover also the case of the nonlocal, fractional Laplace operator being the infinitesimal generator of Lévy processes; see, for instance, [4–7], allowing, contrary to the continuous Brownian motion, for jumps. We prove the analogous stability results as for the Brownian motion with the Laplace operator involved obtained in [1–3].
The problems with the fractional Laplacian attracted in recent years a lot of attention as they naturally arise in various areas of applications to mention only [5–11] and references therein. They appear in probabilistic framework as well as in mathematical finance as infinitesimal generators of stable Lévy processes [4–7]. Moreover one can find the problems involving fractional Laplacian in mechanics and in elastostatics, for example, in Signorini obstacle problem originating from linear elasticity [12–14] as well as in fluid mechanics and in hydrodynamics—appearing in quasi-geostrophic fractional Navier-Stokes equation  and describing some porous media flows in the hydrodynamic model like in . The author considered also global solvability of Hammerstein equations derived from BVPs involving fractional Laplacian in recent paper .
In the theory of boundary value problems (BVPs) and its applications one considers, first of all, the problem of the existence of a solution, next the question of its stability, uniqueness, and smoothness, and finally the issue of asymptotic analysis. One can say that a given problem is well posed if the problem possesses at least one solution or, more generally, one obtains the set of solutions, which continuously changes along with the change of variable parameters of the system which we call stability. Otherwise we refer to the problem as to ill-posed one. The requirement of stability is necessary if the mathematical formulation is to describe observable natural phenomena, which by its very nature cannot possibly be conceived as rigidly fixed: even the mere process of measuring them involves small errors as was noted by Courant and Hilbert in . The theory of ill-posed problems pays most attention to the requirement of the stability of the boundary value problems.
In this paper we formulate some sufficient condition under which the boundary value problem considered here possesses at least one solution which continuously depends on distributed parameters. The problem of controllability of the related evolution equations driven by the anomalous diffusion governed by the fractional Laplacian was considered, for example, in .
The paper is organized as follows. In Section 2 we formulate the problem and list the assumptions appearing throughout the paper. In Section 3, using some variational methods we prove that boundary value problem (1)-(2) is stable with respect to the norm topology in the space of distributed parameters and the norm topology in the fractional Sobolev space of solutions . We can formulate the main result of Section 3 as follows: if in , then in where is the solution of the boundary value problem (1)-(2) with fixed , under suitable conditions imposed on . In the case when (1) is linear with respect to , we can relax the topology in the space . In short, in Section 4, we prove that in provided that weakly in . In the next section, we present a theorem on the existence of an optimal solution to some control problem with the integral cost functional. The proof of this theorem relies in essential way on the continuous dependence results. In the final part of the paper we give a short survey of the results related to the stability of the initial and boundary value problems for the second-order partial differential systems with parameters.
2. Formulation of the Problem, Introduction of the Fractional Laplacian, and Basic Assumptions
For the definition of the fractional Laplacian one can see [19–25]. In particular, we denote by for the system of the eigenfunctions and eigenvalues for the Laplace operator on with the homogeneous Dirichlet condition on . Moreover, by , let us denote the Sobolev space of functions defined on a bounded, smooth domain , , such that and , with the norm in with defined by the equivalent formulas see, for example, [20, 23] and for the last equality, see, for example, . The fractional Laplacian acts on as The fractional Sobolev spaces are also referred to as Gagliardo or Slobodeckij spaces. One can give yet another definition of as follows: with the norm For the definition of the fractional Laplacian operator involving singular integrals consistent with ours when is extended by outside , we refer the readers to , where one can find the following lemma.
Lemma 1. Let and let be the fractional Laplacian operator of the form where . Then for any from the Schwartz space of rapidly decaying functions in we have for all (cf. [23, Lemma 3.5]).
Throughout the paper, we shall assume that satisfies any condition which guarantees a compact embedding of into with where if , for example, may be Lipschitzian; that is, (cf.  for the definition of ). For it is possible to extend by outside and stay in the same space; see [23, Theorem 5.4].
Further, in this paper we shall use the primitive of the mapping , implying to be defined as the derivative with respect to variable of a function ; that is where a.e., , and .
In this case boundary value problem (1)-(2) may be written in the form suitable for variational analysis where , and . It is easily seen that (10)-(11) represent the Euler-Lagrange equation for the following functional of action: where and . It should be underlined that the solutions of Euler-Lagrange equation (10)-(11) are meant in the weak sense; that is, for any the following equality holds:
To obtain the existence of the weak solutions of the boundary value problem with fractional Laplacian (10)-(11) in the fractional Sobolev space and the continuous dependence of solutions on distributed parameters we shall impose on the following conditions.
(A1) regularity: the functions and are measurable with respect to for any and continuous with respect to for a.e. .
(A2) growth: for , there exists a constant such that for a.e. , and , where where ; for and any bounded set there exists a constant such that for a.e. , , , and some , where .
(A3) lower bound: there exist and functions , , such that for a.e. , , and , where and is the principal eigenvalue of the Laplace operator defined on the space .
(A4) convexity: the function is convex in .
Remark 2. The principal eigenvalue of Laplacian appears in the inequality
Indeed, , so infimum on the right hand side of the above inequality is greater or equal to . Moreover, the infimum is attained since is weakly lower semicontinuous, convex, and coercive as the norm in the reflexive space; for details, see [8, 27].
To derive the fractional Poincaré inequality of the form we apply the following theorem with .
Theorem 3. Let be a continuous, increasing, and polynomially bounded real-valued functional on , in particular, for . Then we have the following fractional order Poincaré inequality: compare [28, Theorem 2.8].
For the fractional Poincaré inequality with general measures involving nonlocal quantities on unbounded domain see paper by Mouhot et al. . In what follows we shall also use the following result.
Remark 4. The fractional Sobolev inequality extending the above Poincaré inequality to with, in general, non optimal constant , has the form
for any , , and every . When the best constant in the fractional Sobolev inequality will be denoted by . This constant is explicit and independent of the domain, its exact value is
where is the standard Euler Gamma function defined by , compare .
When we recover the fractional Poincaré inequality without an optimal constant in general.
Remark 5. The fractional Sobolev space is compactly embedded into for and ; see [23, Corollary 7.2].
Under assumptions (A1)-(A2) the functional of action defined in (12) is well defined and Fréchet differentiable and the derivative of acting on has the form
3. Continuous Dependence: Parameters Converging in the Strong Topology
Define to be some sequences of parameters distributed on . For , we denote by , the set of all possible minimizers of the functional ; that is Since each minimizer is a critical point of , that is, for any , it follows that is a weak solution of problem (10)-(11). Inversely, if is a weak solution of (10) satisfying (11), then provided the functional is convex (cf. [30, 31]). It is clear that, in general, the set does not have to be a singleton and hence boundary value problem (10)-(11) does not have to possess a unique solution.
In the following theorem we shall use the definition of the upper Painlevé-Kuratowski limit of the sets (cf. ). We say that a set is an upper limit of the sets , if any point is a cluster point of some sequence in such that for . By , we shall denote the upper Painlevé-Kuratowski limit of the sets , .
Now, we can formulate and prove the main result of this section.
Theorem 6. Assume that(1)the integrand satisfies conditions (A1)–(A3),(2)the sequence of distributed parameters tends to in with .
Then(a)for any , the set is a nonempty subset of , for ,(b)there exists a ball for some such that for ,(c)any sequence such that is relatively compact in and .
Additionally, if the sets are singletons, that is, , , then tends to in .
Before going to the proof, it is worth noting that, if denotes the set of all possible minimizers of the functional defined by (12), then assertion of Theorem 6 states that the set valued mapping is upper semicontinuous with respect to the strong topology of spaces and .
Proof. Consider the following.
Step 1. In the first step we prove assertions and of our theorem.
For , consider the functional By assumption of our theorem, for some . By (A3), we have and therefore the application of the fractional Poincaré inequality (18) gives with from (A3), where , are some constants independent of ; however, depending on and . The functional is weakly lower semicontinuous on as a sum involving the norm in , compare , and the integral term with satisfying the standard regularity and growth conditions (A1) and (A2), compare [33–36], as for suitably chosen in the embedding. Since, by (26), the functionals are coercive, we infer that the sets are nonempty and weakly closed. Moreover, by condition (A2), putting , we get the following estimates due to the boundedness of in where the constants and are independent of . Directly from inequalities (26), (27), and (28) we obtain that for some We have thus proved assertions and of our theorem.
Step 2. For , denote by the minimal value of the functional ; that is where . We shall observe that provided that in as .
We begin by proving that the sequence tends to uniformly on any ball of radius . By definition (24), we have Suppose that, on the contrary, the above integral does not tend to zero uniformly on . It means that there exists and a sequence such that . Passing to a subsequence if necessary, we can assume that tends to some weakly in . From the fractional Sobolev compact embedding theorem, see Remark 5, we deduce that, up to subsequence, tends to in . By assumption , we know that tends to in . Applying the Krasnoselskii theorem (cf. [37, 38]) the continuity of the operator follows and which together with condition (A2) implies as . Thus we have got a contradiction with the inequality . It means that tends to zero uniformly on and consequently converges to uniformly on provided that in .
Consequently, for any and chosen to be sufficiently large, we have Similarly, . We have thus proved equality (31).
Step 3. Finally, we shall prove assertion . Let be a sequence of minimizers; that is, . Since for , the sequence is weakly relatively compact in . We may assume after passing to a subsequence (still denoted by ) that tends to some in the weak topology of . Let us prove now that ; that is, is a minimizer of . Indeed, suppose that . The set is nonempty and therefore there exists some such that . Clearly, since is a minimizer of , and moreover we have Uniform convergence of to on leads to as by . Furthermore, the weak lower semicontinuity of and the weak convergence of to in lead to Thus we have got a contradiction with (31). Consequently, .
What we need to do now is to demonstrate that any sequence such that converges strongly to in . By (22), for , we have where The Hölder inequality and the growth condition (A2) allow us to write the following estimates: where and are some positive constants. Since converges to in and is bounded in we see that as and therefore the first integral tends to zero. Thus the weak convergence of the minimizers to implies the strong convergence of minimizers in , which completes the proof.
Let us return to boundary value problem (10)-(11) and, for , let us denote by the set of solutions to the problem which corresponds to the parameter . It is the well-known fact, see, for instance, [30, 31], that for the convex functional of action the set of minimizers coincides with the set of solutions . Hence for boundary value problem (10)-(11) we have the following corollary.
Corollary 7. If (1)the integrand satisfies conditions (A1)–(A4),(2)the sequence of distributed parameters tends to in with ,
then the sequence satisfies assertions (a)–(c) of Theorem 6 with , .
Moreover, if the functional of action is strictly convex, then for , problem (10)-(11) possesses a unique solution , and in .
4. Continuous Dependence: The Parameters Converging in the Weak Topology
To achieve stronger results which are useful in optimization theory, it is necessary to narrow down the class of equations under considerations. Namely, in this section, we shall assume that the integrand is linear with respect to the distributed parameter ; that is where , , and stands for a scalar product in . In this case, the boundary value problem (10)-(11) takes the form and the functional of action has the form where and with .
We impose the following conditions on , :regularity: the functions , , , and are measurable with respect to for any and continuous with respect to for a.e. ;growth: there exists a constant such that for a.e. , and where and .
Suppose that meets conditions (A3) and (A4). Obviously, assumptions and imply the function to satisfy (A1) and (A2). For this weaker form of the problem, the claim of the theorem on the existence and the continuous dependence can be strengthened. To draw the same conclusion this time, it suffices to assume the weak convergence of parameters.
Theorem 8. Suppose that (1)the integrand is of the form (39) and satisfies conditions , , and ,(2)the sequence of distributed parameters tends to in the weak topology of .
Then the sequence satisfies assertions (a)–(c) of Theorem 6.
Proof. As in the proof of Theorem 6, in the similar manner, we obtain assertions (a) and (b) of our theorem taken from Theorem 6. Let be an arbitrary sequence such that , for , where the sets are defined by formula (23). The sequence is bounded and therefore weakly relatively compact. Passing, if necessary, to a subsequence, we can assume that weakly in . We shall show that , but now we present different approach than in the proof of Theorem 6. By conditions and and formula (22), for and , we have It is easy to observe that since weakly in for any By the fractional Sobolev compact embedding theorem, after passing to a subsequence (still denoted by ) if necessary, we can assume that tends to in for . By , the superposition operator acting on to is continuous; that is, for any Let us consider the integral which can be represented as where Since , we see that where and moreover due to growth estimate we get the bound , since Hence, up to subsequence, we have that as . Similarly, by (2) and we get Thus for all . Taking into account equalities (45), (46), and (53), we infer that . It means that . To complete the proof, we shall verify that the sequence converges to in . By (22), we have Since for by the Hölder inequality and the growth condition , we get and in a similar manner using the Hölder inequality, and can be estimated by the terms involving , , for , and finally . The latter term due to the growth condition imposed on can be estimated as before from the above by for . Since converges to in and converges to in we have as . Consequently, as . Thus, the weak convergence of minimizers to implies the strong convergence of minimizers in . Therefore, the proof of our theorem is complete.
5. Existence of Optimal Solutions
We now formulate the optimal control problem to which this section is dedicated. It transpires that the continuous dependence results from Section 4 enable us to prove a theorem on the existence of optimal processes to some optimal control problem. Specifically, we shall consider control problem governed by boundary value problem (40)-(41) with the integral cost functional where is a given function. Here is the trajectory and is the distributed control where with and being a compact and convex subset of .
Let be the set of all admissible pairs; that is It should be noted that under assumptions of Theorem 8 the set of all admissible pairs is nonempty. In this section, our aim is to find a pair such that On the integrand we impose the following conditions.(A5)The function is measurable with respect to for all , , , continuous with respect to for a.e. , and convex with respect to for all , , and a.e. . Moreover there exists a constant such that for a.e. , all , , , and for some where .(A6)There exists a function and a constant such that for all , , , and a.e. .
Now we prove a theorem on the existence of optimal processes to our optimal control problem (60).
Proof. From (A5), (A6), and classical theorems on semicontinuity of integral functional (cf. [33–36]), we deduce that is lower semicontinuous with respect to the strong topology in the space and the weak topology of , since convergence of any sequence in implies the strong convergence of in with and the strong convergence of in .
Let be a minimizing sequence for optimal control problem (60); that is Since the set is compact and convex, the sequence is compact in the weak topology of . Passing to subsequence, if necessary, we can assume that tends to some weakly in . By assumption (A4) the set of the weak solutions of problem (40)-(41) coincides with the set of minimizers of the functional on the space . By Theorem 8, the sequence , or at least some of its subsequence, tends to in and the pair is an admissible pair for control problem (40)-(41).
Due to the lower semicontinuity of , we have provided tends to in and weakly in . Furthermore, by (63) and (64), we have Thus, . It means that the process is optimal for the problem (60).
Remark 10. From the proof of Theorem 9 one can see that it suffices to assume weaker assumption on controls than to be compact and convex, namely only boundedness of in .
Remark 11. By a direct calculation, one can check that the quadratic functional
is strictly convex for and convex for where is the principal eigenvalue of the operator defined on .
Since . Theorem 9 implies the following.
Corollary 12. The optimal control system (60) possesses at least one optimal process provided the functions of the form (39) satisfy , , and , the integrand meets assumptions (A5) and (A6) and the function is convex in for some , all and a.e. .
Example 13. Let be a cube of the form Note that and are eigenfunction and eigenvalue for on since . Similarly, hence, by (4), is the first eigenvalue for in this case. The equation is of the form for , sufficiently large and the cost is given by where , , and . Obviously, the functional of action for system (69) has the form It is easy to check that the functionals and satisfy all assumptions of Theorems 8 and 9. By Remark 11, is strictly convex. Thus, Theorem 9 implies that for any control there exists exactly one solution of (69) and the solution continuously depends on control . Moreover, by Corollary 12, we infer that there exists optimal control described by (69) with the cost functional given by (70).
In this paper we formulate some sufficient condition under which the boundary value problem considered in the paper possesses at least one solution which continuously depends on distributed parameters. We based our approach on the variational methods and we have investigated the stability problem or continuous dependence problem for the problem involving fractional Laplace operator in the fractional Sobolev space with distributed parameters from the space thus generalizing the stability results obtained for the boundary value problem with the Laplace operator in [1–3]. The stability results enable us to prove the theorem on the existence of optimal processes to some control problem with the integral cost functional.
The question of the existence of a solution for the boundary value problem of the Dirichlet type, periodic, homoclinic or heteroclinic type, and so forth was investigated in many papers and monographs. One can find a wide survey of results and research methods in monographs [30, 31, 38–41] and the references to be found therein. On the contrary to the initial value problem the literature on the stability problems for the boundary value problems governed by the differential equation of the elliptic type is not very vast. The stability of solutions of scalar second-order ordinary differential equation with two-point boundary conditions based on some direct methods related to the implicit function theorem was considered among others in the papers [42–46].
The question of the continuous dependence of solutions of the linear elliptic equations with the variable Dirichlet boundary data and parameters was investigated in the pioneering paper of Oleĭnik compare . In this work sufficient conditions for stability of the linear partial differential equation defined in the classical spaces of smooth solutions were formulated. Analogous results for the scalar linear partial differential equation with the Dirichlet boundary conditions defined on the Sobolev spaces were proved in the paper . The results on the stability of multidimensional nonlinear boundary value problems with variable parameters appeared in papers [49–51] where ordinary differential equations with two-point boundary conditions and variable functional parameters were investigated, and the stability conditions with respect to the strong and weak topology were proved. Similar results for partial differential equation with distributed parameters are given in papers [1–3, 52, 53].
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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