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

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 [

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 [

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 [

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

For the definition of the fractional Laplacian one can see [

Let

Throughout the paper, we shall assume that

Further, in this paper we shall use the primitive

In this case boundary value problem (

To obtain the existence of the weak solutions of the boundary value problem with fractional Laplacian (

(A1) regularity: the functions

(A2) growth: for

(A3) lower bound: there exist

(A4) convexity: the function

The principal eigenvalue

To derive the fractional Poincaré inequality of the form

Let

For the fractional Poincaré inequality with general measures involving nonlocal quantities on unbounded domain see paper by Mouhot et al. [

The fractional Sobolev inequality extending the above Poincaré inequality to

When

The fractional Sobolev space

Under assumptions (A1)-(A2) the functional of action defined in (

Define

In the following theorem we shall use the definition of the upper Painlevé-Kuratowski limit of the sets (cf. [

Now, we can formulate and prove the main result of this section.

Assume that

the integrand

the sequence of distributed parameters

Then

for any

there exists a ball

any sequence

Additionally, if the sets

Before going to the proof, it is worth noting that, if

Consider the following.

For

We begin by proving that the sequence

Consequently, for any

What we need to do now is to demonstrate that any sequence

Let us return to boundary value problem (

If

the integrand

the sequence of distributed parameters

then the sequence

Moreover, if the functional of action is strictly convex, then for

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

We impose the following conditions on

regularity: the functions

growth: there exists a constant

for a.e.

Suppose that

Let

Suppose that

the integrand

the sequence of distributed parameters

Then the sequence

As in the proof of Theorem

We now formulate the optimal control problem to which this section is dedicated. It transpires that the continuous dependence results from Section

Let

The function

for a.e.

There exists a function

for all

Now we prove a theorem on the existence of optimal processes to our optimal control problem (

If the functions

From (A5), (A6), and classical theorems on semicontinuity of integral functional (cf. [

Let

Due to the lower semicontinuity of

From the proof of Theorem

By a direct calculation, one can check that the quadratic functional

Since

The optimal control system (

Let

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

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 [

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 [

The author declares that there is no conflict of interests regarding the publication of this paper.