A Type of Nonlocal Elliptic Problem: Existence and Approximation Through a Galerkin--Fourier Method

Abstract

The aim of this paper is to study a type of nonlocal elliptic equation whose format includes a kernel k and a design function h. We analyze how this equation is connected with the classical elliptic equation that includes h as diffusive term. On one hand, the spectrum of the nonlocal operator that defines the nonlocal equation is studied. Existence and unicity of solutions for the nonlocal equation are proved. On the other hand, the convergence of these solutions to the solution of the classical elliptic equation as the kernel k converges to a Dirac Delta is analyzed. This work is performed by using an spectral theorem on the nonlocal operator and by applying some specific compactness results. The kernel k is assumed to be radial. Dirichlet boundary conditions are assumed for the classical problem, whereas for the nonlocal equation a nonlocal boundary Dirichlet constraint must be defined.

Nonlocal optimal design: A new perspective about the approximation of solutions in optimal design

“…The existence of a basis of (nonlocal) eigenfunctions and the usage of a Galerkin-Fourier Method are the tools that we are going to employ in Section 2 in order to ensure the existence of a solution of the state equation (see [3]). …”

“…Then (v n ) n , up to a subsequence, converges strongly in L 2 0 (Ω δ ) to some v (see [21,Theorem 7.1]). Next, by using the completeness of L 2 (Ω δ × Ω δ ) we easily infer the same property for X (see details in [3]). …”

“…The following result of approximation is the key point we have in mind when we consider the nonlocal variational problem (1.3) instead of the classical local one. [3].) If (u δ ) δ is the sequence of solutions of (P δ ), then (1) There exist u ∈ H 1 0 (Ω) and a subsequence of (u δ ) δ (which is going to be still denoted by (u δ ) δ ) such that…”

“…We carry out the study by analyzing the convergence of the eigenfunctions and eigenvalues: for each minimizing control sequence (h j ) j ⊂ H we perform an asymptotic study of the corresponding sequence of eigenspaces when j → ∞. The procedure we carry out in this analysis is similar to the one given in [3], but here, the limit eigenvalue problem we obtain is of nonlocal nature.…”

Existence and approximation of nonlocal optimal design problems driven by parabolic equations

“…All details for the derivation of these achievements can be found at papers. 30,31,38,42,43 The main result of this paper is the existence of nonlocal optimal design (Theorem 4). The proof in given in Section 3.…”