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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.

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It is well-known from the recent literature that nonlocal integral models are suitable to approximate integral functionals or partial differential equations. In the present work, a nonlocal optimal design model has been considered as approximation of the corresponding classical or local optimal control problem. The new model is driven by a nonlocal elliptic equation and the cost functional belongs to a broad class of nonlocal functional integrals. The purpose of this paper is to prove existence of optimal design for the new model. This work is complemented by showing that the limit of the nonlocal problem is the local one when the cost to minimize is the compliance functional (see [14]).

The aim of this paper is the study of a type of nonlocal parabolic equation. The formulation includes a convolution kernel k in the diffusion term and a design function h that plays the role of the diffusion coefficient. The main goal is twofold: On the one hand, the existence and uniqueness of nonlocal solution are deduced. Also, a comprehensive and rigorous procedure, which is based on the classical Galerkin-Fourier Method, is performed. As in the classical setting, the appropriate choice of the Gelfand triplet will guarantee the differentiation and therefore the operational technique for the study of the parabolic equation. On the other hand, the convergence of the nonlocal solution as the kernel k converges to a Dirac Delta is studied. The series expansion of the nonlocal solution allows us, in an easy way, to show its convergence to the solution of the corresponding local parabolic equation.