Compactness results for a Dirichlet energy of nonlocal gradient with applications

TL;DR

This paper establishes compactness and convergence results for nonlocal gradient function spaces, with applications to diffusion problems and optimal control, advancing the mathematical understanding of nonlocal operators.

Contribution

It provides new compactness theorems for nonlocal gradient spaces, demonstrating their convergence to local spaces and applying these results to diffusion and control problems.

Findings

Proved compactness of nonlocal gradient spaces under certain kernel conditions.

Showed convergence of nonlocal spaces to local function spaces.

Demonstrated applications to diffusion equations and optimal control.

Abstract

We prove two compactness results for function spaces with finite Dirichlet energy of half-space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic compact embedding of the associated nonlocal function spaces into the class of square-integrable functions. Moreover, we will demonstrate that the sequence of nonlocal function spaces converges in an appropriate sense to a limiting function space. As an application, we prove uniform Poincar\'e-type inequalities for sequence of half-space gradient operators. We also apply the compactness result to demonstrate the convergence of appropriately parameterized nonlocal heterogeneous anisotropic diffusion problems. We will construct asymptotically compatible schemes for these type of problems. Another application concerns the convergence and robust discretization of a nonlocal optimal control problem.