$L^{p}$ compactness criteria with an application to variational convergence of some nonlocal energy functionals

TL;DR

This paper establishes new nonlocal compactness criteria for $L^{p}$ vector fields, facilitating the analysis of variational convergence in nonlocal mechanics models with possibly asymmetric kernels.

Contribution

It introduces novel nonlocal $L^{p}$ compactness conditions for vector fields, extending previous scalar criteria and enabling convergence analysis of nonlocal energy minimizers.

Findings

Provided sufficient conditions for $L^{p}$ compactness in nonlocal vector fields.

Applied criteria to demonstrate convergence of minimizers in nonlocal energy problems.

Extended compactness results to non-symmetric interaction kernels.

Abstract

Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of $L^{p}$ vector fields defined on a domain $\Omega$ that is either a bounded domain in $\mathbb{R}^{d}$ or $\mathbb{R}^{d}$ itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The $L^{p}$ compactness criteria are utilized in demonstrating the convergence of minimizers of parametrized nonlocal energy functionals.