# Existence and regularity of minimizers for nonlocal energy functionals

## TL;DR

This paper establishes the existence and regularity of minimizers for nonlocal energy functionals, extending classical results to nonlocal models like peridynamics and nonlocal diffusion without growth restrictions.

## Contribution

It introduces nonlocal Euler-Lagrange equations, proves existence of minimizers under convexity, and demonstrates regularity without growth assumptions, advancing nonlocal calculus of variations.

## Key findings

- Existence of minimizers under convexity assumptions
- Regularity results for solutions of nonlocal Euler-Lagrange equations
- No growth restrictions needed for existence and regularity

## Abstract

In this paper we consider minimizers for nonlocal energy functionals generalizing elastic energies that are connected with the theory of peridynamics \cite{Silling2000} or nonlocal diffusion models \cite{Rossi}. We derive nonlocal versions of the Euler-Lagrange equations under two sets of growth assumptions for the integrand. Existence of minimizers is shown for integrands with joint convexity (in the function and nonlocal gradient components). By using the convolution structure we show regularity of solutions for certain Euler-Lagrange equations. No growth assumptions are needed for the existence and regularity of minimizers results, in contrast with the classical theory.