On non-locality in the Calculus of Variations

TL;DR

This paper explores the fundamental aspects of non-local variational principles, focusing on scalar, one-dimensional cases, revealing unexpected behaviors and establishing a foundation for analyzing more complex, higher-dimensional problems.

Contribution

It introduces a general perspective on non-local variational principles, highlighting key differences from classical problems, such as the use of fractional derivatives and integral equations for optimality.

Findings

Existence of minimizers without convexity of integrands

Optimality characterized by integral rather than differential equations

Different non-local derivatives like fractional Sobolev spaces

Abstract

Non-locality is being intensively studied in various PDE-contexts and in variational problems. The numerical approximation also looks challenging, as well as the application of these models to Continuum Mechanics and Image Analysis, among other areas. Even though there is a growing body of deep and fundamental knowledge about non-locality, for variational principles there are still very basic questions that have not been addressed so far. Taking some of these as a motivation, we describe a general perspective on distinct classes of non-local variational principles setting a program for the analysis of this kind of problems. We start such program with the simplest problem possible: that of scalar, uni-dimensional cases, under a particular class of non-locality. Even in this simple initial scenario, one finds quite unexpected facts to the point that our intuition about local, classic problems can no longer guide us for these new problems. There are three main issues worth highlighting, in the particular situation treated: $\bullet$ natural underlying spaces involve different non-local types of derivatives as, for instance, fractional Sobolev spaces; $\bullet$ no convexity of integrands is required for existence of minimizers; $\bullet$ optimality is formulated in terms of quite special integral equations rather than differential equations. We are thus able to provide some specific answers to the initial questions that motivated our investigation. In subsequent papers, we will move on to consider the higher dimensional situation driven by the possibility that no convexity or quasiconvexity might be involved in weak lower semicontinuity in a full vector, higher dimensional situation.