Quasiconvexity in the fractional calculus of variations: Characterization of lower semicontinuity and relaxation

TL;DR

This paper investigates the role of quasiconvexity in fractional calculus of variations, establishing conditions for lower semicontinuity and providing relaxation formulas for nonlocal variational problems involving Riesz fractional gradients.

Contribution

It characterizes weak lower semicontinuity in fractional variational problems and identifies quasiconvexity as the key notion, extending classical calculus of variations results to the fractional setting.

Findings

Quasiconvexity is the natural notion for fractional variational problems.

A relaxation formula is derived for non-quasiconvex functionals, involving partial quasiconvexification.

Relaxation induces a structural change, transforming the integrand from homogeneous to inhomogeneous.

Abstract

Based on recent developments in the theory of fractional Sobolev spaces, an interesting new class of nonlocal variational problems has emerged in the literature. These problems, which are the focus of this work, involve integral functionals that depend on Riesz fractional gradients instead of ordinary gradients and are considered subject to a complementary-value condition. With the goal of establishing a comprehensive existence theory, we provide a full characterization for the weak lower semicontinuity of these functionals under suitable growth assumptions on the integrands. In doing so, we surprisingly identify quasiconvexity, which is intrinsic to the standard vectorial calculus of variations, as the natural notion also in the fractional setting. In the absence of quasiconvexity, we determine a representation formula for the corresponding relaxed functionals, obtained via partial quasiconvexification outside the region where complementary values are prescribed. Thus, in contrast to classical results, the relaxation process induces a structural change in the functional, turning the integrand from a homogeneous into an inhomogeneous one. Our proofs rely crucially on an inherent relation between classical and fractional gradients, which we extend to Sobolev spaces, enabling us to transition between the two settings.