Superlinear free-discontinuity models: relaxation and phase field approximation

TL;DR

This paper advances the calculus of variations by developing relaxation and phase field approximation methods for superlinear free-discontinuity models with complex jump set structures.

Contribution

It introduces new analytical techniques for lower semicontinuity and relaxation of superlinear free-discontinuity energies on SBV functions, and proposes phase field approximations.

Findings

Established general lower semicontinuity results.

Developed relaxation results for superlinear energies.

Proposed phase field approximation methods.

Abstract

In this paper we develop the Direct Method in the Calculus of Variations for free-discontinuity energies whose bulk and surface densities exhibit superlinear growth, respectively for large gradients and small jump amplitudes. A distinctive feature of this kind of models is that the functionals are defined on $SBV$ functions whose jump sets may have infinite measure. Establishing general lower semicontinuity and relaxation results in this setting requires new analytical techniques. In addition, we propose a variational approximation of certain superlinear energies via phase field models.