Nonconvex minimization related to quadratic double-well energy - approximation by convex problems

TL;DR

This paper investigates a nonconvex double-well energy functional, representing it as a minimum of two quadratic phase energies, and develops a relaxation formula using convex approximations and weak limit analysis.

Contribution

It introduces a novel relaxation formula for a nonconvex double-well energy functional via convex problem approximations and weak limit techniques.

Findings

Derived a relaxation formula for the nonconvex energy

Constructed convex approximations to analyze minimizers

Connected weak limits of solutions and phase indicators

Abstract

A double-well energy expressed as a minimum of two quadratic functions, called phase energies, is studied with taking into account the minimization of the corresponding integral functional. Such integral, as being not sequentially weakly lower semicontinuous, does not admit classical minimizers. To derive the relaxation formula for the infimum, the minimizing sequence consisting of solutions of convex problems appropriately approximating the original nonconvex one is constructed. The weak limit of this sequence together with the weak limit of the sequence of solutions of the corresponding dual problems and the weak limits of the characteristic functions related to the phase energies are involved in the relaxation formula.