Double phase quasiconvex functionals and their partial regularity theory

TL;DR

This paper proves partial regularity results for minimizers of degenerate, nonautonomous energy functionals with double phase growth, using advanced approximation techniques.

Contribution

It introduces new partial regularity results for vector-valued minimizers of double phase functionals under minimal assumptions.

Findings

Established partial Hölder regularity for gradients of minimizers

Developed approximation methods: $ ext{A}$-harmonic and $ ext{phi}$-harmonic approximations

Extended regularity theory to more general double phase functionals

Abstract

We consider degenerate nonautonomous energies $$ \int_\Omega f(x, Dv)\, dx, $$ for vector-valued functions $v \in W^{1,1}(\Omega, \mathbb{R}^N)$, where the integrand $f(x,P)$ satisfies growth and weak uniform quasiconvexity assumption associated with the double phase function $H(x,t)=t^p + a(x)t^q$. We establish partial H\"older regularity for the gradients of minimizers under suitable, and possibly minimal, regularity assumptions on $H$ and $f$. Our approach relies on two approximation results: $\mathcal{A}$-harmonic approximation and a variational version of the $\phi$-harmonic approximation.