Regularity for multi-phase variational problems

TL;DR

This paper establishes $C^{1, u}$ regularity for local minimizers of a multi-phase energy functional involving multiple powers of the gradient, under specific conditions relating exponents and coefficient regularity.

Contribution

It proves regularity results for multi-phase variational problems with complex energy functionals, extending previous regularity theory to more general multi-phase cases.

Findings

Proves $C^{1, u}$ regularity for minimizers of multi-phase energy.

Identifies sharp conditions on exponents and coefficient regularity.

Extends regularity theory to complex multi-phase variational problems.

Abstract

We prove $C^{1,\nu}$ regularity for local minimizers of the \oh{multi-phase} energy: \begin{flalign*} w \mapsto \int_{\Omega}\snr{Dw}^{p}+a(x)\snr{Dw}^{q}+b(x)\snr{Dw}^{s} \ dx, \end{flalign*} under sharp assumptions relating the couples $(p,q)$ and $(p,s)$ to the H\"older exponents of the modulating coefficients $a(\cdot)$ and $b(\cdot)$, respectively.