H\"older continuity of quasiminimizers with nonstandard growth

TL;DR

This paper proves that quasiminimizers of certain energy functionals with nonstandard growth conditions are H"older continuous, extending previous results and applying to solutions of specific harmonic equations.

Contribution

It establishes H"older continuity for quasiminimizers under general nonstandard growth conditions, broadening the scope of regularity results in variational problems.

Findings

Quasiminimizers are H"older continuous under the given conditions.

Weak solutions to certain harmonic equations are quasiminimizers.

Results extend previous regularity theorems to more general growth conditions.

Abstract

We show the H\"older continuity of quasiminimizers of the energy functionals $\int f(x,u,\nabla u)\,dx$ with nonstandard growth under the general structure conditions $$ |z|^{p(x)} - b(x)|y|^{r(x)}-g(x) \leq f(x,y,z) \leq \mu|z|^{p(x)} + b(x)|y|^{r(x)} + g(x). $$ The result is illustrated by showing that weak solutions to a class of $(A,B)$-harmonic equations $$ -{\rm div} A(x,u,\nabla u) = B(x,u,\nabla u), $$ are quasiminimizers of the variational integral of the above type and, thus, are H\"older continuous. Our results extend works by Chiad\`o Piat-Coscia, Fan-Zhao and Giusti-Giaquinta.