Hölder continuity of quasiminimizers with nonstandard growth

Abstract

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

26 pages