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arXiv · 1503.02599

H\"older continuity of quasiminimizers with nonstandard growth

Tomasz Adamowicz·Olli Toivanen
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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.

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Tomasz Adamowicz, Olli Toivanen. 2015-03-09. H\"older continuity of quasiminimizers with nonstandard growth. https://arxiv.org/abs/1503.02599

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