Hölder continuity of quasiminimizers with nonstandard growth

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 $|z|^{p(x)} - b(x)|y|^{r(x)}-g(x) \leq f(x,y,z) \leq μ|z|^{p(x)} + b(x)|y|^{r(x)} + g(x).$ 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 Hölder continuous. Our results extend works by Chiadò Piat-Coscia, Fan-Zhao and Giusti-Giaquinta.

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