Hölder continuity of quasiminimizers with nonstandard growth

We show the Hölder continuity of quasiminimizers of the energy functionals ∫f(x,u,∇u)dx with nonstandard growth under the general structure conditions |z|p(x)−b(x)|y|r(x)−g(x)≤f(x,y,z)≤μ|z|p(x)+b(x)|y|r(x)+g(x).|z|p(x)−b(x)|y|r(x)−g(x)≤f(x,y,z)≤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)(A,B)-harmonic equations −divA(x,u,∇u)=B(x,u,∇u), are quasiminimizers of the variational integral of the above type and, thus, are Hölder continuous. Our results extend work by Chiadò Piat and Coscia (1997), Fan and Zhao (2000) and Giaquinta and Giusti (1984).