Boundary regularity of minimizers of double phase functionals

Abstract

In this paper we treat the functional of double phase with variable exponents:∫(|Du|gp(x)+a(x)|Du|gq(x))dx, where a(x) is a non-negative α-Hölder continuous function with α∈(0,1), p(x) and q(x) Hölder continuous functions with 1<p(x)≤q(x)<p(x)+α, and |ξ|g:=(δijgαβ(x)ξαiξβj)1/2 for a continuous positive definite matrix valued function g(⋅)=(gαβ(⋅)). We prove that the minimizer of the above functional with suitable Dirichlet boundary condition is Hölder continuous up to the boundary. When g is Hölder continuous, we see also that Du is locally Hölder continuous.