Boundary regularity of minimizers of double phase functionals

In this paper we treat the functional of double phase with variable exponents: o(vertical bar Du vertical bar(p(x))(g) + a(x)vertical bar Du vertical bar(q(x))(g)) dx, where a(x) is a non-negative a-Holder continuous function with alpha is an element of (0, 1), p(x) and q(x) Holder continuous functions with 1 < p(x) <= q(x) < p(x) + alpha, and vertical bar xi vertical bar(g) := (delta(ij) g(alpha beta)(x) xi(i)(alpha) xi(j)(beta))(1/2) for a continuous positive definite matrix valued function g(.) = (g(alpha beta)(.)). We prove that the minimizer of the above functional with suitable Dirichlet boundary condition is Holder continuous up to the boundary. When g is Holder continuous, we see also that Du is locally Holder continuous. (C) 2020 Elsevier Inc. All rights reserved.