Boundary Regularity under Generalized Growth Conditions

Abstract

We study the Dirichlet $\varphi$-energy integral with Sobolev boundary values. The function $\varphi$ has generalized Orlicz growth. Special cases include variable exponent and double phase growths. We show that minimizers are regular at the boundary provided a weak capacity fatness condition is satisfied. This condition is satisfied for instance if the boundary is Lipschitz. The results are new even for Orlicz spaces.