Global regularity for a class of quasi-linear local and nonlocal elliptic equations on extension domains

Abstract

We consider the solvability of quasi-linear elliptic equations with either local or nonlocal Neumann, Robin, or Wentzell boundary conditions, defined (in the generalized sense) on a bounded W1,p-extension domain whose boundary is an upper d-set, for an appropriate d≥0. Then, we extend the fine regularity theory for weak solutions of the elliptic equations with the above boundary conditions, known for bounded Lipschitz domains, to bounded W1,p-extension domains whose boundaries are upper d-sets, by showing that such weak solutions are globally Hölder continuous. Consequently, we generalize substantially the class of bounded domains where weak solutions of boundary value problems of type Neumann, Robin, or Wentzell, may be uniformly continuous (up to the boundary).