Global Hölder regularity for the fractional $p$-Laplacian

Abstract

By virtue of barrier arguments we prove $C^{\alpha }$-regularity up to the boundary for the weak solutions of a non-local, non-linear problem driven by the fractional $p$-Laplacian operator. The equation is boundedly inhomogeneous and the boundary conditions are of Dirichlet type. We employ different methods according to the singular ($p<2$) of degenerate ($p>2$) case.