Boundary regularity estimates in Hölder spaces with variable exponent

Abstract

We present a general blow-up technique to obtain local regularity estimates for solutions, and their derivatives, of second order elliptic equations in divergence form in Hölder spaces with variable exponent. The procedure allows to extend the estimates up to a portion of the boundary where Dirichlet or Neumann boundary conditions are prescribed and produces a Schauder theory for partial derivatives of solutions of any order kin {mathbb {N}}. The strategy relies on the construction of a class of suitable regularizing problems and an approximation argument. The given data of the problem are taken in Hölder and Lebesgue spaces, both with variable exponent.