Boundary regularity estimates in H\"older spaces with variable exponent

TL;DR

This paper develops a new blow-up technique to derive local regularity estimates for solutions of elliptic equations in divergence form within H"older spaces with variable exponents, extending boundary regularity results and establishing a Schauder theory for derivatives.

Contribution

It introduces a general blow-up method for boundary regularity in variable exponent H"older spaces, providing sharp estimates and a Schauder theory for derivatives of any order.

Findings

Established boundary regularity estimates in variable exponent H"older spaces.

Extended Schauder theory to derivatives of solutions.

Provided sharp regularity and integrability conditions for coefficients and data.

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 H\"older 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 $k\in\mathbb{N}$. The strategy relies on the construction of a class of suitable regularizing problems and an approximation argument. The estimates we obtain are sharp with respect to the regularity or integrability conditions on variable coefficients, boundaries, boundary data and right hand sides respectively in H\"older and Lebesgue spaces, both with variable exponent