${\cal C}^{1,\beta}$ regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations

TL;DR

This paper establishes Holder regularity of the gradient for solutions to degenerate elliptic equations with Dirichlet boundary conditions, extending previous results to boundary cases and equations with lower order terms.

Contribution

It extends the regularity results to boundary cases and equations with lower order terms, introducing new tools and ideas for a priori estimates.

Findings

Gradient regularity up to the boundary

Holder estimates with boundary conditions

Extension of regularity to equations with lower order terms

Abstract

In this paper we prove Holder regularity of the gradient for solutions of Dirichlet problem associate to degenerate elliptic equations, extending the recent result of Imbert and Silvestre. Indeed we obtain regularity up to the boundary and when the equation has lower order terms.The proof follows their scheme but requires new tools and new ideas. In particular we give some a priori Lipschitz and H\"older estimates in the presence of boundary condition on one part of the boundary.