Second-order boundary estimates for solutions to a class of quasilinear elliptic equations

TL;DR

This paper establishes global second-order regularity for solutions to certain quasilinear elliptic equations, under boundary integrability conditions, with implications for the inverse gradient when the source term has a sign.

Contribution

It introduces new boundary regularity results for quasilinear elliptic equations, including cases with minimal boundary regularity under convexity assumptions.

Findings

Proves second-order regularity with boundary integrability conditions.

Shows inverse gradient integrability when the source term has a sign.

Achieves results without boundary regularity for convex domains.

Abstract

We prove global second-order regularity for a class of quasilinear elliptic equations, both with homogeneous Dirichlet and Neumann boundary conditions. A condition on the integrability of the second fundamental form on the boundary of the domain is required. As a consequence, with the additional assumption that the source term has a sign, we obtain integrability properties of the inverse of the gradient of the solution. Assuming convexity of the domain, no boundary regularity is required.