Regularity up to the boundary for singularly perturbed fully nonlinear elliptic equations

TL;DR

This paper investigates boundary regularity for singularly perturbed fully nonlinear elliptic equations, establishing uniform gradient bounds as the perturbation parameter approaches zero, which advances understanding of solution behavior near boundaries.

Contribution

It provides the first global gradient bounds up to the boundary for a class of singularly perturbed fully nonlinear elliptic problems, independent of the perturbation parameter.

Findings

Established boundary regularity results for singularly perturbed equations.

Proved uniform gradient bounds independent of perturbation parameter.

Enhanced understanding of solution behavior near boundaries in nonlinear elliptic problems.

Abstract

In this article we are interested in studying regularity up to the boundary for one-phase singularly perturbed fully nonlinear elliptic problems, associated to high energy activation potentials, namely $$ F(X, \nabla u^{\varepsilon}, D^2 u^{\varepsilon}) = \zeta_{\varepsilon}(u^{\varepsilon}) \quad \mbox{in} \quad \Omega \subset \R^n $$ where $\zeta_{\varepsilon}$ behaves asymptotically as the Dirac measure $\delta_{0}$ as $\varepsilon$ goes to zero. We shall establish global gradient bounds independent of the parameter $\varepsilon$.