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

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 $$ F(X, \nabla u^{\varepsilon}, D^2 u^{\varepsilon}) = ζ_{\varepsilon}(u^{\varepsilon}) \quad \mbox{in} \quad Ω\subset \R^n $$ where $ζ_{\varepsilon}$ behaves asymptotically as the Dirac measure $δ_{0}$ as $\varepsilon$ goes to zero. We shall establish global gradient bounds independent of the parameter $\varepsilon$.