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

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 $$F(X,\mathrm{\nabla }{u}^{\epsilon },D^2{u}^{\epsilon })={\zeta }_{\epsilon }(u^{\epsilon })in\mathrm{\Omega }\subset {\mathbb{R}}^n$$ where ${\zeta }_{\epsilon }$ behaves asymptotically as the Dirac measure ${\delta }_0$ as $\epsilon$ goes to zero. We shall establish global gradient bounds independent of the parameter $\epsilon$.