Regularity up to the boundary for singularly perturbed fully nonlinear elliptic equations (Q899692)

Let \(\Omega\) be a bounded domain in \(\mathbb R^n\) with a \(C^{1,1}\) boundary \(\partial \Omega,\) and \(F:\Omega\times \mathbb R^n\times\mathrm{Sym}(n) \to\mathbb R\) be a nonlinear uniformly elliptic operator. The paper deals with viscosity solutions of the singular perturbation problem \((E_{\epsilon})\): \[ F(X,\nabla u^{\epsilon}, D^2u^{\epsilon})=\zeta_{\epsilon}(u^{\epsilon}) \text{ in }\Omega, \quad \text{and } u^{\epsilon} =\varphi\text{ on }\partial \Omega, \] where \(\zeta\in C_o^{\infty}([0,1]), \zeta\geq 0, \zeta_{\epsilon}(s)=\frac{1}{\epsilon}\zeta(\frac{s}{\epsilon}),\) \(\varphi\in C^{1,\gamma}(\bar{\Omega}),0<\gamma<1.\) The authors present sufficient conditions, under which there exists a constant \(C\), independent of \(\epsilon\), such that \(\| \nabla u_{\epsilon}\|_{L^{\infty }(\bar{\Omega})}\leq C,\) for viscosity solutions \(u_{\epsilon}\) to problem \(E_{\epsilon}\). As a consequence, they obtain the existence of \(u_o\in C^{0,1}(\bar{\Omega})\), solution of the limiting free boundary problem \[ F(X,\nabla u_o,D^2u_o)=0 \text{ in }\Omega\cap\{u_o>0\},\text{ in the viscosity sense}, \] (Theorem 2.8). They point out that their results can be adapted for the non-homogeneous case.