Optimal gradient regularity to degenerate fully nonlinear elliptic models with oblique boundary condition

Abstract

This paper studies the optimal gradient regularity of viscosity solutions to oblique tangential derivative problems for a class of degenerate fully nonlinear second order uniformly elliptic equation with oblique boundary conditions in form $\left(\begin{array}{ccccc}G(D^{2}u,\nabla u,x)& =& f\left(x\right)& \text{in}& \Omega \\ \beta \cdot \nabla u+\gamma u& =& g\left(x\right)& \text{on}& \partial \Omega ,\end{array}\right)$ where $G$ is a second-order operator that is uniformly elliptic with degeneracy along the set of critical points of the a priori solution. Moreover, we obtain Schauder-type estimates for fully nonlinear elliptic equations with oblique boundary condition with constant coefficients, when $F$ is a quasi-convex operator. As an application of these results, we obtain the optimal regularity of the gradient for the degenerate problem described above.