Boundary $C^{2,\alpha }$ Regularity for the Oblique Boundary Value Problem of Monge-Amp\`ere Equations

We study the good shape property of boundary sections of convex solutions of the oblique boundary value problem for Monge-Amp`ere equations $$\det D^{2}u=f(x)\text{ in }\mathrm{\Omega },D_{\beta }u=\varphi (x)\text{ on }\mathrm{\partial }\mathrm{\Omega }\mathrm{.}$$ In the two-dimensional case, we prove the global $C^{2,\alpha }$ estimate for the solution. When the dimension $n\ge 3$, we show that this estimate still holds if the solution is bounded from above by a quadratic function in the tangent direction. We also obtain an existence result for the convex solution of Monge-Amp`ere equations with Robin oblique boundary conditions.