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 $$\det D^2u =f(x) \text{ in } \Omega , \quad D_{\beta}u = \phi(x) \text{ on } \partial \Omega.$$ In the two-dimensional case, we prove the global $C^{2,\alpha}$ estimate for the solution. When the dimension $n \geq 3$, 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.