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

Abstract

We study the good shape property of boundary sections of convex solutions of the oblique boundary value problem for Monge-Ampère equations $$\det D^{2}u=f(x)\text{ in }\Omega ,D_{\beta }u=\varphi (x)\text{ on }\partial \Omega .$$ In the two-dimensional case, we prove the global $C^{2,\alpha }$ estimate for the solution. When the dimension $n\ge 3$, 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-Ampère equations with Robin oblique boundary conditions.

47 pages

References in corpus (1)

• A Liouville theorem for the Neumann problem of the Monge-Ampere equation