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

TL;DR

This paper establishes boundary regularity results for convex solutions of Monge-Ampère equations with oblique boundary conditions, proving $C^{2,eta}$ regularity in 2D and under certain bounds in higher dimensions.

Contribution

It provides the first global $C^{2,eta}$ regularity results for solutions to Monge-Ampère equations with oblique boundary conditions, including existence results for Robin boundary conditions.

Findings

Global $C^{2,eta}$ regularity in 2D for solutions.

Extension of regularity results to higher dimensions under boundedness assumptions.

Existence of convex solutions with Robin oblique boundary conditions.

Abstract

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.