Boundary $C^{2, \alpha}$ regularity for the oblique boundary value problem of Monge–Ampère equations

Huaiyu Jian; Xushan Tu

doi:10.4171/jems/1780

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Boundary $C^{2, α}$ regularity for the oblique boundary value problem of Monge–Ampère equations

Huaiyu Jian
Beijing Technology and Business University, P. R. China
Xushan Tu
The Hong Kong University of Science and Technology, P. R. China

$Boundary $C^{2, \alpha}$ regularity for the oblique boundary value problem of Monge–Ampère equations cover$

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Abstract

We study a good shape property of boundary sections of convex solutions to the oblique boundary value problem for the Monge–Ampère equation

det D^{2} u = f (x) in Ω, D_{β} u = ϕ (x) on \partial Ω.

In two dimensions, we prove a global $C^{2, α}$ estimate for solutions. For dimensions $n \geq 3$ , we show that this estimate remains valid provided the solution satisfies a quadratic growth condition in tangential directions. We also prove an existence result for convex solutions to the Monge–Ampère equation with an oblique Robin boundary condition.

Cite this article

Huaiyu Jian, Xushan Tu, Boundary $C^{2, α}$ regularity for the oblique boundary value problem of Monge–Ampère equations. J. Eur. Math. Soc. (2026), published online first

DOI 10.4171/JEMS/1780