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

Huaiyu Jian; Xushan Tu

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arXiv · 2111.09765

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

Huaiyu Jian·Xushan Tu

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Abstract

We study the good shape property of boundary sections of convex solutions of the oblique boundary value problem for Monge-Ampère equations $$\det D^2u =f(x) \text{ in } Ω, \quad D_βu = ϕ(x) \text{ on } \partial Ω.$$ In the two-dimensional case, we prove the global $C^{2,α}$ 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-Ampère equations with Robin oblique boundary conditions.

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BibTeX RIS

Huaiyu Jian, Xushan Tu. 2024-02-24. Boundary $C^{2, α}$ Regularity for the Oblique Boundary Value Problem of Monge-Ampère Equations. https://arxiv.org/abs/2111.09765

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