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Global regularity for the Dirichlet problem of Monge-Ampère equation in convex polytopes

Genggeng Huang, Weiming Shen

TL;DR

This work extends the global regularity theory for the Monge-Ampère Dirichlet problem to convex polytopes in all dimensions. By combining a cone/Liouville framework, a careful blow-up analysis, and the A-condition on polytope geometry, the authors establish sharp global $C^2$ and $C^{2,\alpha}$ regularity under the existence of a globally $C^2$, convex sub-solution, and they derive precise regularity up to various skeletons of the polytope. The results reveal that simple polytopes are intrinsic to achieving $A$-condition compatibility, and they provide both necessary and sufficient conditions (in low dimensions) for $C^2$ regularity up to faces of codimension. Additionally, the paper offers dimension-specific sufficiency criteria that weaken sub-solution requirements in $n=2,3$, and it furnishes counterpoints illustrating the sharpness of the structural hypotheses, thus advancing boundary regularity theory in polyhedral domains with geometric constraints.

Abstract

We study the Dirichlet problem for Monge-Ampère equation in bounded convex polytopes. We give sharp conditions for the existence of global $C^2$ and $C^{2,α}$ convex solutions provided that a global $C^2$, convex subsolution exists.

Global regularity for the Dirichlet problem of Monge-Ampère equation in convex polytopes

TL;DR

This work extends the global regularity theory for the Monge-Ampère Dirichlet problem to convex polytopes in all dimensions. By combining a cone/Liouville framework, a careful blow-up analysis, and the A-condition on polytope geometry, the authors establish sharp global and regularity under the existence of a globally , convex sub-solution, and they derive precise regularity up to various skeletons of the polytope. The results reveal that simple polytopes are intrinsic to achieving -condition compatibility, and they provide both necessary and sufficient conditions (in low dimensions) for regularity up to faces of codimension. Additionally, the paper offers dimension-specific sufficiency criteria that weaken sub-solution requirements in , and it furnishes counterpoints illustrating the sharpness of the structural hypotheses, thus advancing boundary regularity theory in polyhedral domains with geometric constraints.

Abstract

We study the Dirichlet problem for Monge-Ampère equation in bounded convex polytopes. We give sharp conditions for the existence of global and convex solutions provided that a global , convex subsolution exists.

Paper Structure

This paper contains 7 sections, 22 theorems, 238 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Global $C^{1,1}$ estimate
  3. Solutions in the cones
  4. $C^{2}$ regularity up to $(n-2)$-face
  5. $C^{2}$ regularity up to $(n-k)$-face, $3\leq k\leq n$
  6. $C^{2,\alpha}$ regularity up to $\partial P$
  7. Appendix

Key Result

Theorem 1.1

Let $\Omega$ be a bounded convex $n-$polytope. Let $u$ be a convex function which solves intro-1. Assume that for some $\beta\in(0,1)$, and there exists a globally $C^2$, convex, sub-solution $\underline u\in C^2(\overline{\Omega})$ to intro-1(that is $\det D^2\underline u\ge f$ in $\overline{\Omega}$ and $\underline u=\varphi$ on $\partial \Omega$). Then, $u\in C^{1,1}(\overline \Omega)\bigcap C

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (52)

  • Definition : $\mathbf{\Theta(u,x_0,\Omega),\theta(u,x_0,\Omega)}$
  • Definition : A-Condition
  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 42 more