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Regularity and classification of the free boundary for a Monge-Ampère obstacle problem

Tianling Jin, Xushan Tu, Jingang Xiong

TL;DR

This work analyzes convex solutions to the Monge–Ampère obstacle problem $\det D^2 v = g\, v^q\,\chi_{\{v>0\}}$ with $q\in[0,n)$ and bounded positive $g$, motivated by the $L_p$ Minkowski problem. The authors establish $C^{1,\alpha}$ regularity for the strictly convex part of the free boundary and, under $g\in C^\alpha$, Schauder-type and $C^{2,\alpha}$ estimates, including a Liouville-type result that leads to a complete classification for $q=0$. They develop a robust framework based on the geometry of the coincidence set $K=\{v=0\}$, exposed points, and centered section normalizations, employing compactness arguments, Pogorelov-type estimates, and Legendre transforms. The results yield precise regularity and convexity properties of the free boundary, with applications to the $L_p$ Minkowski problem, including regularity of the tangent cone and higher-order differentiability of associated dual solutions. Together, these contributions provide a thorough picture of global and local regularity for Monge–Ampère obstacle problems with nonlinear right-hand sides and their links to geometric problems in convex geometry.

Abstract

We study convex solutions to the Monge-Ampère obstacle problem \[ \operatorname{det} D^2 v=g v^qχ_{\{v>0\}}, \quad v \geq 0, \] where $q \in [0,n)$ is a constant and $g$ is a bounded positive function. This problem emerges from the $L_p$ Minkowski problem. We establish $C^{1, α}$ regularity for the strictly convex part of the free boundary $\partial\{v=0\}$. Furthermore, when $g \in C^α$, we prove a Schauder-type estimate. As a consequence, when $g\equiv 1$, we obtain a Liouville theorem for entire solutions with unbounded coincidence sets $\{v=0\}$. Combined with existing results, this provides a complete classification of entire solutions for the case $q=0$.

Regularity and classification of the free boundary for a Monge-Ampère obstacle problem

TL;DR

This work analyzes convex solutions to the Monge–Ampère obstacle problem with and bounded positive , motivated by the Minkowski problem. The authors establish regularity for the strictly convex part of the free boundary and, under , Schauder-type and estimates, including a Liouville-type result that leads to a complete classification for . They develop a robust framework based on the geometry of the coincidence set , exposed points, and centered section normalizations, employing compactness arguments, Pogorelov-type estimates, and Legendre transforms. The results yield precise regularity and convexity properties of the free boundary, with applications to the Minkowski problem, including regularity of the tangent cone and higher-order differentiability of associated dual solutions. Together, these contributions provide a thorough picture of global and local regularity for Monge–Ampère obstacle problems with nonlinear right-hand sides and their links to geometric problems in convex geometry.

Abstract

We study convex solutions to the Monge-Ampère obstacle problem where is a constant and is a bounded positive function. This problem emerges from the Minkowski problem. We establish regularity for the strictly convex part of the free boundary . Furthermore, when , we prove a Schauder-type estimate. As a consequence, when , we obtain a Liouville theorem for entire solutions with unbounded coincidence sets . Combined with existing results, this provides a complete classification of entire solutions for the case .
Paper Structure (8 sections, 46 theorems, 257 equations)

This paper contains 8 sections, 46 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. The structure and $C^{1,\alpha}$ regularity of the free boundary
  3. $C^{1,\alpha}$ regularity of the solution
  4. $C^{1,1}$ regularity and uniform convexity of the free boundary
  5. The upper bound estimates
  6. The lower bound estimates
  7. $C^{2,\alpha}$ regularity of the free boundary
  8. An application to the $L_p$ Minkowski problem

Key Result

Theorem 1.1

Let $n\ge 2$ and $0\le q<n$. Suppose $v\not\equiv 0$ is a convex solution of If the coincidence set $K:=\{v=0\}$ is unbounded, then it must be a paraboloid, and $v$ is affine equivalentTwo functions $u_1, u_2: \mathbb{R}^n \to \mathbb{R}$ are called affine equivalent if there exists an $n \times n$ matrix $A$ with $\det A \neq 0$ and a vector $b = (b_1, \ldots, b_n)^T$ such where $c_{n,q}=\frac{

Theorems & Definitions (95)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Lemma 2.1: Aleksandrov’s Maximum Principle
  • Proposition 2.2
  • proof
  • Lemma 2.3: Comparison Principle
  • ...and 85 more