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Two Necessary and Sufficient Conditions to the Solvability of the Exterior Dirichlet Problem for the Monge-Ampère Equation

Cong Wang, Jiguang Bao

Abstract

The present paper provides two necessary and sufficient conditions for the existence of solutions to the exterior Dirichlet problem of the Monge-Ampère equation with prescribed asymptotic behavior at infinity. By an adapted smooth approximation argument, we prove that the problem is solvable if and only if the boundary value is semi-convex with respect to the inner boundary, which is our first proposed new concept. Along the lines of Perron's method for Laplace equation, we obtain the threshold for solvability in the asymptotic behavior at infinity of the solution, and remove the $C^2$ regularity assumptions on the boundary value and on the inner boundary which are required in the proofs of the corresponding existence theorems in the recent literatures.

Two Necessary and Sufficient Conditions to the Solvability of the Exterior Dirichlet Problem for the Monge-Ampère Equation

Abstract

The present paper provides two necessary and sufficient conditions for the existence of solutions to the exterior Dirichlet problem of the Monge-Ampère equation with prescribed asymptotic behavior at infinity. By an adapted smooth approximation argument, we prove that the problem is solvable if and only if the boundary value is semi-convex with respect to the inner boundary, which is our first proposed new concept. Along the lines of Perron's method for Laplace equation, we obtain the threshold for solvability in the asymptotic behavior at infinity of the solution, and remove the regularity assumptions on the boundary value and on the inner boundary which are required in the proofs of the corresponding existence theorems in the recent literatures.
Paper Structure (6 sections, 16 theorems, 187 equations, 2 figures)

This paper contains 6 sections, 16 theorems, 187 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:necessity']}
  3. Perron's solution of equation \ref{['eq0:MA=1']} in the exterior domain
  4. Asymptotic behavior of the Perron's solution near infinity
  5. Boundary behavior of the Perron's solution
  6. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $\Omega$ be a bounded, strictly convex domain of $\mathbb{R}^n$, $n\geq3$, $\partial\Omega\in C^2$ and let $\varphi\in C^2(\partial\Omega)$. Then for any $A\in\mathbb{\mathcal{A}}$ and $b\in\mathbb{R}^n$, there exists some constant $c_*$, such that eq:DiriProb has a viscosity solution in $C^0(\m

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (39)

  • Theorem 1.1: Caffarelli--Li Caffarelli-Li-2003 and Li--Lu Li-Lu
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Definition 3.1
  • Proposition 3.2
  • ...and 29 more