Uniqueness for the Degenerate Monge-Ampère Equation on Arbitrary Bounded Convex Domains

Abstract

Let $n\ge 2$ and let $\Omega \subset {\mathbb{R}}^n$ be an arbitrary bounded open convex set. The author prove that, for $p>n$, the Dirichlet problem \[ \det D^2u=(-u)^p\quad\text{in }Ω, \qquad u=0\quad\text{on }\partialΩ, \qquad u>0\quad\text{in }Ω\] has at most one convex Alexandrov solution. The proof is based on the affine behavior of the Monge--Ampère energy and on a power-concavity property of the $L^{p+1}$ mass along the Legendre path connecting two solutions. At the homogeneous exponent $p=n$, the same argument shows that any two nonzero solutions with the same coefficient are positive multiples of one another.

arXiv comment9 pages