The eigenvalue problem for the Monge-Amp\`ere operator on general bounded convex domains
Abstract
In this paper, we study the eigenvalue problem for the Monge-Amp\`ere operator on general bounded convex domains. We prove the existence, uniqueness and variational characterization of the Monge-Amp\`ere eigenvalue. The convex Monge-Amp\`ere eigenfunctions are shown to be unique up to positive multiplicative constants. Our results are the singular counterpart of previous results by P-L. Lions and K. Tso in the smooth, uniformly convex setting. Moreover, we prove the stability of the Monge-Amp\`ere eigenvalue with respect to the Hausdorff convergence of the domains. This stability property makes it possible to investigate the Brunn-Minkowski, isoperimetric and reverse isoperimetric inequalities for the Monge-Amp\`ere eigenvalue in their full generality. We also discuss related existence and regularity results for a class of degenerate Monge-Amp\`ere equations.
Cite
@article{arxiv.1701.05165,
title = {The eigenvalue problem for the Monge-Amp\`ere operator on general bounded convex domains},
author = {Nam Q. Le},
journal= {arXiv preprint arXiv:1701.05165},
year = {2017}
}
Comments
v2: typos fixed; to appear in Annali SNS