The eigenvalue problem for the Monge-Ampère operator on general bounded convex domains

In this paper, we study the eigenvalue problem for the Monge-Ampère operator on general bounded convex domains. We prove the existence, uniqueness and variational characterization of the Monge-Ampère eigenvalue. The convex Monge-Ampère 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-Ampère 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-Ampère eigenvalue in their full generality. We also discuss related existence and regularity results for a class of degenerate Monge-Ampère equations.


Publication Date:
Jun 19 2017
Date Submitted:
Mar 01 2019
Citation:
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 18, 4
Note:
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 Record created 2019-03-01, last modified 2019-04-03


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