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