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arXiv · 2501.01358

Global Lipschitz and Sobolev estimates for the Monge-Amp\`ere eigenfunctions of general bounded convex domains

Nam Q. Le
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Abstract

We show that the Monge-Amp\`ere eigenfunctions of general bounded convex domains are globally Lipschitz. The same result holds for convex solutions to degenerate Monge-Amp\`ere equations of the form $\det D^2 u =M|u|^p$ with zero boundary condition on general bounded convex domains in ${\mathbb R}^n$ within the sharp threshold $p>n-2$. As a consequence, we obtain global $W^{2, 1}$ estimates for these solutions.

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Nam Q. Le. 2025-01-02. Global Lipschitz and Sobolev estimates for the Monge-Amp\`ere eigenfunctions of general bounded convex domains. https://arxiv.org/abs/2501.01358

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