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- Title
Remarks on sharp boundary estimates for singular and degenerate Monge-Ampère equations.
- Authors
Le, Nam Q.
- Abstract
By constructing appropriate smooth supersolutions, we establish sharp lower bounds near the boundary for the modulus of nontrivial solutions to singular and degenerate Monge-Ampère equations of the form $ \det D^2 u = |u|^q $ with zero boundary condition on a bounded domain in $ \mathbb R^n $. These bounds imply that currently known global Hölder regularity results for these equations are optimal for all $ q $ negative, and almost optimal for $ 0\leq q\leq n-2 $. Our study also establishes the optimality of global $ C^{\frac{1}{n}} $ regularity for convex solutions to the Monge-Ampère equation with finite total Monge-Ampère measure. Moreover, when $ 0\leq q<n-2 $, the unique solution has its gradient blowing up near any flat part of the boundary. The case of $ q $ being $ 0 $ is related to surface tensions in dimer models. We also obtain new global log-Lipschitz estimates, and apply them to the Abreu's equation with degenerate boundary data.
- Subjects
MONGE-Ampere equations; SURFACE tension; YANG-Baxter equation; DEGENERATE differential equations
- Publication
Communications on Pure & Applied Analysis, 2023, Vol 22, Issue 5, p1
- ISSN
1534-0392
- Publication type
Article
- DOI
10.3934/cpaa.2023043