Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains

Nam Q. Le

doi:10.3934/dcds.2021188

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2022, Volume 42, Issue 5: 2199-2214. Doi: 10.3934/dcds.2021188

This issue Previous Article Global weak solutions to the stochastic Ericksen–Leslie system in dimension two Next Article A dissipative logarithmic-Laplacian type of plate equation: Asymptotic profile and decay rates

Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains

Nam Q. Le^,

Department of Mathematics, Indiana University, Bloomington, 831 E 3rd St, Bloomington, IN 47405, USA

^*Corresponding author: Nam Q. Le
^*Corresponding author: Nam Q. Le

Received: March 31, 2021

Published: May 2022

The author is supported by NSF grant DMS-2054686

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By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as $ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $ with zero boundary data, have unexpected degenerate nature.

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Mathematics Subject Classification: Primary: 35J96, 35J75; Secondary: 53A15.

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