Optimal boundary regularity for some singular Monge-Amp\`ere equations on bounded convex domains

Nam Q. Le

Optimal boundary regularity for some singular Monge-Amp\`ere equations on bounded convex domains

Analysis of PDEs 2021-04-21 v1

Authors: Nam Q. Le

Abstract

By constructing explicit supersolutions, we obtain the optimal global H\"older regularity for several singular Monge-Amp\`ere 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.

Keywords

monge-ampere equations singular equations bounded convex domains holder regularity supersolutions affine spheres

Cite

@article{arxiv.2104.10056,
  title  = {Optimal boundary regularity for some singular Monge-Amp\`ere equations on bounded convex domains},
  author = {Nam Q. Le},
  journal= {arXiv preprint arXiv:2104.10056},
  year   = {2021}
}

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