A Strengthened Alexandrov Maximum Principle or Uniform H\"older Continuity for Solutions of the Monge--Amp\`ere Equation with Bounded Right-Hand Side

arXiv2211.01175MaRDI QIDQ6415938

Lukas Gehring

Publication date: 2 November 2022

Abstract: This article is about the convex solution $u$ of the Monge--Amp`ere equation on an at least 2-dimensional open bounded convex domain with Dirichlet boundary data and nonnegative bounded right-hand side. For convex functions with zero boundary data, an Alexandrov maximum principle $|u(x)|leqCoperatornamedist(x,partialOmega{)}^{a}lpha$ is equivalent to (uniform) H"older continuity with the same constant and exponent. Convex $alpha$-H"older continuous functions are $W^{1,p}$ for $p<1/(1-alpha)$. We prove H"older continuity with the exponent $alpha=2/n$ for $ngeq3$ and any $alphain(0,1)$ for $n=2$, provided that the boundary data satisfy this H"older continuity, and show that these bounds for the exponent are sharp. The only means is to bound the Hessian determinant of a certain explicit function on an $n$-dimensional cylinder and to use the comparison princple.

Mathematics Subject Classification ID

Monge-Ampère equations (35J96)

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