2022/11/02 by Lukas Gehring, Gehring, Lukas
Mathematics · #35J96 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2211.01175
openalex publication_date 2022/11/02 · openalex created_date 2022/11/08 · openalex updated_date 2026/07/28
This article is about the convex solution u of the Monge--Ampère 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)| ≤ C dist(x,∂Ω)α is equivalent to (uniform) Hölder continuity with the same constant and exponent. Convex α-Hölder continuous functions are W1,p for p < 1/(1-α). We prove Hölder continuity with the exponent α=2/n for n ≥ 3 and any α∈ (0,1) for n=2, provided that the boundary data satisfy this Hölder 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.