Global $C^{1,α}$ regularity for Monge-Ampère equations on planar convex domains
Qing Han, Jiakun Liu, Yang Zhou
TL;DR
The paper extends global Hölder gradient estimates for the Monge–Ampère equation from uniformly convex planar domains to non-uniformly convex domains by focusing on boundary degeneracy. It introduces a $k$-order degeneracy framework at a boundary point, employs a local coordinate transform to flatten the degenerate boundary, and constructs barrier functions to derive global $C^{1,\alpha}$ regularity with $\alpha<\beta/k$, under a doubling condition on $f$ and $C^{k,\beta}$ boundary data. It also treats the homogeneous equation $\det D^2u=0$ with the same degeneracy and proves global $C^{1,\beta/k}$ (and $C^{1,1}$ when $\beta=k$) regularity, with extensions to the convex envelope. The results broaden the applicability of global regularity theory to practical domains encountered in imaging and computational geometry, providing new tools for degenerate Monge–Ampère problems. Overall, the work combines barrier methods, boundary-transforms, and careful degeneracy analysis to achieve sharp global gradient estimates in a two-dimensional, non-uniformly convex setting.
Abstract
In this paper, we establish the global Hölder gradient estimate for solutions to the Dirichlet problem of the Monge-Ampère equation $\det D^2u = f$ on strictly convex but not uniformly convex domain $Ω$.