A necessary and sufficient condition for Hölder-class solutions to the complex Monge--Ampère equation
Annapurna Banik
TL;DR
The paper gives a sharp criterion for Hölder regularity of solutions to the complex Monge–Ampère Dirichlet problem on bounded domains in $\mathbb{C}^n$, showing that the existence of a uniformly strictly plurisubharmonic barrier $\rho\in {\rm psh}(\Omega)\cap C^{0,\varepsilon}(\overline{\Omega})$ with $\Omega=\rho^{-1}(-\infty,0)$, $\partial\Omega=\rho^{-1}\{0\}$, and $\rho-\|\cdot\|^2\in{\rm psh}(\Omega)$ is equivalent to having a unique Hölder continuous solution for data with $f\equiv 0$ or $f>0$ and $\phi\in C^{1,1}(\partial\Omega)$. The paper then proves a general regularity result: on bounded domains with such a structure, there exists a modulus of continuity $\omega$ so that a solution lies in $\mathcal{C}_{\omega}(\overline{\Omega})$ whenever $f^{1/n}\in \mathcal{C}_{\omega}(\overline{\Omega})$. For $B$-regular domains, a comparable regularity result holds, but recent work shows that Hölder continuity cannot be guaranteed for all admissible data in general, motivating the refined omega-continuity result. Overall, the work links Li’s Hölder criterion to a robust regularity theory via Perron–Bremermann envelopes and barrier constructions, with implications for complex geometry and PDE applications.
Abstract
We provide a necessary and sufficient condition for the existence of Hölder continuous solutions to the complex Monge--Ampère equation on bounded domains in $\mathbb{C}^n$. This condition is motivated by a paper by S.-Y. Li. We also prove a result on the regularity of solutions to the complex Monge--Ampère equation on general $B$-regular domains in $\mathbb{C}^n$.