Boundary regularity of the solution to the Complex Monge-Amp\`{e}re equation on pseudoconvex domains of infinite type

Abstract

Let $\Omega$ be a bounded, pseudoconvex domain of $\mathbb C^n$ satisfying the "$f$-Property". The $f$-Property is a consequence of the geometric "type" of the boundary; it holds for all pseudoconvex domains of finite type but may also occur for many relevant classes of domains of infinite type. In this paper, we prove the existence, uniqueness and "weak" H\"older-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Amp\`{e}re equation $$ \begin{cases} \det\left[\dfrac{\partial^2(u)}{\partial z_i\partial\bar z_j}\right]=h\ge 0 & \text{in}\quad\Omega,\\ u=\phi & \text{on} \quad b\Omega. \end{cases} $$