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

摘要

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