In the paper "On a On a Generalized Dirichlet Problem for Plurisubharmonic Functions and Pseudo-Convex Domains. Characterization of Silov Boundaries" Theorem 5.3, the following result is obtained for a type of generalized Dirichlet problem:
Set $D = \{z: |z|<1\}\subset \mathbb{C}$. Then, for a continuous $b: \partial D \times \partial D \to \mathbb{R}$, the plurisubharmonic Perron envelope $u(z):=\sup\{ v \in PSH (\bar{D} \times \bar{D}): v_{\partial D \times \partial D} \leq b\}$ is plurisubharmonic and attains the values of $b$ on $\partial D \times \partial D$.
Set $D = \{z: |z|<1\}\subset \mathbb{C}$. Then, for a continuous $b: \partial D \times \partial D \to \mathbb{R}$, the plurisubharmonic Perron envelope $u(z):=\sup\{ v \in\operatorname{ PSH} (\bar{D} \times \bar{D}): v_{\partial D \times \partial D} \leq b\}$ is plurisubharmonic and attains the values of $b$ on $\partial D \times \partial D$.
I am looking for results on the existence and regularity of the solutions of this problem for different types of boundary data. Does anyone know where I can find them?
For clarification, the type of results I am interested in are the following:
Let $b\in L^p(\partial D \times \partial D)$, then the solution $u$ belongs to some smoothness space.
Thanks
