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In the paper "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 $u(z):=\sup\{ v \in\operatorname{ PSH} (\bar{D} \times \bar{D}): v_{\partial D \times \partial D} \leq b\}$ 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

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  • $\begingroup$ Did you try the Google search? Did you pay attention to that article? $\endgroup$
    – user64494
    Commented Jul 18 at 17:00
  • $\begingroup$ @user64494 I do not see how this article answers my question, I have added some additional clarification to my post. $\endgroup$
    – Gamabunto
    Commented Jul 19 at 10:25

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