Given an open domain $\Omega \subseteq \mathbb{C}$ it is known that given $\phi \in C^\infty(\Omega)$ one can solve the partial differential equation $\bar{\partial}u := \frac{\partial u}{\partial \bar{z}} = \phi$ in $\Omega$ to obtain a $u\in C^\infty(\Omega).$ Are there any special domains, say a rectangle or a disk, or extra conditions that can be imposed, say boundedness of $\phi$ so that the solution $u$ can also be taken to be continuous up to the boundary?

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    $\begingroup$ In $\mathbb{C}$, every domain is (strictly) pseudoconvex and hence the $\overline{\partial}$-problem can be solved. So I believe that the condition that you are looking for is not on the domain but rather on $\phi$. \href{books.google.com/…}{Theorem 1.2.2} gives us an explicit solution. So as long as the integral makes sense, we can obtain solution that are continuous up to the boundary. I am not an expert, so this might be wrong/missing important details. $\endgroup$
    – t-rex
    May 31, 2021 at 0:09
  • $\begingroup$ @iota Thanks for the reference! Indeed, this seems to suggest that boundedness of $\phi$ and boundedness of $\Omega$ is good enough for a continuous extension to exist. $\endgroup$
    – Kurosu
    Jun 2, 2021 at 18:49


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