# Solution to the $\bar{\partial}-$ equation in $\mathbb{C}$ continuous up to the boundary

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?

• 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. May 31, 2021 at 0:09
• @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. Jun 2, 2021 at 18:49