Elliptic problem on a domain split in two subdomains Consider the following elliptic problem in a split domain:
$$ (\ast) \quad\begin{cases} -\Delta u=f_1 \quad &\text{ in } U_1\\
-\Delta u =f_2 & \text{ in } 
U_2\\
u=g  & \text{ on } \partial U
\end{cases} $$
where $U = U_1 \cup U_2$ is an open domain.
Where can I find a proof of existence, uniqueness and regularity of solutions for ($\ast$), under suitable assumptons on the regularity of the domain, the boundary data and source terms?
 A: There are many different papers treating such Poisson interface problems. A couple sources have been mentioned in the comments (although the paper of Squier on regularity of solutions differs slightly from your problem and deals specifically with the problem in the plane). One source that is mostly self-contained which handles the question of existence, uniqueness, and regularity is the paper of Xu-Dong Liu and Thomas C. Sideris found on arxiv here, which asserts regularity of solutions via examining finite difference solutions and then letting the mesh size go to zero to establish regularity of the solutions to the continuous problem. They do assume that the boundary is smooth in this paper.
Another approach for regularity on piecewise smooth domains can be found in the thesis of Petzoldt here. In both cases certain jump conditions are needed for uniqueness. One explanation for the type of conditions appearing shows up on page 5 of Chapter 2 of Petzoldt's thesis here. Also in both of these papers the usual weak (variational) forms are considered. If you are interested in classical solutions the papers referenced in the comments may be helpful.
Edit: These problems are also known as transmission problems and optimal piecewise regularity along smooth interfaces is proved in Chapter 5 in the 200 page paper of Martin Costabel, Monique Dauge, and Serge Nicaise here.
