This question is motivated by one that has been previously asked on this website: [Elliptic problem on a domain split in two subdomains](https://mathoverflow.net/questions/319667/elliptic-problem-on-a-domain-split-in-two-subdomains)


Consider an open domain $U$ split in two non-overlapping subdomains: $U = U\_1 \cup U\_2$. 


For a model case, consider a ball split in a smaller ball and an anulus. 


Consider the following elliptic problem: 


\begin{align\*} -&\Delta u=f\_1 \ &\text{ in } U\_1\\
-&\Delta u =f\_2 & \text{ in } 
U\_2\\
& u=g & \text{ on } \partial U
\end{align\*}


* To obtain existence and uniqueness results for this problem, do we need to impose compatibility conditions at the interface between $U\_1$ and $U\_2$?
* What is a reference on this kind of problems?



Boundary condition for elliptic problems and domain decomposition
The place where $U\_1$ and $U\_2$ meet is known as an interface, and so this is a Poisson interface problem which can be read about in the paper *On the Existence and Uniqueness of Solutions of the Poisson Interface Problem* by D. P. Squier found [here](https://www.jstor.org/stable/2373213?seq=1#page_scan_tab_contents).


