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

Consider an open domain $U$ split in two 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?