This question is motivated by one that has been previously asked on this website: 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?