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?