# 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?

• Questions of regularity are considered in the paper of D. P. Squier here msp.org/pjm/1969/30-1/pjm-v30-n1-p16-p.pdf under a normal derivative condition on the interface. This seems to be the only difference in the above question with the one posted here: mathoverflow.net/questions/319715/…. – Josiah Park Dec 30 '18 at 15:25
• Can your problem be put as $-\Delta u=f$ in $U$, $u=g$ on $\partial U$, where $f$ is defined piecewise? – Andrew Dec 31 '18 at 6:41

• These are interesting references. Do you also know about problems with $\mathrm{div}(A_i(x)\nabla u)$ instead of $\Delta u$, with different $A_i$ different in the two parts of the domain? – Dal Jan 1 at 17:23