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*} -&\nabla ( A_1(x)\nabla u ) =f_1 \ &\text{ in } U_1\\ -&\nabla ( A_2(x)\nabla u )=f_2 & \text{ in } U_2\\ & u=g & \text{ on } \partial U \end{align*}

In the previous questions

• Elliptic problem on a domain split in two subdomains
• Boundary condition for elliptic problems and domain decomposition

a related problem was considered in the case of a prescribed jump or Neumann condition at the interface between $U_1$ and $U_2$.

In this question I wander about the general case without prescribed condition on $u$ at the interface.

• What references deal with such problems?
• What are the techniques to obtain existence and uniqueness results in this case (in the weak sense)?
• Indeed can we even get uniqueness without a condition at the interface? Why or why not?
• What are the minimal assumptions on $f_1$, $f_2$ and the domain that make the problem wellposed?