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*}

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In the previous questions 

 - https://mathoverflow.net/questions/319667/elliptic-problem-on-a-domain-split-in-two-subdomains
 - https://mathoverflow.net/questions/319715/boundary-condition-for-elliptic-problems-and-domain-decomposition/319723#319723

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

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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?