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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?
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The place where $U_1$ and $U_2$ meet is known as an interface, and so this is a Poisson interface problem which can be read about in the paper On the Existence and Uniqueness of Solutions of the Poisson Interface Problem by D. P. Squier found here.

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  • $\begingroup$ Thank you. In that paper, the author uses a Neumann boundary condition at the interface. Is it really necessary to impose that to obtain wellposedness? Also has the case of general elliptic operators in divergence form been studied? $\endgroup$
    – user60665
    Commented Dec 29, 2018 at 21:28
  • $\begingroup$ Yes. The question of existence is already answered because of the existence of a solution with added conditions. One is free to change the constant $K$ that appears in the added constraints so there are in fact multiple solutions and so some additional constraints are needed generally to ensure uniqueness. As for general divergence form operators, on the first page of the paper there is mention of work by Olenik in this direction. $\endgroup$
    – Josiah Park
    Commented Dec 30, 2018 at 0:56

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