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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?

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  • $\begingroup$ 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/…. $\endgroup$
    – Josiah Park
    Commented Dec 30, 2018 at 15:25
  • $\begingroup$ Can your problem be put as $-\Delta u=f$ in $U$, $u=g$ on $\partial U$, where $f$ is defined piecewise? $\endgroup$
    – Andrew
    Commented Dec 31, 2018 at 6:41

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There are many different papers treating such Poisson interface problems. A couple sources have been mentioned in the comments (although the paper of Squier on regularity of solutions differs slightly from your problem and deals specifically with the problem in the plane). One source that is mostly self-contained which handles the question of existence, uniqueness, and regularity is the paper of Xu-Dong Liu and Thomas C. Sideris found on arxiv here, which asserts regularity of solutions via examining finite difference solutions and then letting the mesh size go to zero to establish regularity of the solutions to the continuous problem. They do assume that the boundary is smooth in this paper.

Another approach for regularity on piecewise smooth domains can be found in the thesis of Petzoldt here. In both cases certain jump conditions are needed for uniqueness. One explanation for the type of conditions appearing shows up on page 5 of Chapter 2 of Petzoldt's thesis here. Also in both of these papers the usual weak (variational) forms are considered. If you are interested in classical solutions the papers referenced in the comments may be helpful.

Edit: These problems are also known as transmission problems and optimal piecewise regularity along smooth interfaces is proved in Chapter 5 in the 200 page paper of Martin Costabel, Monique Dauge, and Serge Nicaise here.

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  • $\begingroup$ 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? $\endgroup$
    – user60665
    Commented Jan 1, 2019 at 17:23
  • $\begingroup$ @Dal Both the first and second paper deal exactly with divergence form operators of the form you've mentioned. $\endgroup$
    – Josiah Park
    Commented Jan 1, 2019 at 17:40
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    $\begingroup$ I've asked a related question you may contribute to: mathoverflow.net/questions/320226/… Thanks. $\endgroup$
    – user60665
    Commented Jan 6, 2019 at 18:09
  • $\begingroup$ @Dal It seems that question has been answered. You might encourage that user to post something here if the answer there satisfactorily answers this question as well. $\endgroup$
    – Josiah Park
    Commented Jan 7, 2019 at 14:42
  • $\begingroup$ Yes, that was answered. Thanks anyway. Actually, I've just asked another one (which arises from the paper by Squier): mathoverflow.net/questions/321553/… $\endgroup$
    – user60665
    Commented Jan 23, 2019 at 19:01

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