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I want to know if there is some reference on the well-posedness of the following problem :

For $\Omega$ a bounded domain (what results depending on the regularity of the domain?) of $\mathbb{R}^2$ (or $\mathbb{R}^3$, even $\mathbb{R}^n$), containing a hole, i.e. $\Omega$ has an "interior" boundary $\Gamma_1$ and an "exterior" boundary $\Gamma_2$, we look for the solution of :

  1. $\Delta u=0$ (harmonic function)
  2. $u=g_1$ on $\Gamma_1$
  3. $u=g_2$ on $\Gamma_2$

Some theorem, paper, or book that treats this question? Maybe the answer exists even for more general elliptic equations (maybe with a uniform ellipticity hypothesis?)

Also, I guess the results may depend on some "topological" property in the case of arbitrary spatial dimension $n$.

I guess the results also may depend strongly on the regularity of the domain.

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    $\begingroup$ This is the Dirichlet problem. The case for dimension two is frequently treated in books on complex analysis. $\endgroup$
    – Massimo Ortolano
    Commented Jun 4, 2022 at 22:04
  • $\begingroup$ Thank you, Massimo, but I think actually it is different... there are 2 boundaries so two functions are defined on $\partial \Omega=\Gamma_1 \cup\Gamma_2$ $\endgroup$
    – Matthew Binet
    Commented Jun 5, 2022 at 7:53
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    $\begingroup$ There aren’t two boundaries: there is one boundary with two connected components. It’s still a Dirichlet problem. $\endgroup$
    – Massimo Ortolano
    Commented Jun 5, 2022 at 8:06

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