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Suppose $\Omega$ is a domain in $\mathbb{R}^n$ with a sufficiently smooth boundary $\Gamma$. Consider a partition of the boundary $(\Gamma_i)_i, i=1,2,3$, i.e. $\Gamma = \bigcup \limits_{i=1}^3 \Gamma_i$ and $\Gamma_i \cap \Gamma_j = \phi$ for $i\neq j$.

Consider the following boundary value problem: \begin{align} - \nabla \cdot (A \nabla u) + b \cdot u + c u =f \ \ \ in \ \ \Omega \\ u = g \ \ \ \ \ on \ \ \Gamma_1 \\ D \nabla u = h \ \ \ \ \ on \ \ \Gamma_2 \\ D \nabla u + \alpha u = k \ \ \ \ \ on \ \ \Gamma_3 \end{align}

My question is why do we need $\Gamma_i \cap \Gamma_j = \phi$ for $i\neq j$? What if the boundary parts intersect with one another? I am guessing this has something to do with the uniqueness of the solution (if it exists), but I do not know for sure.

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The assumption $\Gamma_i \cap \Gamma_j = \varnothing$ for $i \neq j$ is made so that the problem is well-posed, that is, there exists a unique solution.

If it is not the case that $\Gamma_i \cap \Gamma_j = \varnothing$ for $i \neq j$ then the problem is overdetermined which means that for certain domains $\Omega$ there cannot be a solution. Such problems, aptly named overdetermined problems, have been widely researched. Usually the emphasise in these problems is not so much the solution $u$, but the classification of the domains $\Omega$ which admit a solution.

As an example, one of the classical problems is Serrin's problem, named after James Serrin who answered it in the paper "A symmetry problem in potential theory" in 1971, says that if $\Omega$ is a bounded domain with $C^2$ boundary which admits a solution to $$ \begin{cases} -\Delta u =1,&\text{in } \Omega \\ u=0, &\text{on } \partial \Omega \\ \partial_\nu u = \text{const.}, &\text{on } \partial \Omega \end{cases}$$ then $\Omega$ must be a ball.

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