# Different boundary condition on different boundary parts

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.

## 1 Answer

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.