Elliptic PDE on a Moebius band Let $M \simeq [0,1]^2 / \sim$ be a Moebius band, where we identify $\{0\} \times [0,1]$ and $\{1\} \times [0,1]$ by $(0,y) \sim (1,1-y)$ for all $y \in [0,1]$. 
Let $C = [0,1] \times \{1/2\}$ be the central circle of the Moebius band. We identify $M$ (and $C$) with their smooth embeddings in $\mathbb{R}^3$.
Consider the following mixed boundary value problem on $M \setminus C$:
\begin{cases}
\Delta u + \lambda u = 0  &\text{ in $\mathrm{int} \, M \setminus C$} \\
u = 0 &\text{ on $\partial M$} \\
\frac{\partial u}{\partial \nu} = 0 &\text{ on $C$}
\end{cases}
where $\Delta$ is the metric Laplacian on $M$, $\lambda \in \mathbb{R}$ is a constant. (In other words, $u$ is an eigenfunction of the Laplacian.) Is this is a well-defined problem? If yes, what is the correct weak formulation? If not, what could go wrong?
As far as I can tell, the issue could be that $\partial(M \setminus C) = \partial M \cup C$ is not a Lipschitz boundary in the classical sense, because the interior of $M \setminus C$ lies on both sides of $C$. However, $M \setminus C$ can be isometrically deformed into a "Moebius strip with two twists", and a boundary with two connected components, corresponding to $\partial M$ and $C$. If permissible, such a deformation could get rid of the issue, as after it $M \setminus C$ would lie on one side of $C$.
However, I still can't quite see how one would make sense of the space "$H_0^1(\mathrm{int} \, M\setminus C \cup C)$", which is the right space for the weak formulation according to the answer to this question.
Note: As I didn't get any replies on here, I asked the same question on Math.Overflow.
 A: I assume that we are interested in $u \in H^1(M \setminus C)$. Our space $M \setminus C$ is isometrically equivalent to an ordinary strip $X = S^1 \times I$ (where $S^1$ has length $2$ and $I$ has length $1/2$). The boundary consists of two circles: $\Gamma_1$ (corresponding to $C$) and $\Gamma_2$ (corresponding to $\partial M$). As in this question, take your function space $H$ to be the closure of $C_c^\infty(X \cup \Gamma_1)$ (or $C_0^1(X \cup \Gamma_1)$) wrt. the norm $H^1(X)$. Here $C_c^\infty(X \cup \Gamma_1)$ stands for functions smooth up to $\Gamma_1$ and vanishing on some neighborhood of $\Gamma_2$. 
This space encodes the Dirichlet boundary data $u = 0$ on $\Gamma_2$. The Neumann condition $\frac{\partial u}{\partial \nu} = 0$ on $\Gamma_1$ has to be included in the weak formulation, which can be tested with functions from $C_c^\infty(X \cup \Gamma_1)$, hence also from $H$. 
As a side comment: if one is interested in $u \in H^1(M)$ one has to require additionally "continuity across $C$", i.e. twe two one-sided traces on $C$ should agree. This translates to the property that the trace of $u \in H^1(X)$ on $\Gamma_1 \cong S^1$ is $\mathbb{Z}_2$-invariant. 
