Eigenvalues of the Laplacian on perfored domains

This is based on a discussion I had with some colleagues, after attending the 9th Cascade RAIN, in OSU, Corvallis, OR. In particular, we were talking of the ongoing work of Shivam Patel (under the supervision of Dr. Jeffrey Ovall).

The “cheese grater” problem

(I invented this name). The idea is simple: consider the Laplace eigenvalue problem over $\Omega_n = \Omega(n;\theta)$: $-\Delta u^n = \lambda^n u^n, \text{ in } \Omega^n,$ and $\gamma^n u=0$ on its boundary $\partial \Omega^n$. Here $\Delta$ is the Laplace operator, $\gamma^n$ is a trace operator, $\theta$ is a set of parameters that define the geometry of the domain, and $n\in\mathbb{N}.$

A concrete example is the following:

• Let $\theta = r_0 \in (0,1/2)$ be a fixed radius.
• Let $\Omega(n,\theta) = (0,1)^2 \setminus \bigcup_{i,j=1}^{n-1} B_{i,j}(r_0^n)$, where $B_{i,j}(r_0^n)$ is a ball of radius $r_0^n$ centered at $(i/n,j/n)$.
• $\gamma^n$ corresponds to the normal derivative on the boundary of $\Omega^n$ (i.e., a Neumann boundary condition). In other words, we are removing small balls from the unit square¹.

From Shivam’s presentation, at least in the Neumann case, it was observed that the eigenvalues somewhat stabilize as $n$ increases. Thus, the question is whether convergence can be established in this case, for different settings, and what is the dependence on the geometry of the domain.

Sierpinski carpet in the kitchen?

One concerns is quite natural, and pertains to the geometry of the domain. In what sense does the domain converge to a limit? More over, the limit geometry may not be a domain at all: not open, not simply connected, etc. It may even resemble a Sierpinski carpet, or a Cantor set².

So, my concern is the stability of the formulations and the spaces, as regularity theory stops working as nicely as we would like. Some Sobolev-like spaces can be defined; see this papers³, for example.

What happens in the kitchen stays in the kitchen

I had a quick though I have shared with some friends: suppose we can nest the domains $\iota_n:\Omega_{n+1}\hookrightarrow\Omega_{n}$. Then, naturally, we have a Sobolev-type contravariant functor $V(\cdot)$, such that $V(\Omega_{n}) \hookrightarrow V(\Omega_{n+1}),$ given by $f\mapsto f\circ \iota_n$.

The idea —despite the categorical language— is to use Courant-Fischer min-max principle to show monotonicity of the eigenvalues. The main issue being, of course, that a restriction operator may not preserve subspaces…

$$\lambda_k^n = \min_{V \subset V(\Omega_n), \dim V = k} \max_{u\in V} \frac{\int_{\Omega_n} |\nabla u|^2}{\int_{\Omega_n} |u|^2} \stackrel{?}{\geq} \lambda_k^{n+1} = \min_{V \subset V(\Omega_{n+1}), \dim V = k} \max_{u\in V} \frac{\int_{\Omega_{n+1}} |\nabla u|^2}{\int_{\Omega_{n+1}} |u|^2}.$$

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Footnotes

1. Let me know if this makes sense. Otherwise, the radius can be changed to be $r_n=\frac{1}{2n+1}$, for example. ↩

2. The carpet and the Cantor set. ↩

3. Hu, Jiaxin. “A note on Hajłasz–Sobolev spaces on fractals.” Journal of mathematical analysis and applications 280, no. 1 (2003): 91-101. DOI. ↩