About the problem
For a bounded open set \(\Omega\subset \mathbb{R}^2\) with Lipchitz boundary, let
\[0<\lambda_1(\Omega)\leq \lambda_2(\Omega)\leq \cdots\]be the Dirichlet eigenvalues of the positive Laplacian, counted with multiplicity. Do all eigenvalues satisfy the bound
\[\lambda_k(\Omega)\geq \frac{4\pi k}{|\Omega|}?\]This is the well known Pólya Laplacian conjecture (“Mathematics and plausible reasoning”, Princeton University Press) from 1954.
The conjecture is known to hold for tiling domains (Pólya 1961), for the disc, and most recently for the annulus, but it remains open in general. See the introduction of this paper for a good survey. The conjecture is also known to fail in some cases, for example on cylindrical domains.
This problem asks for a rational polygon with polygonal holes which is a counterexample to Pólya’s Laplacian conjecture. There are two main risks with this problem. First, it may be the case that Pólya’s conjecture is true for rational polygons with polygonal holes, in which case no counterexample of the required form can be found. Second, a counterexample may exist but be too large or complicated for the verifier to successfully compute the eigenvalues.
Prompt
Let Omega be a bounded open set in R^2 with Lipschitz boundary, and let
0 < lambda_1(Omega) <= lambda_2(Omega) <= ...
be the Dirichlet eigenvalues of the positive Laplacian on Omega, counted with
multiplicity. Consider the inequality
lambda_k(Omega) >= 4*pi*k / |Omega|,
where |Omega| is the area of Omega.
## Task
Find a bounded polygonal domain P in R^2 with rational vertices, together with an
index k >= 1, that violates this inequality:
|P| * lambda_k(P) < 4*pi*k.
P is an outer simple polygon with zero or more polygonal holes removed from it, so
it may be multiply connected. Supply a strictly positive rational margin epsilon;
the submission is accepted only if the violation holds with room to spare:
|P| * lambda_k(P) < 4*pi*k - epsilon.
## Submission
Submit the path to a JSON file with this shape:
{
"vertices": [[x1, y1], [x2, y2], ..., [xm, ym]],
"holes": [ [[..], [..], [..]], ... ],
"k": <positive integer>,
"epsilon": <positive rational>,
"refinement": <non-negative integer>
}
* "vertices": the outer simple polygon, vertices in cyclic order.
* "holes" (optional, default none): simple polygons removed from the outer one.
Each hole must lie strictly inside the outer polygon, and holes must be
pairwise disjoint.
* "k", "epsilon": as above.
* "refinement": mesh refinement level used to bound lambda_k(P); larger values
give a tighter bound at higher cost.
Each coordinate and epsilon is an exact rational, given as an integer or as a
string that is a fraction "p/q" or a plain decimal such as "0.25". Floating-point
numbers and scientific notation are not accepted.