On the polygonal Faber-Krahn inequality

Collaboration with Dorin Bucur.

The paper presents a strategy for reducing the proof of the famous Polya-Szego conjecture (regarding the minimality of the first eigenvalue of the Dirichlet-Laplace operator among polygons) to a finite number of numerically validated computing.

Even though a complete proof for some $n\geq 5$ is not available yet, the strategy presented shows that such a proof could be achieved, using a careful study of all estimates involved and large scale certified computations.

In the following, we present some small updates that will be made regarding the initially submitted manuscript.

We were aware of the article: A combined finite element and Bayesian optimization framework for shape optimization in spectral geometry, by Sebastián A. Domínguez-Rivera, Nilima Nigam and Bobak Shahriari, but somehow it got left out. We will add this reference, where the authors give numerical evidence of the validity of the Polya-Szego conjecture for $n=5$.

Created: April 2022, Last modified: April 2022