The diagram $(\lambda_1,\mu_1)$

TL;DR

This paper investigates the possible pairs of first Dirichlet and Neumann eigenvalues of the Laplace operator on plane domains, establishing classical inequalities and new bounds for convex domains, with graphical conjectures.

Contribution

It proves that classical inequalities fully characterize the eigenvalue pairs and introduces new inequalities for convex domains, along with graphical conjectures.

Findings

Classical inequalities form a complete system for eigenvalue pairs.

New bounds for the product of eigenvalues in convex domains.

Graphical conjectures on eigenvalue relationships.

Abstract

In this paper, we are interested in the possible values taken by the pair $(\lambda_1(\Omega), \mu_1(\Omega))$ the first eigenvalues of the Laplace operator with Dirichlet and Neumann boundary conditions respectively of a bounded plane domain $\Omega$. We prove that, without any particular assumption on the class of open sets $\Omega$, the two classical inequalities (the Faber-Krahn inequality and the Weinberger inequality) provide a complete system of inequalities. Then we consider the case of convex plane domains for which we give new inequalities for the product $\lambda_1 \mu_1$. We plot the so-called Blaschke--Santal\'o diagram and give some conjectures.