The diagram $(\lambda_1,\mu_1)$

Ilias Ftouhi; Antoine Henrot

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arXiv · 2501.02283

The diagram $(\lambda_1,\mu_1)$

Ilias Ftouhi·Antoine Henrot

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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.

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Ilias Ftouhi, Antoine Henrot. 2025-01-04. The diagram $(\lambda_1,\mu_1)$. https://arxiv.org/abs/2501.02283

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