The diagram $(\hat{I}{»}_1,\hat{I}{¼}_1)$

Abstract

In this paper, we are interested in the possible values taken by the pair $(\hat{I}{»}_1(\hat{I}\text{c◯}),\hat{I}{¼}_1(\hat{I}\text{c◯}))$ the first eigenvalues of the Laplace operator with Dirichlet and Neumann boundary conditions respectively of a bounded plane domain $\hat{I}\text{c◯}$. We prove that, without any particular assumption on the class of open sets $\hat{I}\text{c◯}$, 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 $\hat{I}{»}_1\hat{I}{¼}_1$. We plot the so-called Blaschke--Santaló diagram and give some conjectures.