Abstract
In this paper we are interested in the possible values taken by the pair (λ1(ω), μ1(ω)) the first eigenvalues of the Laplace operator with Dirichlet and Neumann boundary conditions respectively of a bounded plane domain ω. We prove that, without any particular assumption on the class of open sets ω, 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 λ1μ1. We plot the so-called Blaschke-Santaló diagram and give some conjectures.
| Original language | English |
|---|---|
| Pages (from-to) | 159-177 |
| Number of pages | 19 |
| Journal | Mathematical Reports |
| Volume | 24-74 |
| Issue number | 1-2 |
| State | Published - 2022 |
| Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2022 Editura Academiei Romane. All rights reserved.
Keywords
- Blaschke-Santaló diagrams
- Complete systems of inequalities
- convex sets
- sharp spectral inequalities
ASJC Scopus subject areas
- Analysis
- Algebra and Number Theory
- Geometry and Topology
- Applied Mathematics