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Isoperimetric Inequalities for Eigenvalues of the Laplacian

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Geometry of PDEs and Related Problems

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2220))

Abstract

These lecture notes give an overview of “isoperimetric inequalities”, namely inequalities involving only geometric features, for the eigenvalues of the Laplace operator, with Dirichlet boundary conditions. In other words, we are mainly interested in minimization problems like

$$\displaystyle \begin{aligned} \text{min} \{\lambda_k(\varOmega),\;\varOmega\subset \mathbb{R}^N \mbox{ open set, with some geometric constraints. }\} \end{aligned} $$
(2.1)

Here λ k(Ω) denotes the k-th eigenvalue of the Laplace operator with Dirichlet boundary conditions and the geometric constraints can involve the volume or the perimeter or the diameter or some box constraints or some specific sub-classes like polygons or convex sets. Most of the information contained in these notes are from the book of the author (Henrot, Extremum problems for eigenvalues of elliptic operators. Frontiers in mathematics, Birkhäuser, Basel, 2006), but some more recent results are also presented.

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Acknowledgements

This text is largely inspired by the book [29] of the author supplemented by more recent results. In that context, I am grateful to my collaborators Beniamin Bogosel, Ilaria Lucardesi, Davide Zucco who agree to include recent unpublished results.

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Authors and Affiliations

  1. Institut Élie Cartan, UMR CNRS 7502, Université de Lorraine, Vandœuvre-lès-Nancy, France

    Antoine Henrot

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  1. Antoine Henrot
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Correspondence to Antoine Henrot .

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Editors and Affiliations

  1. Dipt di Matematica e Informatica Ulisse Dini, Università degli Studi di Firenze, Firenze, Italy

    Chiara Bianchini

  2. Institut Elie Cartan – Faculté des Sciences et Technologies, Université de Lorraine, Vandoeuvre-les-Nancy, France

    Antoine Henrot

  3. Dipt di Matematica e Informatica Ulisse Dini, Università degli Studi di Firenze, Firenze, Italy

    Rolando Magnanini

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Henrot, A. (2018). Isoperimetric Inequalities for Eigenvalues of the Laplacian. In: Bianchini, C., Henrot, A., Magnanini, R. (eds) Geometry of PDEs and Related Problems. Lecture Notes in Mathematics(), vol 2220. Springer, Cham. https://doi.org/10.1007/978-3-319-95186-7_2

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