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
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context, I am grateful to my collaborators Beniamin Bogosel, Ilaria Lucardesi, Davide Zucco who agree to include recent unpublished results.
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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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