ETD

Archivio digitale delle tesi discusse presso l'Università di Pisa

Tesi etd-09292019-101550


Tipo di tesi
Tesi di laurea magistrale
Autore
BIONDI, CLAUDIA GINEVRA
URN
etd-09292019-101550
Titolo
On the principal eigenvalue and torsional rigidity of the Laplace operator
Dipartimento
MATEMATICA
Corso di studi
MATEMATICA
Relatori
relatore Prof. Buttazzo, Giuseppe
controrelatore Prof. Pratelli, Aldo
Parole chiave
  • polya
  • Main Laplace eigenvalue
  • kohler-jobin
  • isoperimetric inequality
  • faber-krahn
  • torsional rigidity
  • weyl
Data inizio appello
25/10/2019
Consultabilità
Non consultabile
Data di rilascio
25/10/2089
Riassunto
We discuss some classical results such as Faber-Krahn inequality, K\"{o}hler-Jobin inequality and P\'{o}lya's inequality as well as more recent results.
We start by introducing some basic properties of the first $-\Delta$ eigenvalue and we then proceed to determine the behaviour of the principal frequence on some specific domains (rectangles, balls). The variational characterization of the first eigenvalue (Rayleigh principle) is then recalled and we conclude the chapter by examining the minimization problem on polygons (via Steiner symmetrization for triangles and rectangles) and introducing the conjecture for the minimization problem on polygons with number of sides greater than $4$ (coupled with some numerical results obtained via FEM methods).
In chapter 3 we introduce the notion of torsional rigidity of a domain, giving again both a variational characterization and a non variational one.
Some explicit computations of the torsional rigidity are given in the case of 1-d and particular 2-d domains (long rectangles, balls) via Fourier analysis.\\
In chapter 4 we will discuss problems of the form $inf_{\Omega} \frac{\lambda_1(\Omega)^{\alpha}T(\Omega)^{\beta}}{|\Omega|^{\gamma}}$, $sup_{\Omega} \frac{\lambda_1(\Omega)^{\alpha}T(\Omega)^{\beta}}{|\Omega|^{\gamma}}$ where $\Omega$ is an open bounded domain $\subset \R^n$. The main classical results of this chapter are K\"{o}hler-Jobin and P\'{o}lya's inequalities (Blaschke-Santalò diagrams). We then discuss sharper inequalities in the case of convex domains and give a complete analysis of the problem for 1-d domains as well as a detailed investigation in the case of 2-d domains.
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