• faber-krahn
• isoperimetric inequality
• kohler-jobin
• Main Laplace eigenvalue
• polya
• torsional rigidity
• weyl

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.