Isoperimetric inequalities in the plane

Abstract

[en] The main goal of this work is to study different geometric inequalities in the plane. In particular, we will work on the isoperimetric, the Saint-Venant and the Faber-Krahn inequalities for simple connected domains. We will use two different approaches: first a classic one by complex analysis, and then a more recent one by operator theory, bounding the commutator of Toeplitz operators in the HardySmirnov space $E_2$ and the Bergman space $A^2$. We will also study these spaces and how they relate with geometric quantities. Finally, we will talk about functions of bounded variation in order to extend the classical isoperimetric inequality for any domain in the plane.