Shape Optimization Problems on Polygons

Abstract

In this thesis, we aim to study the Polya -Szego conjecture, which states that the regular polygon with n sides and fixed area minimizes the first Dirichlet eigenvalue among the family of simple polygons with $n$ sides in R^2 and fixed area for n greater than or equal to 3. The conjecture for the cases n = 3,4 was solved using Steiner symmetrization . However, Steiner symmetrization fails to prove the conjecture when we consider n is greater than or equal to 5. In 2022, Beniamin Bogosel and Dorin Bucur showed that the proof of the conjecture for n greater than or equal to 5 can be reduced to a finite number of numerical computations, assuming that the conjecture is true. They used the theory of shape derivatives and the finite element method to prove the local minimality of the regular polygon with a fixed area. They use surgery arguments further to prove the global minimality of the regular polygon with a fixed area.