Federico Villone

A common approach to quantitative Chiti and Kohler–Jobin inequalities

Chiti's reverse Hölder inequality and the Kohler–Jobin inequality are two classical results concerning the first Dirichlet eigenvalue of the Laplacian. The former provides an upper bound for the ratio of different norms of first eigenfunctions, with equality on balls, while the latter states that balls minimize the first Dirichlet eigenvalue among domains with prescribed torsional rigidity.

In this talk, we develop a common strategy to establish quantitative versions of both inequalities. More precisely, we show that the deficit in Chiti's inequality controls, up to translations, the \(L^q\) distance between the first eigenfunction of a domain and that of a ball with the same first eigenvalue. For the Kohler–Jobin inequality, we prove that the deficit controls the distance between the corresponding Dirichlet resolvent operators, where the comparison ball has the same torsional rigidity.