A unified quantitative approach to Chiti and Kohler--Jobin inequalities

We obtain two quantitative results related to classical rearrangement inequalities for the first Dirichlet eigenvalue. The first one is a quantitative version of Chiti’s reverse Hölder inequality. More precisely, for every $1\le q<p<\mathrm{\infty }$, we prove that the deficit controls, up to translations, the $L^q$ distance between a first eigenfunction of a bounded domain and the first eigenfunction of the ball having the same first eigenvalue. The second result concerns the Kohler–Jobin inequality and gives a quantitative estimate in terms of the $L^{\mathrm{\infty }}\text{\hspace{0.17em}⁣}\to L^2$ distance between the corresponding Dirichlet resolvents. In both cases, the main idea is to find a suitable superlevel set of the first eigenfunction which is almost optimal for the Faber-Krahn inequality.