Quantitative Resolvent and Eigenfunction Stability for the Faber-Krahn Inequality

Abstract

For a bounded open set $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ with the same volume as the unit ball, the classical Faber-Krahn inequality says that the first Dirichlet eigenvalue $\hat{I}{»}_1(\hat{I}\text{c◯})$ of the Laplacian is at least that of the unit ball $B$. We prove that the deficit $\hat{I}{»}_1(\hat{I}\text{c◯})-\hat{I}{»}_1(B)$ in the Faber-Krahn inequality controls the square of the distance between the resolvent operator $(-\hat{I}{”}_{\hat{I}}\text{c◯}{)}^{-1}$ for the Dirichlet Laplacian on $\hat{I}\text{c◯}$ and the resolvent operator on the nearest unit ball $B(x_{\hat{I}}\text{c◯})$. The distance is measured by the operator norm from $L^{\infty }$ to $L^2$. As a main application, we show that the Faber-Krahn deficit $\hat{I}{»}_1(\hat{I}\text{c◯})-\hat{I}{»}_1(B)$ controls the squared $L^2$ norm between $k$th eigenfunctions on $\hat{I}\text{c◯}$ and $B(x_{\hat{I}}\text{c◯})$ for every $k\in \mathbb{N}.$ In both of these main theorems, the quadratic power is optimal.

We thank Jimmy Lamboley for showing us a paper of Prunier. Adapting an argument therein allowed us to substantially weaken the hypotheses of our main theorems