Quantitative Resolvent and Eigenfunction Stability for the Faber-Krahn Inequality

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