A Quantitative Robin Faber-Krahn Inequality in Terms of Eigenfunction Distance

We establish a quantitative stability estimate for the first eigenvalue of the Laplacian with positive Robin boundary conditions. For every open set $\mathrm{\Omega }\subset {\mathbb{R}}^n$ of prescribed volume, we prove that $${\lambda }_{1,\beta }(\mathrm{\Omega })-{\lambda }_{1,\beta }(B)\ge c\mathrm{\parallel }{u}_{\mathrm{\Omega }}-u_B{\mathrm{\parallel }}_{L^2({\mathbb{R}}^n)}^4,$$ where $B$ is a ball achieving the Fraenkel asymmetry of $\mathrm{\Omega }$, and $u_{\mathrm{\Omega }}$ and $u_B$ are nonnegative, $L^2$-normalized first Robin eigenfunctions, extended by zero outside their respective domains. The constant $c>0$ depends only on the dimension, the Robin parameter, and the prescribed volume. The exponent $4$ is optimal.