Year: 2026
Type: preprint
Abstract: We establish a quantitative stability estimate for the first eigenvalue of the Laplacian with positive Robin boundary conditions. For every open set $Ω\subset\mathbb R^n$ of prescribed volume, we prove that \[ λ_{1,β}(Ω)-λ_{1,β}(B) \geq c\|u_Ω-u_B\|_{L^2(\mathbb R^n)}^4, \] where $B$ is a ball achieving the Fraenkel asymmetry of $Ω$, and $u_Ω$ 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.
Source: arXiv (Cornell University)
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223108318