Bhattacharya–Weitsman–Nadirashvili sharp quantitative Faber–Krahn conjecture

Let $N\geq 2$ and let $\Omega\subset\mathbb{R}^N$ be an open bounded set. Denote by $\lambda(\Omega)$ the first Dirichlet eigenvalue of the Laplacian on $\Omega$, let $B$ be a ball, and let $\mathcal{A}(\Omega)$ be the Fraenkel asymmetry of $\Omega$.

Bhattacharya–Weitsman–Nadirashvili conjecture. There exists a dimensional constant $C>0$ such that

$$|\Omega|^{2/N}\,\lambda(\Omega)-|B|^{2/N}\,\lambda(B)\geq \frac{1}{C}\,\mathcal{A}(\Omega)^2.$$

This conjecture asks for a sharp quantitative refinement of the Faber–Krahn inequality, asserting that the eigenvalue deficit controls the squared Fraenkel asymmetry. The source presents it as a conjecture of Bhattacharya and Weitsman and, independently, Nadirashvili; no resolution is supplied here.