A reverse quantitative isoperimetric type inequality for the Dirichlet Laplacian

A stability result in terms of the perimeter is obtained for first Dirichlet eigenvalue Laplacian operator. In particular, we prove that, once fix dimension $n\geq2$, there exists a constant $c>0$, depending only on $n$, such every $\Omega\subset\mathbb{R}^n$ open, bounded, and convex set with volume equal to ball $B$ radius $1$, it holds that $\lambda\_1(\Omega)-\lambda\_1(B)\geq c(P(\Omega)-P(B))^2$, where $\lambda\_1(\cdot)$ denotes $P(\cdot)$ its perimeter. The heart present paper sharp estimate Fraenkel asymmetry