*preprint*

**Inserted:** 4 sep 2024

**Last Updated:** 4 sep 2024

**Year:** 2023

**Abstract:**

The main aim of this article is to prove quantitative spectral inequalities for the Laplacian with Dirichlet boundary conditions. More specifically, we prove sharp quantitative stability for the Faber-Krahn inequality in terms of Newtonian capacities and Hausdorff codimension, thus providing an answer to a question posed by De Philippis and Brasco.

One of our results asserts that for any bounded domain $\Omega \subset \mathbb{R}^n$, $n \geq 3$, with Lebesgue measure equal to that of the unit ball $B_0$ and whose first eigenvalue is $\lambda_\Omega$, denoting by $\lambda_{B_0}$ the first eigenvalue for the unit ball, for any $a \in (0,1]$, it holds

\[ \lambda_\Omega - \lambda_{B_0} \geq C(a) \inf_B \left(\sup_{t \in (0,1)} \frac{1}{\mathcal{H}^{n-1}(\partial ((1-t) B))} \int_{\partial ((1-t) B)} \frac{\text{Cap}_{n-2}(B(x, a t r_B) \setminus \Omega)}{(t\,r_B)^{n-3}} \, d\mathcal{H}^{n-1}(x)\right)^2, \]

where the infimum is taken over all balls $B$ with the same Lebesgue measure as $\Omega$ and $\text{Cap}_{n-2}$ is the Newtonian capacity of homogeneity $n-2$. In fact, this holds for bounded subdomains of the sphere and the hyperbolic space, as well.

In a second result, we also apply the new Faber-Krahn type inequality about the characteristics of disjoint domains in the unit sphere. Thirdly, we propose a natural extension of Carleson's $\epsilon^2$-conjecture to higher dimensions in the presence of spherical domains, and we prove the necessity of the finiteness of such square function in the tangential directions via the Alt-Caffarelli-Friedman monotonicity formula. Finally, we answer in the negative a question posed by Allen, Kriventsov and Neumayer in connection to rectifiability and the positivity set of the ACF monotonicity formula.