Faber-Krahn inequalities, the Alt-Caffarelli-Friedman formula, and Carleson's $\varepsilon^2$ conjecture in higher dimensions

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 contents of positive codimension, thus providing an answer to a question posed by De Philippis and Brasco. One of our results asserts that for any bounded domain $\Omega\subset\mathbb R^n$, $n\geq3$, 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 $$\lambda_\Omega - \lambda_{B_0} \geq C(a) \,\inf_B \bigg(\sup_{t\in (0,1)} \frac1{H^{n-1}(\partial ((1-t) B))} \int_{\partial ((1-t) B)} \frac{\operatorname{Cap}_{n-2}(B(x,atr_B)\setminus \Omega)}{(t\,r_B)^{n-3}}\,dH^{n-1}(x)\bigg)^2,$$ where the infimum is taken over all balls $B$ with the same Lebesgue measure as $\Omega$ and $\operatorname{Cap}_{n-2}$ 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 inequalities to quantify the Hayman-Friedland inequality about the characteristics of disjoint domains in the unit sphere. Thirdly, we propose a natural extension of Carleson's $\varepsilon^2$-conjecture to higher dimensions in terms of a square function involving the characteristics of certain spherical domains, and we prove the necessity of the finiteness of such square function in the tangent points 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.