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The main aim of this article is to prove quantitative spectral inequalitiesfor the Laplacian with Dirichlet boundary conditions. More specifically, weprove sharp quantitative stability for the Faber-Krahn inequality in terms ofNewtonian capacities and Hausdorff contents of positive codimension, thusproviding an answer to a question posed by De Philippis and Brasco.  One of our results asserts that for any bounded domain $\Omega\subset\mathbbR^n$, $n\geq3$, with Lebesgue measure equal to that of the unit ball and whosefirst eigenvalue is $\lambda_\Omega$, denoting by $\lambda_B$ the firsteigenvalue for the unit ball, for any $a\in (0,1)$ it holds $$\lambda_\Omega -\lambda_B \geq C(a) \,\inf_B \bigg(\sup_{t\in (0,1)} \frac1{H^{n-1}(\partial((1-t) B))} \int_{\partial ((1-t) B)}\frac{\operatorname{Cap}_{n-2}(B(x,atr_B)\setminus\Omega)}{(t\,r_B)^{n-3}}\,dH^{n-1}(x)\bigg)^2,$$ where the infimum is takenover all balls $B$ with the same Lebesgue measure as $\Omega$ and$\operatorname{Cap}_{n-2}$ is the Newtonian capacity of homogeneity $n-2$. Infact, 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 toquantify the Hayman-Friedland inequality about the characteristics of disjointdomains in the unit sphere. Thirdly, we propose a natural extension ofCarleson's $\varepsilon^2$-conjecture to higher dimensions in terms of a squarefunction involving the characteristics of certain spherical domains, and weprove the necessity of the finiteness of such square function in the tangentpoints via the Alt-Caffarelli-Friedman monotonicity formula. Finally, we answerin the negative a question posed by Allen, Kriventsov and Neumayer inconnection to rectifiability and the positivity set of the ACF monotonicityformula.

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