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Mathematics > Analysis of PDEs
Title: Faber-Krahn inequalities, the Alt-Caffarelli-Friedman formula, and Carleson's $\varepsilon^2$ conjecture in higher dimensions
(Submitted on 9 Jun 2023 (v1), last revised 25 Apr 2024 (this version, v4))
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 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 and whose first eigenvalue is $\lambda_\Omega$, denoting by $\lambda_B$ the first eigenvalue 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 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.
Submission history
From: Xavier Tolsa [view email][v1] Fri, 9 Jun 2023 18:28:14 GMT (40kb)
[v2] Mon, 13 Nov 2023 11:44:50 GMT (49kb)
[v3] Wed, 24 Apr 2024 11:38:46 GMT (49kb)
[v4] Thu, 25 Apr 2024 19:46:25 GMT (49kb)
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