2023/06/09 by Ian Fleschler, Xavier Tolsa, Fleschler, Ian +3
Mathematics · #35P15 28A75 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2306.06187
openalex publication_date 2023/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Ω⊂\mathbb Rn, n≥3, with Lebesgue measure equal to that of the unit ball B0 and whose first eigenvalue is λΩ, denoting by λB0 the first eigenvalue for the unit ball, for any a∈ (0,1) it holds λΩ- λB0 ≥ C(a) infB (supt∈ (0,1) \frac1Hn-1(∂ ((1-t) B)) ∫∂ ((1-t) B) \fracCapn-2(B(x,atrB)∖ Ω)(t rB)n-3 dHn-1(x))2, where the infimum is taken over all balls B with the same Lebesgue measure as Ω and Capn-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 ε2-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.