A quantitative stability estimate for the fractional Faber-Krahn inequality

Abstract

We prove a quantitative version of the Faber-Krahn inequality for the first eigenvalue of the fractional Dirichlet-Laplacian of order s. This is done by using the so-called Caffarelli-Silvestre extension and adapting to the nonlocal setting a trick by Hansen and Nadirashvili. The relevant stability estimate comes with an explicit constant, which is stable as the fractional order of differentiability goes to 1.2010 Mathematics Subject Classification. 47A75, 49Q20, 35R11.

A quantitative dimension free isoperimetric inequality for the fractional Gaussian perimeter

“…Now we prove a technical result similar to [6,Lemma 4.2]. It states that if we are not going too far in the vertical direction, then the level sets of the extension of the characteristic function of a set E are comparable to E itself.…”

“…This section contains some technical results that are the core of the proof of the Main Theorem. Our strategy follows the ideas in [6]: we first estimate D γ s (E) from below with a quantity involving the asymmetry of the superlevel sets of U E (•, z) and then, in a suitable range of values for the function U E and for the vertical variable z, we show that the asymmetry of the superlevel sets is estimated from below by A γ (E).…”

“…The following proposition provides an enhanced version of an inequality proved in [28]. In the spirit of [6,23], given a set E, we apply the Stinga-Torrea extension to the function χ E and exploit the sharp Gaussian quantitative inequality proved in [3]. Proposition 4.1.…”

“…In [28] the authors introduce a notion of Gaussian fractional perimeter using the by now well known extension techniques introduced in [9,30] and they prove a qualitative isoperimetric inequality in the more general setting of abstract Wiener spaces. Inspired by the paper [6], where the authors prove a stability estimate for the fractional Faber-Krahn inequality, and taking into account the extension technique of [30], we prove a quantitative isoperimetric inequality for a fractional perimeter in the Gauss space. Moreover, similarly to the local case (see [3,18]), the constant appearing in the inequality does not depend on the dimension of the ambient space.…”

A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators

“…The optimal shape problems for the first eigenvalue of the fractional Laplacian with homogeneous external datum was addressed in [3,13,56,11], showing that the ball is the optimizer.…”