Abstract
<abstract><p>The aim of this work is to show a non-sharp quantitative stability version of the fractional isocapacitary inequality. In particular, we provide a lower bound for the isocapacitary deficit in terms of the Fraenkel asymmetry. In addition, we provide the asymptotic behaviour of the $ s $-fractional capacity when $ s $ goes to $ 1 $ and the stability of our estimate with respect to the parameter $ s $.</p></abstract>
Highlights
It is well known that the isocapacitary inequality is rigid, in the sense that dcap(Ω) vanishes if and only if Ω is equivalent to a ball up to a set of null Lebesgue measure. It appears as a natural quest the attempt of obtaining a quantitative stability version of (1.3)
We present our main result, which amounts to a quantitative stability inequality for the fractional capacity
We introduce the definition of fractional capacity of a closed subset of Rn
Summary
We introduce some prerequisites that are necessary in order to prove our main result. The notion of s-capacity of a set is in relation with the fractional Laplace operator of order s, which is defined, for u ∈ Cc∞(Rn), as follows (−∆)su(x) : = 2 P.V. u(x) − u(y) |x − y|n+2s dy, where P.V. means that the integral has to be seen in the principal value sense. The proof of our main result strongly relies on an extension procedure for functions in fractional Sobolev spaces, first established in [CS07], which, in some sense, allows to avoid some nonlocal issues and recover a local framework. The symmetric rearrangement of the capacitary potential of Ω coincides with the potential of a ball with the same volume as Ω Following this path, one can prove the fractional isocapacitary inequality using symmetric rearrangements for the extended problem in Rn++1. A proof of the fractional isocapacitary inequality (1.7) can be obtained as a direct consequence the previous result, applied to φ = uΩ, and Remark 2.8
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