A quantitative stability inequality for fractional capacities

Eleonora Cinti; Roberto Ognibene; Berardo Ruffini; Eleonora Cinti; Roberto Ognibene; Berardo Ruffini

doi:10.3934/mine.2022044

Mathematics in Engineering

2022, Volume 4, Issue 5: 1-28. doi: 10.3934/mine.2022044

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A quantitative stability inequality for fractional capacities

1.
Dipartimento di Matematica, Università di Bologna, 40126 Bologna, Italy
2.
Dipartimento di Ingegneria meccanica energetica, gestionale e dei trasporti, Università di Genova, 16145 Genova, Italy

Received: 17 July 2021 Accepted: 04 October 2021 Published: 26 October 2021

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 $.
- capacity,
- fractional Laplacian,
- quantitative inequalities,
- Caffarelli-Silvestre extension
Citation: Eleonora Cinti, Roberto Ognibene, Berardo Ruffini. A quantitative stability inequality for fractional capacities[J]. Mathematics in Engineering, 2022, 4(5): 1-28. doi: 10.3934/mine.2022044

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Abstract

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 $.

References

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