Study of fractional Poincaré inequalities on unbounded domains

Indranil Chowdhury; Gyula Csató; Prosenjit Roy; Firoj Sk

doi:10.3934/dcds.2020394

Article Contents

2021, Volume 41, Issue 6: 2993-3020. Doi: 10.3934/dcds.2020394

This issue Previous Article On global well-posedness of the modified KdV equation in modulation spaces Next Article

Study of fractional Poincaré inequalities on unbounded domains

1.
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway
2.
Universitat de Barcelona, Spain
3.
Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, 208016, India

^* Corresponding author: Gyula Csató
^* Corresponding author: Gyula Csató

Received: April 30, 2020

Revised: August 31, 2020

Early access: December 2020

Published: June 2021

The first author is supported by the ERCIM "Alain Bensoussan" Fellowship program, HRI postdoctoral fellowship grant and Toppforsk project WaNP grant no. 250070. The third author is supported by Inspire grant IFA14-MA43 and Matrix grant MTR/2019/000585.
The second author is member of BGSMath Barcelona, part of the Catalan, research group 2017 SGR 1392, is supported by the MINECO grants MTM2017-83499-P and MTM2017-84214-C2-1-P and by the María de Maeztu Grant MDM-2014-0445

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Abstract

The aim of this paper is to study (regional) fractional Poincaré type inequalities on unbounded domains satisfying the finite ball condition. Both existence and non existence type results on regional fractional inequality are established depending on various conditions on domains and on the range of $ s \in (0,1) $. The best constant in both regional fractional and fractional Poincaré inequality is characterized for strip like domains $ (\omega \times \mathbb{R}^{n-1}) $, and the results obtained in this direction are analogous to those of the local case. This settles one of the natural questions raised by K. Yeressian in [Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89, (2014), no 1-2].

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Mathematics Subject Classification: Primary: 26D10; 35R09; 46E35, Secondary: 45K05, 47G20.

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