# American Institute of Mathematical Sciences

June  2021, 41(6): 2993-3020. doi: 10.3934/dcds.2020394

## 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

Received  May 2020 Revised  September 2020 Published  June 2021 Early access  December 2020

Fund Project: 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

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].

Citation: Indranil Chowdhury, Gyula Csató, Prosenjit Roy, Firoj Sk. Study of fractional Poincaré inequalities on unbounded domains. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2993-3020. doi: 10.3934/dcds.2020394
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