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doi: 10.3934/dcds.2020394

Study of fractional Poincaré inequalities on unbounded domains

Indranil Chowdhury 1, , Gyula Csató 2,, , Prosenjit Roy 3, and  Firoj Sk 3,

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  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].

Keywords: Fractional Poincaré inequality, fractional-Sobolev spaces, unbounded domains, infinite strips like domains, (regional) fractional Laplacian.
Mathematics Subject Classification: Primary: 26D10; 35R09; 46E35, Secondary: 45K05, 47G20.
Citation: Indranil Chowdhury, Gyula Csató, Prosenjit Roy, Firoj Sk. Study of fractional Poincaré inequalities on unbounded domains. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020394
References:
[1]

R. A. Adams and J. Fournier, Sobolev Spaces, Second Edition, Pure and Applied Mathematics (Amsterdam), Vol. 140 2003, Elsevier/Academic Press, xiv+305 pp.  Google Scholar

[2]

V. Ambrosio, L. Freddi and R. Musina, Asymptotic analysis of the Dirichlet fractional Laplacian in domains becoming unbounded, J. Math. Anal. Appl., 485 (2020), 123845, 17 pp. doi: 10.1016/j.jmaa.2020.123845.  Google Scholar

[3]

L. Brasco and A. Salort, A note on homogeneous Sobolev space of fractional order, Ann. Mat. Pura Appl. (4), 198 (2019), 1295-1330.  doi: 10.1007/s10231-018-0817-x.  Google Scholar

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L. Brasco, E. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458.  doi: 10.4171/IFB/325.  Google Scholar

[5]

L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Mathematical Journal, 37 (2014), 769-799.  doi: 10.2996/kmj/1414674621.  Google Scholar

[6]

H. Chen, The Dirichlet elliptic problem involving regional fractional Laplacian, J. Math. Physics, 59 (2018), 071504, 19 pp. doi: 10.1063/1.5046685.  Google Scholar

[7]

M. Chipot, A. Mojsic and P. Roy, On some variational problems set on domains tending to infinity, Discrete Contin. Dyn. Syst., 36 (2016), 3603-3621.  doi: 10.3934/dcds.2016.36.3603.  Google Scholar

[8]

M. Chipot, P. Roy and I. Shafrir, Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity, Asymptot. Anal., 85 (2013), 199-227.  doi: 10.3233/ASY-131182.  Google Scholar

[9]

M. Chipot and K. Yeressian, On the asymptotic behavior of variational inequalities set in cylinders, Discrete Contin. Dyn. Syst., 33 (2013), 4875-4890.  doi: 10.3934/dcds.2013.33.4875.  Google Scholar

[10]

I. Chowdhury and P. Roy, On the asymptotic analysis of problems involving fractional Laplacian in cylindrical domains tending to infinity, Commun. Contemp. Math., 19 (2017), 21 pp. doi: 10.1142/S0219199716500358.  Google Scholar

[11]

I. Chowdhury and P. Roy, Fractional Poincaré inequality for unbounded domains with finite ball condition: A Counter Example, arXiv: 2001.04441 (2020). Google Scholar

[12]

E. Cinti, J. Serra and E. Valdinoci, Quantitative flatness results and BV-estimates for stable nonlocal minimal surfaces, J. Differential Geom., 112 (2019), 447-504.  doi: 10.4310/jdg/1563242471.  Google Scholar

[13]

B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588.  doi: 10.1215/ijm/1258138400.  Google Scholar

[14]

B. Dyda and R. L. Frank, Fractional Hardy–Sobolev–Maz'ya inequality for domains, Studia Math., 208 (2012), 151-166.  doi: 10.4064/sm208-2-3.  Google Scholar

[15]

B. Dyda, J. Lehrbäck and A. V. Vähäkangas, Fractional Hardy-Sobolev type inequalities for half spaces and John domains, Proc. Amer. Math. Soc., 146 (2018), 3393-3402.  doi: 10.1090/proc/14051.  Google Scholar

[16]

B. Dyda, L. Ihnatsyeva and A. Vähäkangas, On improved fractional Sobolev-Poincaré inequalities, Ark. Mat., 54 (2016), 437-454.  doi: 10.1007/s11512-015-0227-x.  Google Scholar

[17]

L. Esposito, P. Roy and F. Sk, On the asymptotic behavior of the eigenvalues of nonlinear elliptic problems in domains becoming unbounded, Asymptot. Anal., (2020), 1–16. doi: 10.3233/ASY-201626.  Google Scholar

[18]

M. Felsinger, M. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Mathematische Zeitschrift, 279 (2015), 779-809.  doi: 10.1007/s00209-014-1394-3.  Google Scholar

[19]

A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.  doi: 10.5186/aasfm.2015.4009.  Google Scholar

[20]

R. L. Frank, Eigenvalue bounds for the fractional Laplacian: A review,, Recent Developments in Nonlocal Theory, 210–235, De Gruyter, Berlin, 2018. doi: 10.1515/9783110571561-007.  Google Scholar

[21]

R. L. Frank, T. Jin and J. Xiong, Minimizers for the fractional Sobolev inequality on domains, Calc. Var. Partial Differential Equations, 57 (2018), Art. 43, 31 pp. doi: 10.1007/s00526-018-1304-3.  Google Scholar

[22]

R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 225 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.  Google Scholar

[23]

R. Hurri-Syrjanen and A. V. Vähäkangas, Fractional Sobolev-Poincaré and fractional Hardy inequalities in unbounded John domains, Mathematika, 61 (2015), 385-401.  doi: 10.1112/S0025579314000230.  Google Scholar

[24]

R. Hurri-Syrjanen and A. V. Vähäkangas, On fractional Poincaré inequalities, J. Anal. Math., 120 (2013), 85-104.  doi: 10.1007/s11854-013-0015-0.  Google Scholar

[25]

D. Li and K. Wang, Symmetric radial decreasing rearrangement can increase the fractional Gagliardo norm in domains, Commun. Contemp. Math., 21 (2019), 1850059, 9 pp. doi: 10.1142/S0219199718500591.  Google Scholar

[26]

J.-L. Lions and E. Magenes, Non Homogeneous Boundary Value Problems and Applications, Springer, Volume 1, 1972.  Google Scholar

[27]

M. Loss and C. Sloane, Hardy inequalities for fractional integrals on general domains, J. Funct. Anal., 259 (2010), 1369-1379.  doi: 10.1016/j.jfa.2010.05.001.  Google Scholar

[28]

G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math., 12 (2010), 1055-1068.  doi: 10.1142/S0219199710004111.  Google Scholar

[29]

V. G. Maz'ja, Sobolev Spaces, Springer Ser. Soviet Math., Springer, Berlin, (1985). doi: 10.1007/978-3-662-09922-3.  Google Scholar

[30]

C. Mouhot, E. Russ and Y. Sire, Fractional Poincaré inequalities for general measures, J. Math. Pures Appl., 95 (2011), 72-84.  doi: 10.1016/j.matpur.2010.10.003.  Google Scholar

[31]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[32]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.  doi: 10.5565/PUBLMAT_60116_01.  Google Scholar

[33]

X. Ros-Oton and J. Serra, Regularity theory for general stable operators, J. Diff. Equations, 260 (2016), 8675-8715.  doi: 10.1016/j.jde.2016.02.033.  Google Scholar

[34]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445.  Google Scholar

[35]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.  Google Scholar

[36]

R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publicacions Matemàtiques, 58 (2014), 133-154.  doi: 10.5565/PUBLMAT_58114_06.  Google Scholar

[37]

R. Servadei and E. Valdinoci, The Brezis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.  Google Scholar

[38]

K. Yeressian, Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal., 89 (2014), 21-35.  doi: 10.3233/ASY-141224.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. Fournier, Sobolev Spaces, Second Edition, Pure and Applied Mathematics (Amsterdam), Vol. 140 2003, Elsevier/Academic Press, xiv+305 pp.  Google Scholar

[2]

V. Ambrosio, L. Freddi and R. Musina, Asymptotic analysis of the Dirichlet fractional Laplacian in domains becoming unbounded, J. Math. Anal. Appl., 485 (2020), 123845, 17 pp. doi: 10.1016/j.jmaa.2020.123845.  Google Scholar

[3]

L. Brasco and A. Salort, A note on homogeneous Sobolev space of fractional order, Ann. Mat. Pura Appl. (4), 198 (2019), 1295-1330.  doi: 10.1007/s10231-018-0817-x.  Google Scholar

[4]

L. Brasco, E. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458.  doi: 10.4171/IFB/325.  Google Scholar

[5]

L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Mathematical Journal, 37 (2014), 769-799.  doi: 10.2996/kmj/1414674621.  Google Scholar

[6]

H. Chen, The Dirichlet elliptic problem involving regional fractional Laplacian, J. Math. Physics, 59 (2018), 071504, 19 pp. doi: 10.1063/1.5046685.  Google Scholar

[7]

M. Chipot, A. Mojsic and P. Roy, On some variational problems set on domains tending to infinity, Discrete Contin. Dyn. Syst., 36 (2016), 3603-3621.  doi: 10.3934/dcds.2016.36.3603.  Google Scholar

[8]

M. Chipot, P. Roy and I. Shafrir, Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity, Asymptot. Anal., 85 (2013), 199-227.  doi: 10.3233/ASY-131182.  Google Scholar

[9]

M. Chipot and K. Yeressian, On the asymptotic behavior of variational inequalities set in cylinders, Discrete Contin. Dyn. Syst., 33 (2013), 4875-4890.  doi: 10.3934/dcds.2013.33.4875.  Google Scholar

[10]

I. Chowdhury and P. Roy, On the asymptotic analysis of problems involving fractional Laplacian in cylindrical domains tending to infinity, Commun. Contemp. Math., 19 (2017), 21 pp. doi: 10.1142/S0219199716500358.  Google Scholar

[11]

I. Chowdhury and P. Roy, Fractional Poincaré inequality for unbounded domains with finite ball condition: A Counter Example, arXiv: 2001.04441 (2020). Google Scholar

[12]

E. Cinti, J. Serra and E. Valdinoci, Quantitative flatness results and BV-estimates for stable nonlocal minimal surfaces, J. Differential Geom., 112 (2019), 447-504.  doi: 10.4310/jdg/1563242471.  Google Scholar

[13]

B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588.  doi: 10.1215/ijm/1258138400.  Google Scholar

[14]

B. Dyda and R. L. Frank, Fractional Hardy–Sobolev–Maz'ya inequality for domains, Studia Math., 208 (2012), 151-166.  doi: 10.4064/sm208-2-3.  Google Scholar

[15]

B. Dyda, J. Lehrbäck and A. V. Vähäkangas, Fractional Hardy-Sobolev type inequalities for half spaces and John domains, Proc. Amer. Math. Soc., 146 (2018), 3393-3402.  doi: 10.1090/proc/14051.  Google Scholar

[16]

B. Dyda, L. Ihnatsyeva and A. Vähäkangas, On improved fractional Sobolev-Poincaré inequalities, Ark. Mat., 54 (2016), 437-454.  doi: 10.1007/s11512-015-0227-x.  Google Scholar

[17]

L. Esposito, P. Roy and F. Sk, On the asymptotic behavior of the eigenvalues of nonlinear elliptic problems in domains becoming unbounded, Asymptot. Anal., (2020), 1–16. doi: 10.3233/ASY-201626.  Google Scholar

[18]

M. Felsinger, M. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Mathematische Zeitschrift, 279 (2015), 779-809.  doi: 10.1007/s00209-014-1394-3.  Google Scholar

[19]

A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.  doi: 10.5186/aasfm.2015.4009.  Google Scholar

[20]

R. L. Frank, Eigenvalue bounds for the fractional Laplacian: A review,, Recent Developments in Nonlocal Theory, 210–235, De Gruyter, Berlin, 2018. doi: 10.1515/9783110571561-007.  Google Scholar

[21]

R. L. Frank, T. Jin and J. Xiong, Minimizers for the fractional Sobolev inequality on domains, Calc. Var. Partial Differential Equations, 57 (2018), Art. 43, 31 pp. doi: 10.1007/s00526-018-1304-3.  Google Scholar

[22]

R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 225 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.  Google Scholar

[23]

R. Hurri-Syrjanen and A. V. Vähäkangas, Fractional Sobolev-Poincaré and fractional Hardy inequalities in unbounded John domains, Mathematika, 61 (2015), 385-401.  doi: 10.1112/S0025579314000230.  Google Scholar

[24]

R. Hurri-Syrjanen and A. V. Vähäkangas, On fractional Poincaré inequalities, J. Anal. Math., 120 (2013), 85-104.  doi: 10.1007/s11854-013-0015-0.  Google Scholar

[25]

D. Li and K. Wang, Symmetric radial decreasing rearrangement can increase the fractional Gagliardo norm in domains, Commun. Contemp. Math., 21 (2019), 1850059, 9 pp. doi: 10.1142/S0219199718500591.  Google Scholar

[26]

J.-L. Lions and E. Magenes, Non Homogeneous Boundary Value Problems and Applications, Springer, Volume 1, 1972.  Google Scholar

[27]

M. Loss and C. Sloane, Hardy inequalities for fractional integrals on general domains, J. Funct. Anal., 259 (2010), 1369-1379.  doi: 10.1016/j.jfa.2010.05.001.  Google Scholar

[28]

G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math., 12 (2010), 1055-1068.  doi: 10.1142/S0219199710004111.  Google Scholar

[29]

V. G. Maz'ja, Sobolev Spaces, Springer Ser. Soviet Math., Springer, Berlin, (1985). doi: 10.1007/978-3-662-09922-3.  Google Scholar

[30]

C. Mouhot, E. Russ and Y. Sire, Fractional Poincaré inequalities for general measures, J. Math. Pures Appl., 95 (2011), 72-84.  doi: 10.1016/j.matpur.2010.10.003.  Google Scholar

[31]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[32]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.  doi: 10.5565/PUBLMAT_60116_01.  Google Scholar

[33]

X. Ros-Oton and J. Serra, Regularity theory for general stable operators, J. Diff. Equations, 260 (2016), 8675-8715.  doi: 10.1016/j.jde.2016.02.033.  Google Scholar

[34]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445.  Google Scholar

[35]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.  Google Scholar

[36]

R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publicacions Matemàtiques, 58 (2014), 133-154.  doi: 10.5565/PUBLMAT_58114_06.  Google Scholar

[37]

R. Servadei and E. Valdinoci, The Brezis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.  Google Scholar

[38]

K. Yeressian, Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal., 89 (2014), 21-35.  doi: 10.3233/ASY-141224.  Google Scholar

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