-
Previous Article
Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains
- DCDS Home
- This Issue
-
Next Article
Martin boundary of brownian motion on Gromov hyperbolic metric graphs
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 |
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].
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. |
[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. |
[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. |
[4] |
L. Brasco, E. Lindgren and E. Parini,
The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458.
doi: 10.4171/IFB/325. |
[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. |
[6] |
H. Chen, The Dirichlet elliptic problem involving regional fractional Laplacian, J. Math. Physics, 59 (2018), 071504, 19 pp.
doi: 10.1063/1.5046685. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
B. Dyda,
A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588.
doi: 10.1215/ijm/1258138400. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
J.-L. Lions and E. Magenes, Non Homogeneous Boundary Value Problems and Applications, Springer, Volume 1, 1972. |
[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. |
[28] |
G. Mancini and K. Sandeep,
Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math., 12 (2010), 1055-1068.
doi: 10.1142/S0219199710004111. |
[29] |
V. G. Maz'ja, Sobolev Spaces, Springer Ser. Soviet Math., Springer, Berlin, (1985).
doi: 10.1007/978-3-662-09922-3. |
[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. |
[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. |
[32] |
X. Ros-Oton,
Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.
doi: 10.5565/PUBLMAT_60116_01. |
[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. |
[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. |
[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. |
[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. |
[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. |
[38] |
K. Yeressian,
Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal., 89 (2014), 21-35.
doi: 10.3233/ASY-141224. |
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. |
[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. |
[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. |
[4] |
L. Brasco, E. Lindgren and E. Parini,
The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458.
doi: 10.4171/IFB/325. |
[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. |
[6] |
H. Chen, The Dirichlet elliptic problem involving regional fractional Laplacian, J. Math. Physics, 59 (2018), 071504, 19 pp.
doi: 10.1063/1.5046685. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
B. Dyda,
A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588.
doi: 10.1215/ijm/1258138400. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
J.-L. Lions and E. Magenes, Non Homogeneous Boundary Value Problems and Applications, Springer, Volume 1, 1972. |
[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. |
[28] |
G. Mancini and K. Sandeep,
Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math., 12 (2010), 1055-1068.
doi: 10.1142/S0219199710004111. |
[29] |
V. G. Maz'ja, Sobolev Spaces, Springer Ser. Soviet Math., Springer, Berlin, (1985).
doi: 10.1007/978-3-662-09922-3. |
[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. |
[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. |
[32] |
X. Ros-Oton,
Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.
doi: 10.5565/PUBLMAT_60116_01. |
[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. |
[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. |
[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. |
[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. |
[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. |
[38] |
K. Yeressian,
Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal., 89 (2014), 21-35.
doi: 10.3233/ASY-141224. |
[1] |
María J. Garrido-Atienza, Bohdan Maslowski, Jana Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088 |
[2] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
[3] |
Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113 |
[4] |
Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks & Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151 |
[5] |
Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 |
[6] |
Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511 |
[7] |
Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281 |
[8] |
Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261 |
[9] |
Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881 |
[10] |
Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 |
[11] |
Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363 |
[12] |
Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119 |
[13] |
Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065 |
[14] |
Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]