Trace Hardy--Sobolev--Maz'ya inequalities for the half fractional Laplacian

Previous Article
Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type
CPAA Home
This Issue
Next Article
Global solutions of a Keller--Segel system with saturated logarithmic sensitivity function

March 2015, 14(2): 373-382. doi: 10.3934/cpaa.2015.14.373

Trace Hardy--Sobolev--Maz'ya inequalities for the half fractional Laplacian

Stathis Filippas ^1, , Luisa Moschini ^2, and Achilles Tertikas ^1,

1.	Department of Mathematics and Applied Mathematics, University of Crete, 70013 Heraklion, Greece, Greece
2.	Dipartimento di Scienze di Base ed Applicate per l'Ingegneria, University of Rome "La Sapienza"', 00185 Rome, Italy

Received September 2013 Revised April 2014 Published December 2014

Abstract
Related Papers

In this work we establish trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for weakly mean convex domains. We accomplish this by obtaining a new weighted Hardy type estimate which is of independent inerest. We then produce Hardy-Sobolev-Maz'ya inequalities for the spectral half Laplacian. This covers a critical case left open in [9].

Keywords: fractional Sobolev inequality, Hardy inequality, best constant, fractional Laplacian, trace inequality., critical exponent.

Mathematics Subject Classification: Primary: 35J60, 42B20, 46E35; Secondary: 26D10, 35J15, 35P15, 47G3.

Citation: Stathis Filippas, Luisa Moschini, Achilles Tertikas. Trace Hardy--Sobolev--Maz'ya inequalities for the half fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (2) : 373-382. doi: 10.3934/cpaa.2015.14.373

References:

[1]	D. H. Armitage and U. Kuran, The convexity and the superharmonicity of the signed distance function,, \emph{Proc. Amer. Math. Soc.}, 93 (1985), 598. doi: 10.2307/2045528.
[2]	K. Bogdan, K. Burdzy and Z-Q. Chen, Censored stable processes,, \emph{Prob. theory related fields}, 127 (2003), 89. doi: 10.1007/s00440-003-0275-1.
[3]	X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, \emph{Adv. Math.}, 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025.
[4]	L. Caffarelli, J. M. Roquejoffre and O. Savin, Non local minimal surfaces,, \emph{Comm. Pure Appl. Math.}, 63 (2010), 1111. doi: 10.1002/cpa.20331.
[5]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. P.D.E.} \textbf{32} (2007), 32 (2007), 1245. doi: 10.1080/03605300600987306.
[6]	B. Dyda, A fractional order Hardy inequality,, \emph{Illinois J. Math.}, 48 (2004), 575.
[7]	B. Dyda and R. L. Frank, Fractional Hardy-Sobolev-Maz'ya inequality for domains,, \emph{Studia Math.}, 208 (2012), 151. doi: 10.4064/sm208-2-3.
[8]	S. Filippas, V. Maz'ya and A. Tertikas, Critical Hardy-Sobolev inequalities,, \emph{J. Math. Pures Appl.}, 87 (2007), 37. doi: 10.1016/j.matpur.2006.10.007.
[9]	S. Filippas, L. Moschini and A. Tertikas, Sharp Trace Hardy-Sobolev-Maz'ya inequalities and the fractional Laplacian,, \emph{Arch. Ration. Mech. Anal.}, 208 (2013), 109. doi: 10.1007/s00205-012-0594-4.
[10]	R. L. Frank and M. Loss, Hardy-Sobolev-Maz'ya inequalities for arbitrary domains,, \emph{J. Math. Pures Appl.}, 97 (2012), 39. doi: 10.1016/j.matpur.2011.04.004.
[11]	R. T. Lewis, J. Li and Y. Y. Li, A geometric characterization of a sharp Hardy inequality,, \emph{J. Funct. Anal.}, 262 (2012), 3159. doi: 10.1016/j.jfa.2012.01.015.
[12]	V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations,, Second, (2011). doi: 10.1007/978-3-642-15564-2.
[13]	G. Psaradakis, $L^1$ Hardy inequalities with weights,, \emph{J. Geom. Anal.}, 23 (2013), 1703. doi: 10.1007/s12220-012-9302-8.
[14]	O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions,, \emph{Ann. Inst. H. Poincare Anal. Non Linire}, 29 (2012), 479. doi: 10.1016/j.anihpc.2012.01.006.

show all references

References:

[1]	D. H. Armitage and U. Kuran, The convexity and the superharmonicity of the signed distance function,, \emph{Proc. Amer. Math. Soc.}, 93 (1985), 598. doi: 10.2307/2045528.
[2]	K. Bogdan, K. Burdzy and Z-Q. Chen, Censored stable processes,, \emph{Prob. theory related fields}, 127 (2003), 89. doi: 10.1007/s00440-003-0275-1.
[3]	X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, \emph{Adv. Math.}, 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025.
[4]	L. Caffarelli, J. M. Roquejoffre and O. Savin, Non local minimal surfaces,, \emph{Comm. Pure Appl. Math.}, 63 (2010), 1111. doi: 10.1002/cpa.20331.
[5]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. P.D.E.} \textbf{32} (2007), 32 (2007), 1245. doi: 10.1080/03605300600987306.
[6]	B. Dyda, A fractional order Hardy inequality,, \emph{Illinois J. Math.}, 48 (2004), 575.
[7]	B. Dyda and R. L. Frank, Fractional Hardy-Sobolev-Maz'ya inequality for domains,, \emph{Studia Math.}, 208 (2012), 151. doi: 10.4064/sm208-2-3.
[8]	S. Filippas, V. Maz'ya and A. Tertikas, Critical Hardy-Sobolev inequalities,, \emph{J. Math. Pures Appl.}, 87 (2007), 37. doi: 10.1016/j.matpur.2006.10.007.
[9]	S. Filippas, L. Moschini and A. Tertikas, Sharp Trace Hardy-Sobolev-Maz'ya inequalities and the fractional Laplacian,, \emph{Arch. Ration. Mech. Anal.}, 208 (2013), 109. doi: 10.1007/s00205-012-0594-4.
[10]	R. L. Frank and M. Loss, Hardy-Sobolev-Maz'ya inequalities for arbitrary domains,, \emph{J. Math. Pures Appl.}, 97 (2012), 39. doi: 10.1016/j.matpur.2011.04.004.
[11]	R. T. Lewis, J. Li and Y. Y. Li, A geometric characterization of a sharp Hardy inequality,, \emph{J. Funct. Anal.}, 262 (2012), 3159. doi: 10.1016/j.jfa.2012.01.015.
[12]	V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations,, Second, (2011). doi: 10.1007/978-3-642-15564-2.
[13]	G. Psaradakis, $L^1$ Hardy inequalities with weights,, \emph{J. Geom. Anal.}, 23 (2013), 1703. doi: 10.1007/s12220-012-9302-8.
[14]	O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions,, \emph{Ann. Inst. H. Poincare Anal. Non Linire}, 29 (2012), 479. doi: 10.1016/j.anihpc.2012.01.006.

[1]	Jianqing Chen. Best constant of 3D Anisotropic Sobolev inequality and its applications. Communications on Pure & Applied Analysis, 2010, 9 (3) : 655-666. doi: 10.3934/cpaa.2010.9.655
[2]	Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951
[3]	James Scott, Tadele Mengesha. A fractional Korn-type inequality. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3315-3343. doi: 10.3934/dcds.2019137
[4]	Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935
[5]	José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138
[6]	Pablo Raúl Stinga, Chao Zhang. Harnack's inequality for fractional nonlocal equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3153-3170. doi: 10.3934/dcds.2013.33.3153
[7]	Boumediene Abdellaoui, Fethi Mahmoudi. An improved Hardy inequality for a nonlocal operator. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1143-1157. doi: 10.3934/dcds.2016.36.1143
[8]	YanYan Li, Tonia Ricciardi. A sharp Sobolev inequality on Riemannian manifolds. Communications on Pure & Applied Analysis, 2003, 2 (1) : 1-31. doi: 10.3934/cpaa.2003.2.1
[9]	Igor E. Verbitsky. The Hessian Sobolev inequality and its extensions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6165-6179. doi: 10.3934/dcds.2015.35.6165
[10]	Julián Fernández Bonder, Julio D. Rossi. Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains. Communications on Pure & Applied Analysis, 2002, 1 (3) : 359-378. doi: 10.3934/cpaa.2002.1.359
[11]	Jinggang Tan, Jingang Xiong. A Harnack inequality for fractional Laplace equations with lower order terms. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 975-983. doi: 10.3934/dcds.2011.31.975
[12]	Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987
[13]	Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171
[14]	Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[15]	Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143
[16]	Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure & Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567
[17]	Pradeep Boggarapu, Luz Roncal, Sundaram Thangavelu. On extension problem, trace Hardy and Hardy's inequalities for some fractional Laplacians. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2575-2605. doi: 10.3934/cpaa.2019116
[18]	Aleksandra Čižmešija, Iva Franjić, Josip Pečarić, Dora Pokaz. On a family of means generated by the Hardy-Littlewood maximal inequality. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 223-231. doi: 10.3934/naco.2012.2.223
[19]	Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121
[20]	Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527

2017 Impact Factor: 0.884

American Institute of Mathematical Sciences