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

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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
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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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	B. Dyda, A fractional order Hardy inequality,, \emph{Illinois J. Math.}, 48 (2004), 575. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations,, Second, (2011). doi: 10.1007/978-3-642-15564-2. Google Scholar
[13]	G. Psaradakis, $L^1$ Hardy inequalities with weights,, \emph{J. Geom. Anal.}, 23 (2013), 1703. doi: 10.1007/s12220-012-9302-8. Google Scholar
[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. Google Scholar

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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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	B. Dyda, A fractional order Hardy inequality,, \emph{Illinois J. Math.}, 48 (2004), 575. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations,, Second, (2011). doi: 10.1007/978-3-642-15564-2. Google Scholar
[13]	G. Psaradakis, $L^1$ Hardy inequalities with weights,, \emph{J. Geom. Anal.}, 23 (2013), 1703. doi: 10.1007/s12220-012-9302-8. Google Scholar
[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. Google Scholar

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