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Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type
Trace Hardy--Sobolev--Maz'ya inequalities for the half fractional Laplacian
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 |
References:
[1] |
D. H. Armitage and U. Kuran, The convexity and the superharmonicity of the signed distance function, Proc. Amer. Math. Soc., 93 (1985), 598-600.
doi: 10.2307/2045528. |
[2] |
K. Bogdan, K. Burdzy and Z-Q. Chen, Censored stable processes, Prob. theory related fields, 127 (2003), 89-152.
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, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[4] |
L. Caffarelli, J. M. Roquejoffre and O. Savin, Non local minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.
doi: 10.1002/cpa.20331. |
[5] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. P.D.E. 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[6] |
B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588. |
[7] |
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. |
[8] |
S. Filippas, V. Maz'ya and A. Tertikas, Critical Hardy-Sobolev inequalities, J. Math. Pures Appl., 87 (2007), 37-56.
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, Arch. Ration. Mech. Anal., 208 (2013), 109-161.
doi: 10.1007/s00205-012-0594-4. |
[10] |
R. L. Frank and M. Loss, Hardy-Sobolev-Maz'ya inequalities for arbitrary domains, J. Math. Pures Appl., 97 (2012), 39-54.
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, J. Funct. Anal., 262 (2012), 3159-3185.
doi: 10.1016/j.jfa.2012.01.015. |
[12] |
V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Second, revised and augmented edition. Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-15564-2. |
[13] |
G. Psaradakis, $L^1$ Hardy inequalities with weights, J. Geom. Anal., 23 (2013), 1703-1728.
doi: 10.1007/s12220-012-9302-8. |
[14] |
O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions, Ann. Inst. H. Poincare Anal. Non Linire, 29 (2012), 479-500.
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, Proc. Amer. Math. Soc., 93 (1985), 598-600.
doi: 10.2307/2045528. |
[2] |
K. Bogdan, K. Burdzy and Z-Q. Chen, Censored stable processes, Prob. theory related fields, 127 (2003), 89-152.
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, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[4] |
L. Caffarelli, J. M. Roquejoffre and O. Savin, Non local minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.
doi: 10.1002/cpa.20331. |
[5] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. P.D.E. 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[6] |
B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588. |
[7] |
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. |
[8] |
S. Filippas, V. Maz'ya and A. Tertikas, Critical Hardy-Sobolev inequalities, J. Math. Pures Appl., 87 (2007), 37-56.
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, Arch. Ration. Mech. Anal., 208 (2013), 109-161.
doi: 10.1007/s00205-012-0594-4. |
[10] |
R. L. Frank and M. Loss, Hardy-Sobolev-Maz'ya inequalities for arbitrary domains, J. Math. Pures Appl., 97 (2012), 39-54.
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, J. Funct. Anal., 262 (2012), 3159-3185.
doi: 10.1016/j.jfa.2012.01.015. |
[12] |
V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Second, revised and augmented edition. Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-15564-2. |
[13] |
G. Psaradakis, $L^1$ Hardy inequalities with weights, J. Geom. Anal., 23 (2013), 1703-1728.
doi: 10.1007/s12220-012-9302-8. |
[14] |
O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions, Ann. Inst. H. Poincare Anal. Non Linire, 29 (2012), 479-500.
doi: 10.1016/j.anihpc.2012.01.006. |
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