Sharp constants in weighted trace inequalities on Riemannian manifolds

Calculus of Variations and Partial Differential Equations

Volumn 48, Issue 3-4, 2013, Pages 555-585

  Jin, Tianling ^a   Xiong, Jingang ^a,b  

^a RUTGERS UNIVERSITY   (United States)

^b BEIJING NORMAL UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84885586318     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00526-012-0562-8     Document Type: Article

References (43)

1. Adimurthi, Yadava S. L.: Some remarks on Sobolev type inequalities. Calc. Var. Partial Differ. Equ. 2, 427-442 (1994).
2. Aubin T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573-598 (1976).
3. Aubin T.: Some Nonlinear Problems in Riemannian Geometry. Springer Monographs in Mathematics. Springer, Berlin (1998).
4. Aubin T., Li Y. Y.: On the best Sobolev inequality. J. Math. Pures Appl. 78, 353-387 (1999).
5. Beckner W.: Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. Ann. Math. (2) 138, 213-242 (1993).
6. Boccardo L., Gallouët T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149-169 (1989).
7. 1. J. Math. Soc. Jpn. 25, 565-590 (1973).
8. Cabre, X.; Sire, Y.: Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates. arXiv: 1012. 0867.
9. Caffarelli L., Silvestre L.: An extension problem related to the fractional Laplacian. Commun. Partial. Differ. Equ. 32, 1245-1260 (2007).
10. Chang S.-Y., González M.: Fractional Laplacian in conformal geometry. Adv. Math. 226, 1410-1432 (2011).
11. Druet O.: The best constants problem in Sobolev inequalities. Math. Ann. 314, 327-346 (1999).
12. Druet O.: Isoperimetric inequalities on compact manifolds. Geom. Dedicata 90, 217-236 (2002).
13. Druet, O.; Hebey, E. (2002) The AB program in geometric analysis: sharp Sobolev inequalities and related problems. Mem. Amer. Math. Soc. 160.
14. Escobar J. F.: Sharp constant in a Sobolev trace inequality. Indiana Univ. Math. J. 37, 687-698 (1988).
15. Escobar J. F.: Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary. Ann. Math. (2) 136, 1-50 (1992).
16. Fabes E., Jerison D., Kenig C.: The Wiener test for degenerate elliptic equations. Annales de l'institut Fourier 32, 151-182 (1982).
17. Fabes E., Kenig C., Serapioni R.: The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differ. Equ. 7, 77-116 (1982).
18. Gallouët, T.; Sire, Y.: Some possibly degenerate elliptic problems with measure data and non linearity on the boundary. arXiv: 1002. 4982v1.
19. González M.: Gamma convergence of an energy functional related to the fractional Laplacian. Calc. Var. Partial Differ. Equ. 36, 173-210 (2009).
20. González M., Mazzeo R., Sire Y.: Singular solutions of fractional order conformal laplacians. J. Geom. Anal. 22, 845-863 (2012).
21. González, M.; Qing, J.: Fractional conformal Laplacians and fractional Yamabe problems. arXiv: 1012. 0579v1.
22. Graham C. R., Zworski M.: Scattering matrix in conformal geometry. Invent. Math. 152, 89-118 (2003).
23. Gurka P., Opic B.: Continuous and compact imbeddings of weighted Sobolev spaces II. Czechoslovak Math. J. 39, 78-94 (1989).
24. Hebey, E.: Nonlinear analysis on manifolds: Sobolev spaces and inequalities. Courant Lecture Notes in Math., 5. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI (1999).
25. Hebey E., Vaugon M.: Meilleures constantes dans le théorème d'inclusion de Sobolev. Ann. Inst. H. Poincare 13, 57-93 (1996).
26. Jin, T., Li, Y. Y., Xiong, J.: On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions. arXiv: 1111. 1332v1.
27. Jin, T., Li, Y. Y., Xiong, J.: On a fractional Nirenberg problem, part II: existence of solutions. in preparation (2012).
28. Jin, T., Xiong, J.: A fractional Yamabe flow and some applications. arXiv: 1110. 5664v1.
29. Kenig C., Pipher J.: The Neumann problem for elliptic equations with non-smooth coefficients. Invent. Math. 113, 447-509 (1993).
30. Kufner A.: Weighted Sobolev Spaces. Wiley, New York (1985).
31. Li Y. Y., Ricciardi T.: A sharp Sobolev inequality on Riemannian manifolds. Commun. Pure Appl. Anal. 2, 1-31 (2003).
32. Li Y. Y., Zhu M.: Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries. Commun. Pure Appl. Math. 50, 449-487 (1997).
33. Li Y. Y., Zhu M.: Sharp Sobolev inequalities involving boundary terms. Geom. Funct. Anal. 8, 59-87 (1998).
34. Lieb E. H.: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. Math. (2) 118, 349-374 (1983).
35. Lions P.-L.: The concentration-compactness principle in the calculus of variations, The limit case, II. Rev. Mat. Iberoamericana 1, 45-121 (1985).
36. Maz'ya, V.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Second, revised and augmented edition. Springer, Heidelberg (2011).
37. Mazzeo R.: The Hodge cohomology of a conformally compact metric. J. Differ. Geom. 28, 309-339 (1988).
38. Mazzeo R., Melrose R.: Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal. 75, 260-310 (1987).
39. M) on M. Czechoslovak Math. J. 43, 695-711 (1993).
40. Palatucci G., Sire Y.: Γ-convergence of some super quadratic functionals with singular weights. Math. Z. 266, 533-560 (2010).
41. Qing, J., Raske, D.: On positive solutions to semilinear conformally invariant equations on locally conformally flat manifolds. Int. Math. Res. Not. Art. ID 94172 (2006).
42. Schoen, R., Yau, S. T.: Lectures on Differential Geometry. International Press, Cambridge (1994).
43. Tan J., Xiong J.: A Harnack inequality for fractional Laplace equations with lower order terms. Discret. Contin. Dyn. Syst. 31, 975-983 (2011).