Prescribing scalar and boundary mean curvature on the three dimensional half sphere

Journal of Geometric Analysis

Volumn 13, Issue 2, 2003, Pages 255-289

  Djadli, Zindine ^a   Malchiodi, Andrea ^b   Ahmedou, Ould ^c  

^a UNIVERSITÉ DE CERGY PONTOISE   (France)

^b INSTITUTE FOR ADVANCED STUDY   (United States)

^c UNIVERSITY OF BONN   (Germany)

Author keywords

blow up analysis; conformally invariant equations; continuity method; critical Sobolev exponent; degree arguments

Indexed keywords

EID: 84867968819     PISSN: 10506926     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02930697     Document Type: Article

References (36)

1. S. Agmon A. Douglis L. Nirenberg 1959 Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I Comm. Pure Appl. Math. 12 623 727 0093.10401 10.1002/cpa. 3160120405 125307
2. A. Ambrosetti M. Homoclinics Badiale 1998 Poincaré-Melnikov type results via a variational approach Ann. Inst. Henri. Poincaré Analyse Non Linéaire 15 233 252 1004.37043 10.1016/S0294-1449(97)89300-6 1614571
3. N and related topics J. Funct. Anal. 165 117 149 0938.35056 10.1006/jfan.1999.3390 1696454
4. A. Ambrosetti Y.Y. Li A. Malchiodi 2000 Yamabe and scalar curvature problems under boundary conditions, preprint S.I.S.S.A. Preliminary note on C.R.A.S. 330 1 1013 1018 0969.58005 1779696
5. Aubin, T.Some Nonlinear Problems in Differential Geometry, Springer-Verlag, Berlin, (1998).
6. Bahri, A. Critical points at infinity in some variational problems,Research Notes in Mathematics,182, Longman-Pitman, London, (1989).
7. A. Bahri 1996 An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimension, A celebration of John F. Nash, Jr. Duke Math. J. 81 323 466 0856.53028 10.1215/S0012-7094-96-08116-8 1395407
8. A. Bahri J.M. Coron 1991 The scalar-curvature problem on the standard three-dimensional sphere J. Funct. Anal. 95 106 172 0722.53032 10.1016/0022-1236(91)90026-2 1087949
9. M. Ben Ayed Y. Chen H. Chtioui M. Hammami 1996 On the prescribed scalar curvature problem on 4-manifolds Duke Math. J. 84 633 677 0862.53034 10.1215/S0012-7094-96-08420-3 1408540
10. G. Bianchi X.B. Pan 1999 Yamabe equations on half-spaces Nonlinear Anal. 37 161 186 10.1016/S0362-546X(98)00041-8 1689744
11. K.C. Chang J.Q. Liu 1993 On Nirenberg's problem Int. J. Math. 4 35 58 0786.58010 10.1142/S0129167X93000042 1209959
12. S.A. Chang M.J. Gursky P. Yang 1993 The scalar curvature equation on 2- and 3-spheres Calc. Var. 1 205 229 0822.35043 10.1007/BF01191617 1261723
13. S.A. Chang X. Xu P. Yang 1998 A perturbation result for prescribing mean curvature Math. Ann. 310-3 473 496 1612266
14. 2 Acta Math. 159 215 259 0636.53053 10.1007/BF02392560 908146
15. 2 J. Diff. Geom. 27 256 296 925123
16. n Duke Math. J. 64 27 69 0739.53027 10.1215/S0012-7094-91-06402-1 1131392
17. P. Cherrier 1984 Problémes de Neumann non linéaires sur les variétés Riemanniennes J. Funct. Anal. 57 154 207 0552.58032 10.1016/0022-1236(84)90094-6 749522
18. n with a nonlinear boundary condition J. Math. Anal. Appl. 223 429 471 0932.35086 10.1006/jmaa.1998.5958 1629293
19. Djadli, Z.; Malchiodi, A.; and Ould Ahmedou, M. Prescribing mean curvature on the standard ball, in preparation.
20. J. Escobar 1988 Sharp constant in a Sobolev trace inequality Indiana Univ Math. J. 37 687 698 0666.35014 10.1512/iumj.1988.37.37033 962929
21. J. Escobar 1992 Conformal deformation of a Riemannian metric to a scalar fla metric with constant mean curvature on the boundary Ann. of Math. 136 1 50 10.2307/2946545 1173925
22. J. Escobar 1996 Conformal metrics with prescribed mean curvature on the boundary Cal. Var. 4 559 592 0867.53034 1416000
23. Garcia, G. On conformal metrics on the Euclidean ball, Cornell University, PhD. Thesis, (2000).
24. B. Gidas J. Spruck 1981 Global and local behavior of positive solutions of nonlinear elliptic equations Comm. Pure Appl. Math. 34 525 598 0465.35003 10.1002/cpa.3160340406 615628
25. Z.C. Han Y.Y. Li 1999 The Yamabe problem on manifolds with boundaries: existence and compactness results Duke Math. J. 99 489 542 0945.53023 10.1215/S0012-7094-99-09916-7 1712631
26. Z.C. Han Y.Y. Li 2000 The existence of conformal metrics with constant scalar curvature and constant boundary mean curvature Comm. Anal. Geom. 8 809 869 0990.53033 1792375
27. E. Hebey 1990 Changements de métriques conformes sur la sphère-Le problème de Nirenberg Bull. Sci. Math. 114 215 242 0713.53023 1056162
28. P.L. Li J.Q. Liu 1993 Nirenberg's problem on the two-dimensional hemi-sphere Int. J. Math. 4 927 939 0819.35044 10.1142/S0129167X9300042X
29. n and related topics Part I J. Diff. Eq. 120 319 410 0827.53039 10.1006/jdeq.1995.1115
30. n and related topics Part II, Existence and compactness Comm. Pure Appl. Math. 49 437 477 10.1002/(SICI)1097-0312(199611)49:11<1175::AID-CPA2>3.0.CO;2-9
31. Y.Y. Li 1995 The Nirenberg problem in a domain with boundary Top. Meth. Nonlin. Anal. 6 309 329 0870.35036
32. Li, Y.Y. and Zhang, L. Liouville type theorems and Harnack type inequalities for semilinear elliptic equations, preprint, (2001).
33. Y.Y. Li M.J. Zhu 1995 Uniqueness theorems through the method of moving spheres Duke Math. J. 80 383 417 0846.35050 10.1215/S0012-7094-95-08016-8 1369398
34. Schoen, R. On the number of constant scalar curvature metrics in a conformal class, inDifferential Geometry: A Symposium in Honor of Manfredo Do Carmo, Lawson, H.B. and Tenenblat, F. Eds.; John Wiley & Sons, New York, 331-320, (1991).
35. R. Schoen S.T. Yau 1988 Conformally flat manifolds, Kleinian groups, and scalar curvature Invent. Math. 92 47 71 0658.53038 10.1007/BF01393992 931204
36. R. Schoen D. Zhang 1996 Prescribed scalar curvature on then-sphere Calculus of Variations and Partial Differential Equations 4 1 25 0843.53037 1379191 10.1007/BF01322307