A Harnack type inequality for the Yamabe equation in low dimensions

Calculus of Variations and Partial Differential Equations

Volumn 20, Issue 2, 2004, Pages 133-151

  Li, Yan Yan ^a   Zhang, Lei ^b  

^a RUTGERS UNIVERSITY   (United States)

^b TEXAS A AND M UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 2442690399     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00526-003-0230-0     Document Type: Article

References (42)

1. Alexandrov, A.D.: Uniqueness theorems for surfaces in the large. V, Vestnik Leningrad Univ. Mat. Mekh. Astronom 13, 5-8 (1958); Amer. Math. Soc. Transl. 21, 412-416 (1962)
2. Alexandrov, A.D.: Uniqueness theorems for surfaces in the large. V, Vestnik Leningrad Univ. Mat. Mekh. Astronom 13, 5-8 (1958); Amer. Math. Soc. Transl. 21, 412-416 (1962)
3. Aubin, T.: Équations différentielles non linéaires et probléme de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55, 269-296 (1976)
4. Bahri, A., Brezis, H.: Non-linear elliptic equations on Riemannian manifolds with the Sobolev critical exponent. In: Topics in geometry, pp. 1-100. Progr. Nonlinear Differential Equations Appl., 20, Birkhäuser Boston, Boston, MA, 1996
5. Berestycki, H., Nirenberg, L.: On the method of moving planes and the sliding method. Bol. Soc. Brasil. Mat. (N.S.) 22, 1-37 (1991)
6. Blanchi, G.: Non-existence and symmetry of solutions to the scalar curvature equation. Comm. Partial Differential Equations 21, 229-234 (1996)
7. n. Math. Z. 210, 137-166 (1992)
8. u in two dimension. Comm. Partial Differential Equation 16, 1223-1253 (1991)
9. Brezis, H., Li, Y.Y., Shafrir, I.: A sup + inf inequality for some nonlinear elliptic equations involving exponential nonlinearities. J. Functional Analysis 115, 344-358 (1993)
10. Caffarelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42, 271-297 (1989)
11. Chang, S.Y,. Gursky, M.J., Yang, P.: The scalar curvature equation on 2- and 3-sphere. Calc. Var. 1, 205-229 (1993)
12. Chen, C.C., Lin, C.S.: Estimates of the conformal scalar curvature equation via the method of moving planes. Comm. Pure Appl. Math. 50, 971-1017 (1997)
13. Chen, C.C., Lin, C.S.: Estimates of the conformal scalar curvature equation via the method of moving planes, II. J. Diff. Geom. 49, 115-178 (1998)
14. 2. Comm. Anal. Geom. 6, 1-19 (1998)
15. n. Comm. Partial Differential Equations 24, 785-799 (1999)
16. Escobar, J., Schoen, R.: Conformal metrics with prescribed scalar curvature, Invent. Math. 86, 243-254 (1986)
17. Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209-243 (1979)
18. Lee, J., Parker, T.: The Yamabe problem. Bull. Amer. Math. Soc. (N.S.) 17, 37-91 (1987)
19. Lelong-Ferrand, J.: Transformations conformes et quasi conformes des variétés riemanniennes compactes. Mem. Acad. Royal de Belgique 39, 1-44 (1971)
20. Li, A., Li, Y.Y.: On some conformally invariant fully nonlinear equations. Preprint
21. Li, A., Li, Y.Y.: A fully nonlinear version of the Yamabe problem and a Harnack type inequality. arXiv:math.AP/0212031 v1 2 Dec 2002
22. Li, A., Li, Y.Y.: General Liouville type theorem for some conformally invariant fully nonlinear equations. arXiv:math.AP/0301239 v1 21 Jan 2003
23. Li, A., Li, Y.Y.: Further results on Liouville type theorems for some conformally invariant fully nonlinear equations. arXiv:math.AP/0301254 v1 22 Jan 2003
24. Li, A., Li, Y.Y.: On some conformally invariant fully nonlinear equations, Part II. Liouville, Harnack and Yamabe (in preparation)
25. n and related problems, Part I. J. Diff. Equations 120, 319-410 (1995)
26. n and related problems, Part II: Existence and compactness. Comm. Pure Appl. Math. 49, 541-597 (1996)
27. Li, Y.Y.: A Harnack type inequality: the method of moving planes. Comm. Math. Phys. 200, 421-444 (1999)
28. Li, Y.Y., Zhang, L.: Liouville type theorems and Harnack type inequalities for semilinear elliptic equations. Journal d'Analyse Mathematique (to appear)
29. Li, Y.Y., Zhu, M.: Yamabe type equations on three dimensional Riemannian manifolds. Communications in Contemporary Math. 1, 1-50 (1999)
30. Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom. 20, 479-495 (1984)
31. Schoen, R.: Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In: Giaquinta, M. (ed.) Topics in Calculus of Variations (Lecture notes in mathematics, No. 1365), pp. 120-154. Springer-Verlag 1989
32. Schoen, R.: Courses at Stanford University (1988) and New York University (1989) (unpublished)
33. Schoen, R.: On the number of constant scalar curvature metrics in a conformal class. Differential Geometry: A symposium in honor of Manfredo Do Carmo (H.B. Lawson and K. Tenenblat, eds), Wiley, 1991, pp. 311-320
34. Schoen, R.: On the conformal and CR automorphism groups. Geom. Funct. Anal. 5, 464-481 (1995)
35. Schoen, R., Yau, S.T.: On the proof of the positive mass conjecture in General Relativity. Comm. Math. Phys. 65, 45-76 (1979)
36. Schoen, R., Yau, S.T.: Conformally flat manifolds, Kleinian groups, and scalar curvature. Invent. Math. 92, 47-71 (1988)
37. Schoen, R., Zhang, D.: Prescribed scalar curvature on the n-sphere. Calc. Var. 4, 1-25 (1996)
38. Serrin, J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43, 304-18 (1971)
39. Siu, Y-T: The existence of Ks̈hler-Einstein metrics on manifolds with positive anti-canonical line bundle and a suitable finite symmetry group. Ann. Math. 127, 585-627 (1988)
40. Tian, G.: A Harnack type inequality for certain complex Monge-Ampère equations. J. Differ. Geom. 29, 481-488 (1989)
41. Trudinger, N.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Cl. Sci. 22, 265-274 (1968)
42. Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21-37 (1960)