On Some Conformally Invariant Fully Nonlinear Equations

Communications on Pure and Applied Mathematics

Volumn 56, Issue 10, 2003, Pages 1416-1464

  Li, Aobing ^a   Li, Yanyan ^a  

^a RUTGERS UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0041857923     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.10099     Document Type: Article

References (37)

1. Caffarelli, L.; Gidas, B.; Spruck, J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42 (1989), no. 3, 271-297.
2. Caffarelli, L.; Nirenberg, L.; Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math. 155 (1985), no. 3-4, 261-301.
3. Chang, S.-Y. A.; Gursky, M. J.; Yang, P. C. An a priori estimate for a fully nonlinear equation on four-manifolds. J. Anal. Math. 87 (2002), 151-186.
4. Chang, S.-Y. A.; Gursky, M. J.; Yang, P. C. An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature. Ann. of Math. (2) 155 (2002), no. 3, 709-787.
5. Chang, S.-Y. A.; Gursky, M. J.; Yang, P. C. Entire solutions of a fully nonlinear equation. Preprint.
6. Chen, C.-C.; Lin, C.-S. Estimates of the conformal scalar curvature equation via the method of moving planes. Comm. Pure Appl. Math. 50 (1997), no. 10, 971-1017.
7. Cheng, S. Y.; Yau, S. T. Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28 (1975), no. 3, 333-354.
8. Chipot, M., Shafrir, I.; Fila, M. On the solutions to some elliptic equations with nonlinear Neumann boundary conditions. Adv. Differential Equations 1 (1996), no. 1, 91-110.
9. Evans, L. C. Classical solutions of fully nonlinear, convex, second-order elliptic equations. Comm. Pure Appl. Math. 35 (1982), no. 3, 333-363.
10. Gidas, B.; Ni, W. M.; Nirenberg, L. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), no. 3, 209-243.
11. n for surfaces of constant Gauss curvature. Ann. of Math. (2) 138 (1993), no. 3, 601-624.
12. Guan, P.; Viaclovsky, J.; Wang, G. Some properties of the Schouten tensor and applications to conformal geometry. Trans. Amer. Math. Soc. 355 (2003), no. 3, 925-933.
13. Guan, P.; Wang, G. A fully nonlinear conformal flow on locally conformally flat manifolds. Preprint.
14. Guan, P.; Wang, G. Local estimates for a class of fully nonlinear equations arising from conformal geometry. Preprint.
15. Gursky, M. J.; Viaclovsky, J. Fully nonlinear equations on Riemannian manifolds with negative curvature. Preprint.
16. Krylov, N. V. Boundedly inhomogeneous elliptic and parabolic equations in a domain. Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 1, 75-108.
17. Li, A.; Li, Y.Y. On some conformally invariant fully nonlinear equations. C. R. Math. Acad. Sci. Paris 334 (2002), no. 4, 305-310.
18. Li, A.; Li, Y.Y. A fully nonlinear version of the Yamabe problem and a Harnack type inequality. Available online at: http://arxiv.org/abs/math.AP/0212031
19. Li, A.; Li, Y.Y. Further results on Liouville type theorems for some conformally invariant fully nonlinear equations. Available online at: http://arxiv.org/pdf/math.AP/0301254
20. Li, A.; Li, Y.Y. A general Liouville type theorem for some conformally invariant fully nonlinear equations. Available online at: http://arxiv.org/pdf/math.AP/0301239
21. Li, A.; Li, Y.Y. On some conformally invariant fully nonlinear equations, Part II: Liouville, Harnack and Yamabe. In preparation.
22. Li, Y. Y. Degree theory for second order nonlinear elliptic operators and its applications. Comm. Partial Differential Equations 14 (1989), no. 11, 1541-1578.
23. Li, Y. Y. Some existence results for fully nonlinear elliptic equations of Monge-Ampère type. Comm. Pure Appl. Math. 43 (1990), no. 2, 233-271.
24. 3. Comm. Pure Appl. Math. 46 (1993), no. 3, 303-340.
25. Li, Y. Y.; Zhang, L. Liouville type theorems and Harnack type inequalities for semilinear elliptic equations. J. Anal. Math., to appear.
26. Li, Y.; Zhu, M. Uniqueness theorems through the method of moving spheres. Duke Math. J. 80 (1995), no. 2, 383-417.
27. Obata, M. The conjectures on conformal transformations of Riemannian manifolds. J. Differential Geometry 6 (1971/72), 247-258.
28. Reilly, R. C. On the Hessian of a function and the curvatures of its graph. Michigan Math. J. 20 (1973), 373-383.
29. Schoen, R. Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom. 20 (1984), no. 2, 479-495.
30. Schoen, R. Courses at Stanford University, 1988, and New York University, 1989.
31. Schoen, R. M. On the number of constant scalar curvature metrics in a conformal class. Differential geometry, 311-320. Pitman Monographs and Surveys in Pure and Applied Mathematics, 52. Longman, Harlow, 1991.
32. Schoen, R.; Yau, S.-T. Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math. 92 (1988), no. 1, 47-71.
33. Trudinger, N. S. On the Dirichlet problem for Hessian equations. Acta Math. 175 (1995), no. 2, 151-164.
34. Trudinger, N. S.; Wang, X.-J. Hessian measures. II. Ann. of Math. (2) 150 (1999), no. 2, 579-604.
35. Viaclovsky, J. A. Conformal geometry, contact geometry, and the calculus of variations. Duke Math. J. 101 (2000), no. 2, 283-316.
36. Viaclovsky, J. A. Conformally invariant Monge-Ampère equations: global solutions. Trans. Amer. Math. Soc. 352 (2000), no. 9, 4371-4379.
37. Viaclovsky, J. A. Estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. Comm. Anal. Geom. 10 (2002), no. 4, 815-846.