On some conformally invariant fully nonlinear equations
Comptes Rendus. Mathématique, Volume 334 (2002) no. 4, pp. 305-310.

We outline proofs of our results in [7] on Liouville type theorems, Harnack type inequalities, and existence and compactness of solutions to some conformally invariant fully nonlinear elliptic equations of second order on locally conformally flat Riemannian manifolds. Details will appear in [7].

On présente des résultats de type Liouville, des inégalités de type Harnack ainsi que d'existence et de compacité de solutions pour certaines équations elliptiques du second ordre, complètement nonlinéaires, sur des variétés Riemanniennes localement conformément plates. Les démonstrations détaillées sont contenues dans [7].

Received:
Published online:
DOI: 10.1016/S1631-073X(02)02264-1
Li, Aobing 1; Li, YanYan 1

1 Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA
@article{CRMATH_2002__334_4_305_0,
     author = {Li, Aobing and Li, YanYan},
     title = {On some conformally invariant fully nonlinear equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {305--310},
     publisher = {Elsevier},
     volume = {334},
     number = {4},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02264-1},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02264-1/}
}
TY  - JOUR
AU  - Li, Aobing
AU  - Li, YanYan
TI  - On some conformally invariant fully nonlinear equations
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 305
EP  - 310
VL  - 334
IS  - 4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(02)02264-1/
DO  - 10.1016/S1631-073X(02)02264-1
LA  - en
ID  - CRMATH_2002__334_4_305_0
ER  - 
%0 Journal Article
%A Li, Aobing
%A Li, YanYan
%T On some conformally invariant fully nonlinear equations
%J Comptes Rendus. Mathématique
%D 2002
%P 305-310
%V 334
%N 4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(02)02264-1/
%R 10.1016/S1631-073X(02)02264-1
%G en
%F CRMATH_2002__334_4_305_0
Li, Aobing; Li, YanYan. On some conformally invariant fully nonlinear equations. Comptes Rendus. Mathématique, Volume 334 (2002) no. 4, pp. 305-310. doi : 10.1016/S1631-073X(02)02264-1. http://www.numdam.org/articles/10.1016/S1631-073X(02)02264-1/

[1] Caffarelli, L.; Gidas, B.; Spruck, J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., Volume 42 (1989), pp. 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., Volume 155 (1985), pp. 261-301

[3] S.Y.A. Chang, M. Gursky, P. Yang, An equation of Monge–Ampere type in conformal geometry, and four-manifolds of positive Ricci curvature, Ann. of Math. (to appear)

[4] S.Y.A. Chang, M. Gursky, P. Yang, An a priori estimate for a fully nonlinear equation on four-manifolds, Preprint

[5] Gidas, B.; Ni, W.M.; Nirenberg, L. Symmetry and related properties via the maximum principle, Comm. Math. Phys., Volume 68 (1979), pp. 209-243

[6] P. Guan, G. Wang, Local estimates for a class of fully nonlinear equations arising from conformal geometry, Preprint

[7] A. Li, Y.Y. Li, On some conformally invariant fully nonlinear equations (in preparation)

[8] Li, Y.Y. Some existence results of fully nonlinear elliptic equations of Monge–Ampere type, Comm. Pure Appl. Math., Volume 43 (1990), pp. 233-271

[9] Y.Y. Li, L. Zhang, Liouville type theorems and Harnack type inequalities for semilinear elliptic equations, J. Anal. Math. (to appear)

[10] Obata, M. The conjecture on conformal transformations of Riemannian manifolds, J. Differential Geom., Volume 6 (1971), pp. 247-258

[11] Schoen, R. Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., Volume 20 (1984), pp. 479-495

[12] Schoen, R. On the number of constant scalar curvature metrics in a conformal class (Lawson, H.B.; Tenenblat, K., eds.), Differential Geometry: A Symposium in Honor of Manfredo Do Carmo, Wiley, 1991, pp. 311-320

[13] R. Schoen, Courses at Stanford University, 1988, and New York University, 1989

[14] Trudinger, N.; Wang, X. Hessian measures II, Ann. of Math., Volume 150 (1999), pp. 579-604

[15] J. Viaclovsky, Estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds, Comm. Anal. Geom. (to appear)

[16] Viaclovsky, J. Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J., Volume 101 (2000), pp. 283-316

[17] Viaclovsky, J. Conformally invariant Monge–Ampere equations: global solutions, Trans. Amer. Math. Soc., Volume 352 (2000), pp. 4371-4379

Cited by Sources: