Duke Mathematical Journal
previous :: next

The Yamabe problem on manifolds with boundary: Existence and compactness results

Zheng-Chao Han and Yanyan Li

Source: Duke Math. J. Volume 99, Number 3 (1999), 489-542.

First Page PDF: View first page of article (PDF, 34 KB)

Primary Subjects: 53C21
Secondary Subjects: 35J60, 58J60

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077227912
Mathematical Reviews number (MathSciNet): MR1712631
Zentralblatt MATH identifier: 0945.53023
Digital Object Identifier: doi:10.1215/S0012-7094-99-09916-7

References

[1] Thierry Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), no. 3, 269–296.
Mathematical Reviews (MathSciNet): MR55:4288
Zentralblatt MATH: 0159.23801
[2] A. Bahri, Proof of the Yamabe conjecture without the positive mass conjecture for locally conformally flat manifolds, Nonlinear variational problems and partial differential equations (Isola d'Elba, 1990), Pitman Res. Notes Math. Ser., vol. 320, Longman Sci. Tech., Harlow, 1995, pp. 13–43.
Mathematical Reviews (MathSciNet): MR96d:53038
Zentralblatt MATH: 0839.53029
[3] A. Bahri and H. Brezis, Non-linear elliptic equations on Riemannian manifolds with the Sobolev critical exponent, Topics in geometry, Progr. Nonlinear Differential Equations Appl., vol. 20, Birkhäuser Boston, Boston, MA, 1996, pp. 1–100.
Mathematical Reviews (MathSciNet): MR97c:53056
Zentralblatt MATH: 0863.35037
[4] H. Brezis, Semilinear equations in $\bf R\sp N$ without condition at infinity, Appl. Math. Optim. 12 (1984), no. 3, 271–282.
Mathematical Reviews (MathSciNet): MR86f:35076
Zentralblatt MATH: 0562.35035
Digital Object Identifier: doi:10.1007/BF01449045
[5] Haïm Brezis, Yan Yan Li, and Itai Shafrir, A $\rm sup+\rm inf$ inequality for some nonlinear elliptic equations involving exponential nonlinearities, J. Funct. Anal. 115 (1993), no. 2, 344–358.
Mathematical Reviews (MathSciNet): MR94g:35080
Zentralblatt MATH: 0794.35048
Digital Object Identifier: doi:10.1006/jfan.1993.1094
[6] Luis A. Caffarelli, Basilis Gidas, and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297.
Mathematical Reviews (MathSciNet): MR90c:35075
Zentralblatt MATH: 0702.35085
Digital Object Identifier: doi:10.1002/cpa.3160420304
[7] Sun-Yung A. Chang, Matthew J. Gursky, and Paul C. Yang, The scalar curvature equation on $2$- and $3$-spheres, Calc. Var. Partial Differential Equations 1 (1993), no. 2, 205–229.
Mathematical Reviews (MathSciNet): MR94k:53055
Zentralblatt MATH: 0822.35043
Digital Object Identifier: doi:10.1007/BF01191617
[8] Wen Xiong Chen and Congming Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), no. 3, 615–622.
Mathematical Reviews (MathSciNet): MR93e:35009
Zentralblatt MATH: 0768.35025
Digital Object Identifier: doi:10.1215/S0012-7094-91-06325-8
Project Euclid: euclid.dmj/1077296071
[9] Chiun-Chuan Chen and Chang-Shou Lin, Estimates of the conformal scalar curvature equation via the method of moving planes, Comm. Pure Appl. Math. 50 (1997), no. 10, 971–1017.
Mathematical Reviews (MathSciNet): MR98k:35051
Zentralblatt MATH: 0958.35013
[10] Pascal Cherrier, Problèmes de Neumann non linéaires sur les variétés riemanniennes, J. Funct. Anal. 57 (1984), no. 2, 154–206.
Mathematical Reviews (MathSciNet): MR86c:58154
Zentralblatt MATH: 0552.58032
Digital Object Identifier: doi:10.1016/0022-1236(84)90094-6
[11] M. Chipot, I. Shafrir, and M. Fila, On the solutions to some elliptic equations with nonlinear Neumann boundary conditions, Adv. Differential Equations 1 (1996), no. 1, 91–110.
Mathematical Reviews (MathSciNet): MR96h:35059
Zentralblatt MATH: 0839.35042
[12] José F. Escobar, Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and an eigenvalue estimate, Comm. Pure Appl. Math. 43 (1990), no. 7, 857–883.
Mathematical Reviews (MathSciNet): MR92f:58038
Zentralblatt MATH: 0713.53024
Digital Object Identifier: doi:10.1002/cpa.3160430703
[13] José F. Escobar, Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary, Ann. of Math. (2) 136 (1992), no. 1, 1–50.
Mathematical Reviews (MathSciNet): MR93e:53046
Zentralblatt MATH: 0766.53033
Digital Object Identifier: doi:10.2307/2946545
[14] José F. Escobar, The Yamabe problem on manifolds with boundary, J. Differential Geom. 35 (1992), no. 1, 21–84.
Mathematical Reviews (MathSciNet): MR93b:53030
Zentralblatt MATH: 0771.53017
Project Euclid: euclid.jdg/1214447805
[15] José F. Escobar, Conformal deformation of a Riemannian metric to a constant scalar curvature metric with constant mean curvature on the boundary, Indiana Univ. Math. J. 45 (1996), no. 4, 917–943.
Mathematical Reviews (MathSciNet): MR98d:53051
Zentralblatt MATH: 0881.53037
Digital Object Identifier: doi:10.1512/iumj.1996.45.1344
[16] B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243.
Mathematical Reviews (MathSciNet): MR80h:35043
Zentralblatt MATH: 0425.35020
Digital Object Identifier: doi:10.1007/BF01221125
Project Euclid: euclid.cmp/1103905359
[17] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598.
Mathematical Reviews (MathSciNet): MR83f:35045
Zentralblatt MATH: 0465.35003
Digital Object Identifier: doi:10.1002/cpa.3160340406
[18] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983.
Mathematical Reviews (MathSciNet): MR86c:35035
Zentralblatt MATH: 0562.35001
[19] Z. C. Han and Y. Y. Li, The existence of conformal metrics with constant scalar curvature and constant boundary mean curvature, to appear in Comm. Anal. Geom.
Mathematical Reviews (MathSciNet): MR1792375
Zentralblatt MATH: 0990.53033
[20] Z. C. Han and Y. Y. Li, Further results on the Yamabe problem with boundary, in preparation.
[21] Jerry L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry 10 (1975), 113–134.
Mathematical Reviews (MathSciNet): MR51:1661
Zentralblatt MATH: 0296.53037
Project Euclid: euclid.jdg/1214432678
[22] Oliver Dimon Kellogg, Foundations of potential theory, Reprint from the first edition of 1929. Die Grundlehren der Mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin, 1967.
Mathematical Reviews (MathSciNet): MR36:5369
Zentralblatt MATH: 0152.31301
[23] John M. Lee and Thomas H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91.
Mathematical Reviews (MathSciNet): MR88f:53001
Zentralblatt MATH: 0633.53062
Digital Object Identifier: doi:10.1090/S0273-0979-1987-15514-5
Project Euclid: euclid.bams/1183553962
[24] YanYan Li, The Nirenberg problem in a domain with boundary, Topol. Methods Nonlinear Anal. 6 (1995), no. 2, 309–329.
Mathematical Reviews (MathSciNet): MR97i:35040
Zentralblatt MATH: 0870.35036
[25] Yan Yan Li, Prescribing scalar curvature on $S\sp n$ and related problems. I, J. Differential Equations 120 (1995), no. 2, 319–410.
Mathematical Reviews (MathSciNet): MR98b:53031
Zentralblatt MATH: 0827.53039
Digital Object Identifier: doi:10.1006/jdeq.1995.1115
[26] Yanyan Li, Prescribing scalar curvature on $S\sp n$ and related problems. II. Existence and compactness, Comm. Pure Appl. Math. 49 (1996), no. 6, 541–597.
Mathematical Reviews (MathSciNet): MR98f:53036
Zentralblatt MATH: 0849.53031
[27] Yanyan Li and Meijun Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J. 80 (1995), no. 2, 383–417.
Mathematical Reviews (MathSciNet): MR96k:35061
Zentralblatt MATH: 0846.35050
Digital Object Identifier: doi:10.1215/S0012-7094-95-08016-8
Project Euclid: euclid.dmj/1077246088
[28] Yanyan Li and Meijun Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math. 1 (1999), no. 1, 1–50.
Mathematical Reviews (MathSciNet): MR2000m:53051
Zentralblatt MATH: 0973.53029
Digital Object Identifier: doi:10.1142/S021919979900002X
[29] L. Nirenberg, Topics in nonlinear functional analysis, Courant Institute of Mathematical Sciences New York University, New York, 1974, lecture notes, 1973–1974.
Mathematical Reviews (MathSciNet): MR58:7672
Zentralblatt MATH: 0286.47037
[30] Morio Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry 6 (1971/72), 247–258.
Mathematical Reviews (MathSciNet): MR46:2601
Zentralblatt MATH: 0236.53042
Project Euclid: euclid.jdg/1214430407
[31] Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479–495.
Mathematical Reviews (MathSciNet): MR86i:58137
Zentralblatt MATH: 0576.53028
Project Euclid: euclid.jdg/1214439291
[32] R. Schoen, Courses at Stanford University, 1988, and New York University, 1989.
[33] Richard M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987), Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120–154.
Mathematical Reviews (MathSciNet): MR90g:58023
Zentralblatt MATH: 0702.49038
Digital Object Identifier: doi:10.1007/BFb0089180
[34] Richard M. Schoen, On the number of constant scalar curvature metrics in a conformal class, Differential geometry, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 311–320.
Mathematical Reviews (MathSciNet): MR94e:53035
Zentralblatt MATH: 0733.53021
[35] Richard Schoen and Shing Tung Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), no. 1, 45–76.
Mathematical Reviews (MathSciNet): MR80j:83024
Zentralblatt MATH: 0405.53045
Digital Object Identifier: doi:10.1007/BF01940959
Project Euclid: euclid.cmp/1103904790
[36] R. Schoen and S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), no. 1, 47–71.
Mathematical Reviews (MathSciNet): MR89c:58139
Zentralblatt MATH: 0658.53038
Digital Object Identifier: doi:10.1007/BF01393992
[37] Richard Schoen and Dong Zhang, Prescribed scalar curvature on the $n$-sphere, Calc. Var. Partial Differential Equations 4 (1996), no. 1, 1–25.
Mathematical Reviews (MathSciNet): MR97j:58027
Zentralblatt MATH: 0843.53037
Digital Object Identifier: doi:10.1007/BF01322307
[38] Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 265–274.
Mathematical Reviews (MathSciNet): MR39:2093
Zentralblatt MATH: 0159.23801
[39] Hidehiko Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21–37.
Mathematical Reviews (MathSciNet): MR23:A2847
Zentralblatt MATH: 0096.37201
previous :: next

2010 © Duke University Press