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Curvature functions for 2-manifolds with negative Euler characteristic. (English) Zbl 0255.53028


MSC:

53A99 Classical differential geometry
35J25 Boundary value problems for second-order elliptic equations
35J60 Nonlinear elliptic equations
53C99 Global differential geometry
53C20 Global Riemannian geometry, including pinching
58J99 Partial differential equations on manifolds; differential operators
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References:

[1] Donald S. Cohen, Multiple stable solutions of nonlinear boundary value problems arising in chemical reactor theory, SIAM J. Appl. Math. 20 (1971), 1 – 13. · Zbl 0219.34027 · doi:10.1137/0120001
[2] R. Courant, Methods of mathematical physics. Vol. II: Partial differential equations, Interscience, New York, 1962. MR 25 # 4216. · Zbl 0099.29504
[3] Herman Gluck, The generalized Minkowski problem in differential geometry in the large, Ann. of Math. (2) 96 (1972), 245 – 276. · Zbl 0243.53046 · doi:10.2307/1970788
[4] Jerry L. Kazdan and F. W. Warner, Integrability conditions for \Delta u = k - Ke, Bull. Amer. Math. Soc. 77 (1971), 819-823. · Zbl 0218.35030
[5] Jerry L. Kazdan and F. W. Warner, Curvature functions for 2-manifolds, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1973, pp. 387 – 392.
[6] Jerry L. Kazdan and F. W. Warner, Curvature functions for 2-manifolds, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1973, pp. 387 – 392.
[7] Jerry L. Kazdan and F. W. Warner, Curvature functions for 2-manifolds, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1973, pp. 387 – 392.
[8] Jerry L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry 10 (1975), 113 – 134. · Zbl 0296.53037
[9] Jerry L. Kazdan and F. W. Warner, Remarks on some nonlinear elliptic equations (to appear). · Zbl 0325.35038
[10] M. A. Krasnosel\(^{\prime}\)skiĭ, Positive solutions of operator equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd. Groningen, 1964.
[11] J. Moser, On a nonlinear problem in differential geometry, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 273 – 280.
[12] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 979 – 1000. · Zbl 0223.35038 · doi:10.1512/iumj.1972.21.21079
[13] R. Bruce Simpson and Donald S. Cohen, Positive solutions of nonlinear elliptic eigenvalue problems, J. Math. Mech. 19 (1969/70), 895 – 910. · Zbl 0195.39502
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