×

The Christoffel-Minkowski problem. III: Existence and convexity of admissible solutions. (English) Zbl 1284.35174

From the text: This paper is a sequel to Part I [Invent. Math. 151, No. 3, 553–577 (2003; Zbl 1213.35213)] on geometric fully nonlinear partial differential equations associated to the Christoffel-Minkowski problem. Here the authors establish a convexity criterion for the main equation (full rank theorem), a necessary and sufficient condition for the existence of admissible solutions and an existence theorem imposing a group-invariant condition.
For Part II, see Chin. Ann. Math., Ser. B 27, No. 6, 595–614 (2006; Zbl 1151.35349).
In the original title Christofel should be corrected to Christoffel.

MSC:

35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
53C65 Integral geometry
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Alexandrov, Mat Sbornik NS 3 pp 27– (1938)
[2] Alexandrov, Vestnik Leningrad Univ 11 pp 5– (1956)
[3] Amer Math Soc Trans (2) 21 pp 341– (1962) · doi:10.1090/trans2/021/09
[4] Caffarelli, Duke Math J 52 pp 431– (1985)
[5] Caffarelli, Acta Math 155 pp 261– (1985)
[6] ; ; Nonlinear second order elliptic equations. IV. Starshaped compact Weingarten hypersurfaces. Current topics in partial differential equations, 1–26. Kinokuniya, Tokyo, 1986.
[7] Cheng, Comm Pure Appl Math 29 pp 495– (1976)
[8] Chern, J Math Mech 8 pp 947– (1959)
[9] Christoffel, J Reine Angew Math 64 pp 193– (1865) · ERAM 064.1676cj · doi:10.1515/crll.1865.64.193
[10] Firey, Mathematika 14 pp 1– (1967)
[11] G??rding, J Math Mech 8 pp 957– (1959)
[12] Guan, Ann of Math (2) 156 pp 655– (2002)
[13] Guan, Duke Math J 86 pp 323– (1997)
[14] Guan, Comm Pure Appl Math 50 pp 789– (1997)
[15] Guan, Math Z 247 pp 1– (2004)
[16] Guan, Invent Math 151 pp 553– (2003)
[17] Guan, Int Math Res Not 2003 pp 1413–
[18] Ivochkina, Soviet Math Dokl 37 pp 322– (1988)
[19] Korevaar, Arch Rational Mech Anal 97 pp 19– (1987)
[20] Krylov, Trans Amer Math Soc 347 pp 857– (1995)
[21] Li, Comm Partial Differential Equations 14 pp 1541– (1989)
[22] Nirenberg, Comm Pure Appl Math 6 pp 337– (1953)
[23] Pogorelov, Uspekhi Mat Nauk 8 pp 127– (1953)
[24] Extrinsic geometry of convex surfaces. Translations of Mathematical Monographs, 35. American Mathematical Society, Providence, R.I., 1973.
[25] The Minkowski multidimensional problem. Scripta Series in Mathematics. Translated from the Russian by Vladimir Oliker. Winston, Washington, D.C.; Halsted, New York–Toronto–London, 1978.
[26] Urbas, J Differential Geom 33 pp 91– (1991)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.