We give a new proof of a classical uniqueness theorem of Alexandrov [4] using the weak uniqueness continuation theorem of Bers–Nirenberg [8]. We prove a version of this theorem with the minimal regularity assumption: the spherical Hessians of the corresponding convex bodies as Radon measures are nonsingular.
Mots clés : Uniqueness, Convex surfaces, Nonlinear elliptic equations, Unique continuation, Alexandrov theorem
@article{AIHPC_2016__33_2_329_0,
author = {Guan, Pengfei and Wang, Zhizhang and Zhang, Xiangwen},
title = {A proof of {Alexandrov's} uniqueness theorem for convex surfaces in $ {\mathbb{R}}^{3}$},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {329--336},
publisher = {Elsevier},
volume = {33},
number = {2},
year = {2016},
doi = {10.1016/j.anihpc.2014.09.011},
zbl = {1335.53092},
mrnumber = {3465378},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.09.011/}
}
TY - JOUR
AU - Guan, Pengfei
AU - Wang, Zhizhang
AU - Zhang, Xiangwen
TI - A proof of Alexandrov's uniqueness theorem for convex surfaces in $ {\mathbb{R}}^{3}$
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2016
SP - 329
EP - 336
VL - 33
IS - 2
PB - Elsevier
UR - http://www.numdam.org/articles/10.1016/j.anihpc.2014.09.011/
DO - 10.1016/j.anihpc.2014.09.011
LA - en
ID - AIHPC_2016__33_2_329_0
ER -
%0 Journal Article
%A Guan, Pengfei
%A Wang, Zhizhang
%A Zhang, Xiangwen
%T A proof of Alexandrov's uniqueness theorem for convex surfaces in $ {\mathbb{R}}^{3}$
%J Annales de l'I.H.P. Analyse non linéaire
%D 2016
%P 329-336
%V 33
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2014.09.011/
%R 10.1016/j.anihpc.2014.09.011
%G en
%F AIHPC_2016__33_2_329_0
Guan, Pengfei; Wang, Zhizhang; Zhang, Xiangwen. A proof of Alexandrov's uniqueness theorem for convex surfaces in $ {\mathbb{R}}^{3}$. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 2, pp. 329-336. doi : 10.1016/j.anihpc.2014.09.011. http://www.numdam.org/articles/10.1016/j.anihpc.2014.09.011/
[1] Zur Theorie der gemischten Volumina von konvexen Körpern, II. Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen, Mat. Sb. (N.S.), Volume 2 (1937), pp. 1205–1238 (in Russian) | JFM
[2] Zur Theorie der gemischten Volumina von konvexen Körpern, III. Die Erweiterung zweeier Lehrsätze Minkowskis über die konvexen Polyeder auf beliebige konvexe Flächen, Mat. Sb. (N.S.), Volume 3 (1938), pp. 27–46 (in Russian) | JFM | Zbl
[3] Sur les théorèmes d'unicité pour les surfaces fermeès, C. R. (Dokl.) Acad. Sci. URSS, N.S., Volume 22 (1939), pp. 99–102 (translation Selected Works. Part I. Selected Scientific Papers Class. Sov. Math., vol. 4, 1996, 149–153) | JFM | MR | Zbl
[4] Uniqueness theorems for surfaces in the large. I, Vestn. Leningr. Univ., Volume 11 (1956) no. 19, pp. 5–17 (in Russian) | MR
[5] Uniqueness theorems for surfaces in the large. II, Vestn. Leningr. Univ., Volume 12 (1957) no. 7, pp. 15–44 (in Russian) | MR
[6] On the curvature of surfaces, Vestn. Leningr. Univ., Volume 21 (1966) no. 19, pp. 5–11 (in Russian) | MR
[7] , Edizioni cremonese dells S.A. editrice perrella, Roma (1954), pp. 111–140 | MR | Zbl
[8] Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali (1954), pp. 141–167 (Trieste) | MR | Zbl
[9] Measure Theory and Fine Properties of Functions, CRC Press Inc., 1992 | MR
[10] Mengenfunktionen und konvexe Körper, Det. Kgl. Danske Videnskab. Selskab, Math.-fys. Medd., Volume 16 (1938) no. 3, pp. 1–31 | JFM | Zbl
[11] Convex hypersurfaces of prescribed curvatures, Ann. Math., Volume 256 (2002) no. 2, pp. 655–673 | DOI | MR | Zbl
[12] Elliptic Partial Differential Equations of Second Order, Springer, 1998 | MR | Zbl
[13] Linearity of homogeneous order-one solutions to elliptic equations in dimension three, Commun. Pure Appl. Math., Volume 56 (2003), pp. 425–432 | MR | Zbl
[14] On the third fundamental form of a surface, Am. J. Math., Volume 75 (1953), pp. 298–334 | DOI | MR | Zbl
[15] Contre-example à une caracteérisation conjecturée de; as sphére, C. R. Acad. Sci. Paris Sér. I Math., Volume 332 (2001), pp. 41–44 | DOI | MR | Zbl
[16] On the solutions of quasi-linear elliptic partial differential equations, Trans. Am. Math. Soc., Volume 43 (1938) no. 1, pp. 126–166 | DOI | JFM | MR
[17] The Weyl and Minkowski problems in differential geometry in the large, Commun. Pure Appl. Math., Volume 6 (1953), pp. 337–394 | DOI | MR | Zbl
[18] On nonlinear elliptic partial differential equations and Hölder continuity, Commun. Pure Appl. Math., Volume 6 (1953), pp. 103–156 | MR | Zbl
[19] New counterexamples to A. D. Alexandrov's hypothesis, Adv. Geom., Volume 5 (2005), pp. 301–317 | DOI | MR | Zbl
[20] Extension of a general uniqueness theorem of A. D. Aleksandrov to the case of nonanalytic surfaces, Dokl. Akad. Nauk SSSR (N.S.), Volume 62 (1948), pp. 297–299 (in Russian) | MR | Zbl
[21] Extrinsic Geometry of Convex Surfaces, Transl. Math. Monogr., vol. 35, American Mathematical Society, Providence, RI, 1973 (translated from Russian by Israel Program for Scientific Translations) | MR | Zbl
[22] Solution of a problem of A. D. Aleksandrov, Dokl. Akad. Nauk, Volume 360 (1998), pp. 317–319 (in Russian) | MR | Zbl
[23] Uniqueness theorems for closed convex surfaces, Dokl. Akad. Nauk, Volume 366 (1999), pp. 602–604 (in Russian) | MR | Zbl
[24] Convex Bodies: The Brunn–Minkowski Theory, Encycl. Math. Appl., vol. 44, Cambridge Univ. Press, Cambridge, 1993 | MR | Zbl
[25] Über die Bestimmung einer geschlossenen konvexen Fläche durch ihr Linienelement, Vierteljahrschrift Naturforsch. Gesell. Zurich, Volume 3 (1916) no. 2, pp. 40–72 | JFM
Cité par Sources :
