Calculus of variations
Regularity of weak constant anisotropic mean curvature surfaces
Comptes Rendus. Mathématique, Volume 344 (2007) no. 9, pp. 603-606.

In this Note a definition of weak constant anisotropic mean curvature surfaces and the expression in conformal coordinates of the anisotropic mean curvature of surfaces in R3 are obtained. Moreover, we prove that all weak constant anisotropic mean curvature surfaces in R3 are continuous.

Dans cette Note on donne une définition des solutions faibles du problème des surfaces à courbure moyenne anisotropique constante ; dans R3 on donne une représentation en coodonnées conformes des solutions. De plus, dans la cas de R3, nous démontrons la continuité de toutes les solutions.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.03.004
Zhai, Jian 1

1 Department of Mathematics, Zhejiang University, Hangzhou 310027, PR China
@article{CRMATH_2007__344_9_603_0,
     author = {Zhai, Jian},
     title = {Regularity of weak constant anisotropic mean curvature surfaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {603--606},
     publisher = {Elsevier},
     volume = {344},
     number = {9},
     year = {2007},
     doi = {10.1016/j.crma.2007.03.004},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2007.03.004/}
}
TY  - JOUR
AU  - Zhai, Jian
TI  - Regularity of weak constant anisotropic mean curvature surfaces
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 603
EP  - 606
VL  - 344
IS  - 9
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2007.03.004/
DO  - 10.1016/j.crma.2007.03.004
LA  - en
ID  - CRMATH_2007__344_9_603_0
ER  - 
%0 Journal Article
%A Zhai, Jian
%T Regularity of weak constant anisotropic mean curvature surfaces
%J Comptes Rendus. Mathématique
%D 2007
%P 603-606
%V 344
%N 9
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2007.03.004/
%R 10.1016/j.crma.2007.03.004
%G en
%F CRMATH_2007__344_9_603_0
Zhai, Jian. Regularity of weak constant anisotropic mean curvature surfaces. Comptes Rendus. Mathématique, Volume 344 (2007) no. 9, pp. 603-606. doi : 10.1016/j.crma.2007.03.004. http://www.numdam.org/articles/10.1016/j.crma.2007.03.004/

[1] Douglas, J. Solution of the problem of Plateau, Trans. Amer. Math. Soc., Volume 33 (1931), pp. 263-321

[2] Y. Giga, Surface evolution equations – a level set method, Hokkaido University Technical Report Series in Mathematics #71, 2002

[3] Y. Giga, J. Zhai, Uniqueness of constant weakly anisotropic mean curvature immersion of sphere S2 to R3, Hokkaido University Preprint Series in Mathematics #794, 2006

[4] Grüter, M. Regularity of weak H-surfaces, J. Reine Angew. Math., Volume 329 (1981), pp. 1-15

[5] Heinz, E. Über die Existenz einer Fläche konstanter mittlerer Krümmung bei vorgegebener Berandung, Math. Ann., Volume 127 (1954), pp. 258-287

[6] Hildebrandt, S. On the Plateau problem for surfaces of constant mean curvature, Comm. Pure Appl. Math., Volume 23 (1970), pp. 97-114

[7] Hildebrandt, H.; Kaul, H. Two dimensional variational problems with obstructions, and Plateau's problem for H-surfaces in a Riemannian manifold, Comm. Pure Appl. Math., Volume 25 (1972), pp. 187-223

[8] Jost, J. Two-Dimensional Geometric Variational Problems, John Wiley & Sons, 1991

[9] Rado, T. On the problem of Plateau, Ann. of Math., Volume 31 (1930), pp. 457-469

[10] Taylor, J.E. Mean curvature and weighted mean curvature, Acta Metall. Mater., Volume 40 (1992) no. 7, pp. 1475-1485

[11] Wente, H.C. An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl., Volume 26 (1969), pp. 318-344

[12] Wente, H.C. A general existence theorem for surfaces of constant mean curvature, Math. Z., Volume 120 (1971), pp. 277-288

[13] Werner, H. Problem von Douglas für Flächen konstanter mittlerer Krümmung, Math. Ann., Volume 135 (1957), pp. 303-319

Cited by Sources: