Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

A generalized Minkowski problem with Dirichlet boundary condition

Author(s): Oliver C. Schnurer
Journal: Trans. Amer. Math. Soc. 355 (2003), 655-663.
MSC (2000): Primary 35J65; Secondary 53C42
Posted: September 6, 2002
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: We prove the existence of hypersurfaces with prescribed boundary whose Weingarten curvature equals a given function that depends on the normal of the hypersurface.

References:

1.
L. Caffarelli, L. Nirenberg, J. Spruck: The Dirichlet Problem for Nonlinear Second-Order Elliptic Equations I. Monge-Ampère Equation, Comm. Pure Applied Math. 37 (1984), 369-402. MR 87f:35096
2.
L. Caffarelli, L. Nirenberg, J. Spruck: The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), 261-301. MR 87f:35098
3.
S.-Y. Cheng, S.-T. Yau: On the Regularity of the Solution of the $n$-Dimensional Minkowski Problem, Comm. Pure Appl. Math. 29 (1976), 495-516. MR 54:11247
4.
C. Gerhardt: Closed Weingarten hypersurfaces in Riemannian manifolds, J. Differential Geom. 43 (1996), 612-641. MR 97g:53067
5.
B. Guan, P. Guan: Convex hypersurfaces of prescribed curvature, to appear in Ann. of Math..
6.
B. Guan: The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauß curvature, Trans. Amer. Math. Soc. 350, No. 12, (1998), 4955-4971. MR 99b:53055
7.
B. Guan, J. Spruck: Boundary-value problems on $\mathbb{S} ^n$ for surfaces of constant Gauss curvature, Ann. Math. 138 (1993), 601-624. MR 94i:53039
8.
N. M. Ivochkina, F. Tomi: Locally convex hypersurfaces of prescribed curvature and boundary, Calc. Var. and PDEs 7 (1998), 293-314. MR 2000b:53043
9.
T. Nehring: Hypersurfaces of prescribed Gauß curvature and boundary in Riemannian manifolds, J. reine angew. Math. 501 (1998), 143-170. MR 99i:53041
10.
O. C. Schnürer: The Dirichlet problem for Weingarten hypersurfaces in Lorentz manifolds, Math. Z. 242 (2002), 159-181.
11.
M. E. Taylor: Partial differential equations. III. Nonlinear equations, Applied Mathematical Sciences, 117, Springer, New York, 1997, xxii+608 pp. MR 98k:35001
12.
N. S. Trudinger: On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995), 151-164. MR 96m:35113

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J65, 53C42

Retrieve articles in all Journals with MSC (2000): 35J65, 53C42

Additional Information:

Oliver C. Schnurer
Affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22--26, D-04103 Leipzig, Germany
Address at time of publication: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA
Email: Oliver.Schnuerer@mis.mpg.de, schnuere@math.harvard.edu

DOI: 10.1090/S0002-9947-02-03135-5
PII: S 0002-9947(02)03135-5
Keywords: Minkowski problem, Dirichlet problem, prescribed curvature, convex hypersurfaces
Received by editor(s): November 8, 2000
Posted: September 6, 2002
Copyright of article: Copyright 2002, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google