Abstract
Abstract The k-Hessian is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k ≥ 2, the k-Hessian equation is a fully nonlinear partial differential equations. It is elliptic when restricted to k-admissible functions. In this paper we establish the existence and regularity of k-admissible solutions to the Dirichlet problem of the k-Hessian equation. By a gradient flow method we prove a Sobolev type inequality for k-admissible functions vanishing on the boundary, and study the corresponding variational problems. We also extend the definition of k-admissibility to non-smooth functions and prove a weak continuity of the k-Hessian operator. The weak continuity enables us to deduce a Wolff potential estimate. As an application we prove the Hölder continuity of weak solutions to the k-Hessian equation. These results are mainly from the papers [CNS2, W2, CW1, TW2, Ld] in the references of the paper.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Z. Balogh; Size of rectifiable sets and functions of prescribed gradient, J. Reine Angew. Math. 564 (2003), 63–83.
J.-H. Cheng, J.-F. Hwang, A. Malchiodi and P. Yang; “Minimal surfaces in Pseudo-hermitian geometry,” Ann. Scuola Normal Sup., Pisa, I. Sci, (5) 2005, 129–177.
J.-H. Cheng, J.-F Hwang and P. Yang; “Existence and Uniqueness for P-area minimizers in the Heisenberg group,” Math Ann., (2007), 337 = 253 – 293.
S. Chanillo and P. Yang; “Isoperimetric inequality and volume comparison on CR manifolds (WIP).
P. Pansu; “Une inegalité isoperimétrique sur le group de Heisenberg,” C.R. 295, 127–130.
S. Pauls, “Minimal surfaces in the Heisenberg group,” Geometry Dedicata, 104 (2004), 201–231.
N. Tanaka, “A differetial geometric study on strongly pseudo-convex manifolds,” Kinokuniya, Tokyo, (1975).
S. Webster, “Pseudo-hermitian structures of a real hyper-surface,” JDG 13 (1978), 25–41.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Yang, P. (2009). Minimal Surfaces in CR Geometry. In: Chang, SY., Ambrosetti, A., Malchiodi, A. (eds) Geometric Analysis and PDEs. Lecture Notes in Mathematics(), vol 1977. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01674-5_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-01674-5_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-01673-8
Online ISBN: 978-3-642-01674-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)