Minimal surfaces in CR geometry

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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.

Original languageEnglish (US)
Title of host publicationGeometric Analysis and PDEs
PublisherSpringer Verlag
Pages253-273
Number of pages21
ISBN (Print)9783642016738
DOIs
StatePublished - Jan 1 2009

Publication series

NameLecture Notes in Mathematics
Volume1977
ISSN (Print)0075-8434

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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    Yang, P. (2009). Minimal surfaces in CR geometry. In Geometric Analysis and PDEs (pp. 253-273). (Lecture Notes in Mathematics; Vol. 1977). Springer Verlag. https://doi.org/10.1007/978-3-642-01674-5_6