Global C2‐Estimates for Convex Solutions of Curvature Equations. Issue 8 (8th July 2014)

• Record Type: Journal Article
• Authors: Guan, Pengfei; Ren, Changyu; Wang, Zhizhang
• Is Part Of: Communications on pure and applied mathematics. Volume 68:Issue 8(2015:Aug.)
• Page Start: 1287
• Page End: 1325
• Publication Date: 2014-07-08
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.21528
• Languages: English
• ISSNs: 0010-3640

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>We establish a <italic>C</italic>² a priori estimate for convex hypersurfaces whose principal curvatures κ=(κ₁, …, κ_{<italic>n</italic>}) satisfy σ_{<italic>k</italic>}(κ(<italic>X</italic>))=<italic>f</italic>(<italic>X</italic>, ν(<italic>X</italic>)), the Weingarten curvature equation. We also obtain such an estimate for admissible 2‐convex hypersurfaces in the case <italic>k</italic>=2. Our estimates resolve a longstanding problem in geometric fully nonlinear elliptic equations.© 2015 Wiley Periodicals, Inc.</p> </abstract>