Convexity of solutions and $C^{1,1}$ estimates for fully nonlinear elliptic equations

Abstract : The starting point of this work is a paper by Alvarez, Lasry and Lions (1997) concerning the convexity and the partial convexity of viscosity solutions of fully nonlinear degenerate elliptic equations. We extend their results in two directions. First, we deal with possibly sublinear (but epi-pointed) solutions instead of $1$-coercive ones; secondly, the partial convexity of $C^2$ solutions is extended to the class of continuous viscosity solutions. A third contribution of this paper concerns $C^{1,1}$ estimates for convex viscosity solutions of strictly elliptic nonlinear equations. To finish with, all the tools and techniques introduced here permit us to give a new proof of the Alexandroff estimate obtained by Trudinger (1988) and Caffarelli (1989).
Document type :
Journal articles
Complete list of metadatas

Cited literature [13 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-00012969
Contributor : Cyril Imbert <>
Submitted on : Monday, May 4, 2009 - 2:51:29 PM
Last modification on : Friday, April 12, 2019 - 4:46:03 PM
Long-term archiving on: Wednesday, March 29, 2017 - 4:49:02 PM

File

visconv-hal2.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-00012969, version 2

Citation

Cyril Imbert. Convexity of solutions and $C^{1,1}$ estimates for fully nonlinear elliptic equations. Journal de Mathématiques Pures et Appliquées, Elsevier, 2006, 85, pp.791-807. ⟨hal-00012969v2⟩

Share

Metrics

Record views

239

Files downloads

130