 |
|
|
|
| HAL: hal-00012969, version 2 |
| Detailed view |  Export this paper |
|
|
| Journal de mathématiques pures et appliquées 85 (2006) 791-807 |
|
|
| Available versions: | v1 (2005-11-02) | v2 (2009-05-05) |
|
|
|
|
| Convexity of solutions and $C^{1,1}$ estimates for fully nonlinear elliptic equations |
|
|
| Cyril Imbert 1 |
|
|
| (2006) |
|
|
|
| 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). |
|
|
|
|
|
|
|
|
|
|
|
| 1: | Institut de Mathématiques et de Modélisation de Montpellier (I3M) |
|
|
|
CNRS : UMR5149 – Université Montpellier II - Sciences et Techniques du Languedoc |
|
|
|
|
|
|
|
|
|
| ACSIOM |
|
|
|
|
| Subject | : | Mathematics/Analysis of PDEs |
|
|
| convexity and partial convexity of viscosity solutions – subjet of a convex envelope of epi-pointed functions – $C^{1 – 1}$ estimate – Alexandroff estimate |
|
|
|
| Attached file list to this document: | |||||
|
|||||
|
|
|
| hal-00012969, version 2 | |
| http://hal.archives-ouvertes.fr/hal-00012969 | |
| oai:hal.archives-ouvertes.fr:hal-00012969 | |
| From: Cyril Imbert | |
| Submitted on: Monday, 4 May 2009 14:51:29 | |
| Updated on: Tuesday, 5 May 2009 08:19:00 | |
