# A symmetry result for the Ornstein-Uhlenbeck operator

created by novaga on 03 Apr 2012
modified on 07 Jan 2016

[BibTeX]

Published Paper

Inserted: 3 apr 2012
Last Updated: 7 jan 2016

Journal: Discrete Contin. Dyn. Syst. A
Volume: 34
Number: 6
Pages: 2451-2467
Year: 2014

Abstract:

In 1978 E. De Giorgi formulated a conjecture concerning the one-dimensional symmetry of bounded solutions to the elliptic equation $\Delta u=F'(u)$, which are monotone in some direction. In this paper we prove the analogous statement for the equation $\Delta u - \langle x,\nabla u\rangle u=F'(u)$, where the Laplacian is replaced by the Ornstein-Uhlenbeck operator. Our theorem holds without any restriction on the dimension of the ambient space, and this allows us to obtain a similar result in infinite dimensions, by a limit procedure.