A symmetry result  for the Ornstein-Uhlenbeck operator

Annalisa Cesaroni; Matteo Novaga; Enrico Valdinoci

doi:10.3934/dcds.2014.34.2451

Article Contents

2014, Volume 34, Issue 6: 2451-2467. Doi: 10.3934/dcds.2014.34.2451

This issue Previous Article DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface Next Article A new critical curve for the Lane-Emden system

A symmetry result for the Ornstein-Uhlenbeck operator

1.
Dipartimento di Matematica, Università di Padova, Via Trieste 63, I-35121 Padova
2.
Dipartimento di Matematica, Universitá di Pisa, Largo Bruno Pontecorvo 5, I-56127 Pisa
3.
Weierstraß Institut für Angewandte Analysis und Stochastik, Hausvogteiplatz 11A, D-10117 Berlin

Received: August 2012

Revised: December 2012

Published: June 2014

Abstract / Introduction Related Papers Cited by

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 an similar result in infinite dimensions by a limit procedure.

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Mathematics Subject Classification: Primary: 35J15, 35J20; Secondary: 35J61.

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