A symmetry result for the Ornstein-Uhlenbeck operator

Annalisa Cesaroni,Enrico Valdinoci,Matteo Novaga

doi:10.3934/dcds.2014.34.2451

Abstract

In 1978 E. De Giorgi formulated a conjecture concerning the one-dimensional symmetryof 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 holdswithout any restriction on the dimension of the ambient space, and this allows us to obtain an similar resultin infinite dimensions by a limit procedure.

Highlights

A celebrated conjecture by De Giorgi [6] asks if bounded entire solutions to the equation
∆u = u3 − u which are strictly increasing in some direction are one-dimensional, in the sense that the level sets {u = λ} are hyperplanes, at least if n ≤ 8
This conjecture has been proved by Ghoussoub and Gui [14] in dimension n = 2, and by Ambrosio and Cabre [2] in dimension n = 3, and a counterexample has been given by del Pino, Kowalczyk and Wei in [7] for n ≥ 9

Summary

Introduction

A celebrated conjecture by De Giorgi [6] asks if bounded entire solutions to the equation (1.1). As in the case of the Laplacian, Theorem 1.1 is closely related to the Bernstein problem in the Gauss space, which asks for flatness of entire minimal surfaces which are graphs in some direction. Differently from the Euclidean case, the result holds without any restriction on the dimension of the ambient space, and there is no such restriction in Theorem 1.1. This is due to the exponential decay of the Gaussian measure associated to the Ornstein-Uhlenbeck operator which allows for better estimates than the corresponding Euclidean ones. The proof that we perform exploits and generalizes some geometric ideas of [18, 19, 10, 11]

Notation

Heteroclinic solutions