A symmetry result for the Ornstein-Uhlenbeck operator

TL;DR

This paper proves a symmetry result for solutions to an Ornstein-Uhlenbeck type elliptic equation, extending De Giorgi's conjecture to infinite-dimensional spaces and removing dimension restrictions.

Contribution

It establishes a one-dimensional symmetry for solutions of the Ornstein-Uhlenbeck equation, generalizing De Giorgi's conjecture to all dimensions including infinite-dimensional spaces.

Findings

Symmetry result holds in all finite dimensions.

Extension to infinite-dimensional spaces via limit procedures.

No restrictions on the dimension of the ambient space.

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 - <x, Du> =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.