A symmetry result for the Ornstein-Uhlenbeck operator
Annalisa Cesaroni, Matteo Novaga, Enrico Valdinoci · Discrete and Continuous Dynamical Systems · 2013
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 $\Delta u- \langle x, abla u\rangle u=F'(u)$,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.