In 1978 E. De Giorgi formulated a conjecture concerning the onedimensional symmetry of bounded solutions to the elliptic equation u = F0(u), which are monotone in some direction. In this paper we prove the analogous statement for the equation uδ hx;ruiu = F0(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 innite dimensions by a limit procedure.
A symmetry result for the ornstein-uhlenbeck operator / A. Cesaroni, M. Novaga, E. Valdinoci. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - 34:6(2014), pp. 2451-2467.
A symmetry result for the ornstein-uhlenbeck operator
E. ValdinociUltimo
2014
Abstract
In 1978 E. De Giorgi formulated a conjecture concerning the onedimensional symmetry of bounded solutions to the elliptic equation u = F0(u), which are monotone in some direction. In this paper we prove the analogous statement for the equation uδ hx;ruiu = F0(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 innite dimensions by a limit procedure.
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