# A symmetry result for degenerate elliptic equations on the Wiener space with nonlinear boundary conditions and applications

created by pallara on 25 Oct 2014
modified by novaga on 17 Apr 2016

[BibTeX]

Published Paper

Inserted: 25 oct 2014
Last Updated: 17 apr 2016

Journal: Discrete Contin. Dyn. Syst. S
Volume: 9
Number: 3
Pages: 815-831
Year: 2016

Abstract:

The purpose of this paper is to study a boundary reaction problem on the space $X \times \mathbb R$, where $X$ is an abstract Wiener space. We prove that smooth bounded solutions enjoy a symmetry property, i.e., are one-dimensional in a suitable sense. As a corollary of our result, we obtain a symmetry property for some solutions of the following equation $(-\Delta_\gamma)^s u= f(u)$, with $s\in (0,1)$, where $(-\Delta_\gamma)^s$ denotes a fractional power of the Ornstein-Uhlenbeck operator, and we prove that for any $s \in (0,1)$ monotone solutions are one-dimensional.