Inserted: 25 oct 2014
Last Updated: 17 apr 2016
Journal: Discrete Contin. Dyn. Syst. S
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.