Calculus of Variations and Geometric Measure Theory
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M. Novaga - D. Pallara - Y. Sire

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


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


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.


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