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

Matteo Novaga; Diego  Pallara; Yannick  Sire

doi:10.3934/dcdss.2016030

Article Contents

2016, Volume 9, Issue 3: 815-831. Doi: 10.3934/dcdss.2016030

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

1.
Dipartimento di Matematica, Universitá di Pisa, Largo Bruno Pontecorvo 5, I-56127 Pisa
2.
Dipartimento di Matematica e Fisica "Ennio De Giorgi" & INFN, Università del Salento, P.O.B. 193, 73100, Lecce
3.
Université Aix-Marseille, I2M, UMR CNRS 7353, Marseille

Received: October 31, 2014

Revised: March 31, 2015

Published: May 2016

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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.

Keywords:

Fractional Ornstein-Uhlenbeck operator,
Wiener spaces.

Mathematics Subject Classification: Primary: 35R15, 35R11, 35J61; Secondary: 35J70.

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References

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