Saddle-shaped solutions for the fractional Allen-Cahn equation

Eleonora Cinti

doi:10.3934/dcdss.2018024

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2018, Volume 11, Issue 3: 441-463. Doi: 10.3934/dcdss.2018024

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Saddle-shaped solutions for the fractional Allen-Cahn equation

Eleonora Cinti

Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5,40126 Bologna, Italy

Received: April 30, 2017

Revised: July 31, 2017

Published: June 2018

The author is supported by MINECO grant MTM2014-52402-C3-1-P, the ERC Advanced Grant 2013 n. 339958 Complex Patterns for Strongly Interacting Dynamical Systems, the GNAMPA project Metodi variazionali per problemi nonlocali and is part of the Catalan research group 2014 SGR 1083.

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Abstract

We establish existence and qualitative properties of solutions to the fractional Allen-Cahn equation, which vanish on the Simons cone and are even with respect to the coordinate axes. These solutions are called saddle-shaped solutions.

More precisely, we prove monotonicity properties, asymptotic behaviour, and instability in dimensions $2m=4, 6$. We extend to any fractional power $s$ of the Laplacian, some results obtained for the case $s=1/2$ in [19].

The interest in the study of saddle-shaped solutions comes in connection with a celebrated De Giorgi conjecture on the one-dimensional symmetry of monotone solutions and of minimizers for the Allen-Cahn equation. Saddle-shaped solutions are candidates to be (not one-dimensional) minimizers in high dimension, a property which is not known to hold yet.

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Mathematics Subject Classification: Primary: 35J61; Secondary: 35B08, 35J20.

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