Calculus of Variations and Geometric Measure Theory
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E. Cinti

Saddle-shaped solutions for the fractional Allen-Cahn equation

created by cinti on 31 Aug 2017


Accepted Paper

Inserted: 31 aug 2017
Last Updated: 31 aug 2017

Journal: DCDS-S
Year: 2017


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 = 12 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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