Calculus of Variations and Geometric Measure Theory
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E. Cinti

Flatness results for nonlocal phase transitions

created by cinti on 11 Feb 2018
modified on 20 Mar 2019


Accepted Paper

Inserted: 11 feb 2018
Last Updated: 20 mar 2019

Journal: Springer INDAM Series
Year: 2018


We consider a nonlocal version of the Allen-Cahn equation, which models phase transitions problems. In the classical setting, the connection between the Allen-Cahn equation and the classification of entire minimal surfaces is well known and motivates a celebrated conjecture by E. De Giorgi on the one-dimensional symmetry of bounded monotone solutions to the (classical) Allen-Cahn equation up to dimension 8. In this note, we present some recent results in the study of the nonlocal analogue of this phase transition problem. In particular we describe the results obtained in several contributions 8, 9, 13, 14, 25, 41, 44, 46 where the classification of certain entire bounded solutions to the fractional Allen-Cahn equation has been obtained. Moreover we describe the connection between the fractional Allen-Cahn equation and the fractional perimeter functional, and we present also some results in the classifications of nonlocal minimal surfaces obtained in 16, 42, 10, 21.


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