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Flatness Results for Nonlocal Phase Transitions

  • Eleonora CintiEmail author
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 33)

Abstract

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 work, 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 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.

Keywords

Fractional Laplacian Symmetry properties Nonlocal minimal surfaces 

Notes

Acknowledgements

The author is supported by MINECO grants MTM2014-52402-C3-1-P and MTM2017-84214-C2-1-P, and is part of the Catalan research group 2014 SGR 1083.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di BolognaBolognaItaly

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