Abstract
We consider in this chapter a nonlocal phase transition model, in particular described by the Allen-Cahn equation. We deal here with a two-phase transition model, in which a fluid can reach two pure phases forming an interface of separation. The aim is to describe the pattern and the separation of the two phases, focusing on the study of long range interactions that naturally leads to the analysis of phase transitions and interfaces of nonlocal type. The formation of the interface is driven by a variational principle, and here the kinetic energy is modified to take into account far away changes in phase (though the influence is weaker and weaker towards infinity). A fractional analogue of a conjecture of De Giorgi, that deals with possible one-dimensional symmetry of entire solutions naturally arises from treating this model, and will be consequently presented.
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Notes
- 1.
We would like to thank Alberto Farina who, during a summer-school in Cortona (2014), gave a beautiful introduction on phase transitions in the classical case.
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Bucur, C., Valdinoci, E. (2016). Nonlocal Phase Transitions. In: Nonlocal Diffusion and Applications. Lecture Notes of the Unione Matematica Italiana, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-319-28739-3_4
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DOI: https://doi.org/10.1007/978-3-319-28739-3_4
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