A three-dimensional symmetry result for a phase transition equation in the genuinely nonlocal regime

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Abstract

We consider bounded solutions of the nonlocal Allen–Cahn equation (Formula Presented.)limxn→-∞u(x′,xn)=-1andlimxn→+∞u(x′,xn)=1,it has been recently shown in Dipierro et al. (Improvement of flatness for nonlocal phase transitions, 2016) that u is necessarily 1D, i.e. it depends only on one Euclidean variable. The goal of this paper is to obtain a similar result without assuming such limit conditions. This type of results can be seen as nonlocal counterparts of the celebrated conjecture formulated by De Giorgi (Proceedings of the international meeting on recent methods in nonlinear analysis (Rome, 1978), Pitagora, Bologna, pp 131–188, 1979).

Original languageEnglish
Article number15
JournalCalculus of Variations and Partial Differential Equations
Volume57
Issue number1
DOIs
Publication statusPublished - 1 Feb 2018

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Nonlinear analysis
Phase Transition
Phase transitions
Allen-Cahn Equation
Symmetry
Three-dimensional
Nonlocal Equations
Flatness
Bounded Solutions
Nonlinear Analysis
Euclidean

Cite this

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