One-dimensional symmetry and Liouville type results for the fourth order Allen-Cahn equation in ℝ N

Denis Bonheure 1, 2 François Hamel 3
1 MEPHYSTO - Quantitative methods for stochastic models in physics
Inria Lille - Nord Europe, ULB - Université Libre de Bruxelles [Bruxelles], LPP - Laboratoire Paul Painlevé - UMR 8524
Abstract : In this paper, we prove an analogue of Gibbons' conjecture for the extended fourth order Allen-Cahn equation in R N , as well as Liouville type results for some solutions converging to the same value at infinity in a given direction. We also prove a priori bounds and further one-dimensional symmetry and rigidity results for semilinear fourth order elliptic equations with more general nonlinearities.
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Submitted on : Friday, March 11, 2016 - 9:27:10 AM
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Denis Bonheure, François Hamel. One-dimensional symmetry and Liouville type results for the fourth order Allen-Cahn equation in ℝ N. Chinese Annals of Mathematics - Series B, Springer Verlag, 2017, Special Issue in Honor of Haim Brezis, 38 (1), pp.149 - 172. ⟨10.1007/s11401-016-1065-2⟩. ⟨hal-01182688v2⟩

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