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
2 Département de mathématiques Université Libre de Bruxelles
3 I2M - Institut de Mathématiques de Marseille

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.

Mots clés: Gibbons' conjecture; one-dimensional symmetry; Liouville type results; Fourth order Allen-Cahn equation

Domaines: Equations aux dérivées partielles [math.AP]

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, 2017, Special Issue in Honor of Haim Brezis, 38 (1), pp.149 - 172. ⟨10.1007/s11401-016-1065-2⟩. ⟨hal-01182688v2⟩