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| One-Dimensional Symmetry and Liouville Type Results for the Fourth Order Allen-Cahn Equation in \RN |
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Citation： |
Denis BONHEURE,Fran{c{c}}ois HAMEL.One-Dimensional Symmetry and Liouville Type Results for the Fourth Order Allen-Cahn Equation in \RN[J].Chinese Annals of Mathematics B,2017,38(1):149~172 |
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Net amount： 799 |
Authors： |
Denis BONHEURE; Fran{c{c}}ois HAMEL |
Foundation： |
This work was carried out in the framework of the Labex
Archim\`ede (ANR-11-LABX-0033) and of the A*MIDEX project
(ANR-11-IDEX-0001-02), funded by the ``Investissements d'Avenir"
French Government program managed by the French National Research
Agency (ANR). The research leading to these results has also
received funding from the European Research Council under the
European Union's Seventh Framework Programme (FP/2007-2013) ERC
Grant Agreement n.~321186-ReaDi-Reaction-Diffusion Equations,
Propagation and Modelling and from the ANR NONLOCAL project
(ANR-14-CE25-0013). Part of this work was carried out during visits
by D. Bonheure and F.~Hamel to Aix-Marseille University and to the
Universit\'e Libre de Bruxelles, whose hospitality is thankfully
acknowledged. D.~Bonheure was partially supported by INRIA-Team
MEPHYSTO, MIS F.4508.14 (FNRS), PDR T.1110.14F (FNRS) and ARC
AUWB-2012-12/17-ULB1-IAPAS. |
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| Abstract： |
In this paper, the authors 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. The authors also prove a
priori bounds and further one-dimensional symmetry and rigidity
results for semilinear fourth order elliptic equations with more
general nonlinearities. |
Keywords： |
Fourth order elliptic equation, Allen-Cahn equation, ExtendedFisher-Kolmogorov equation, One-dimensional symmetry, Liouville typeresults |
Classification： |
35B51, 35B53, 35J30, 35J47, 35J61 |
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