Multiple end solutions to the Allen-Cahn equation in \mathbb{R}^2
نویسندگان
چکیده
منابع مشابه
Multiple-end Solutions to the Allen-cahn Equation in R
We construct new class of entire solutions of the Allen-Cahn equation ∆u+(1−u2)u = 0, in R2(∼ C). Given k ≥ 1, we find a family of solutions whose zero level sets are, away from a compact set, asymptotic to 2k straight lines (which we call the ”ends”). These solutions have the property that there exists θ0 < θ1 < . . . < θ2k = θ0 + 2π such that limr→+∞ u(re iθ) = (−1)j uniformly in θ on compact...
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has been studied for several decades and is an important nonlinear PDE due to the fact that it lies at the interface of several different mathematical fields. The famous De Giorgi conjecture states that any entire solution to (1) in R which is monotone in one direction should be one dimensional, at least for n ≤ 8. The conjecture was proved in dimension n = 2 by Ghoussoub-Gui ([13]) and dimensi...
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The discrete Allen-Cahn equation is a spatially discrete analogue of the Allen-Cahn equation, a parabolic partial differential equation proposed as a simple model for phase separation in materials. In some sense, the solutions of the discrete equation display a richer variety of behaviors than do the corresponding solutions of the continuous equation. In particular, the number of stationary sol...
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ژورنال
عنوان ژورنال: Séminaire Laurent Schwartz — EDP et applications
سال: 2013
ISSN: 2266-0607
DOI: 10.5802/slsedp.55