Multiple Catenoidal End Solutions to the Allen-cahn Equation in R
نویسندگان
چکیده
Contents 1. Introduction 1 2. Geometrical setting near a dilated catenoid 6 3. Jacobi-Toda system on the Catenoid 9 4. Jacobi operator and the linear Jacobi-Toda operator on the catenoid. 16 5. Approximation of the solution of the theorem 1 23 6. Proof of theorem 1. 30 7. gluing reduction and solution to the projected problem.
منابع مشابه
Catenoidal Layers for the Allen-cahn Equation in Bounded Domains
In this paper we present a new family of solutions to the singularly perturbed Allen-Cahn equation α∆u + u(1 − u) = 0, in Ω ⊂ R where N = 3, Ω is a smooth bounded domain and α > 0 is a small parameter. We provide asymptotic behavior which shows that, as α → 0, the level sets of the solutions collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature that ...
متن کاملThe existence of global attractor for a Cahn-Hilliard/Allen-Cahn equation
In this paper, we consider a Cahn-Hillard/Allen-Cahn equation. By using the semigroup and the classical existence theorem of global attractors, we give the existence of the global attractor in H^k(0
متن کاملMultiple-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...
متن کاملA Generalization of the Allen-cahn Equation
Our aim in this paper is to study generalizations of the Allen-Cahn equation based on a modification of the Ginzburg-Landau free energy proposed in [25]. In particular, the free energy contains an additional term called Willmore regularization. We prove the existence, uniqueness and regularity of solutions, as well as the existence of the global attractor. Furthermore, we study the convergence ...
متن کاملSolutions of the fractional Allen-Cahn equation which are invariant under screw motion
We establish existence and non-existence results for entire solutions to the fractional Allen–Cahn equation in R, which vanish on helicoids and are invariant under screw motion. In addition, we prove that helicoids are surfaces with vanishing non-local mean curvature.
متن کامل