Higher Dimensional Catenoid, Liouville Equation and Allen-cahn Equation
نویسندگان
چکیده
We build a family of entire solutions to the Allen-Cahn equation in RN+1 for N ≥ 3, whose level set approaches the higher dimensional catenoid in a compact region and has two logarithmic ends governed by the solutions to the Liouville equation.
منابع مشابه
. A P ] 9 N ov 2 01 7 A GLUING CONSTRUCTION FOR FRACTIONAL ELLIPTIC EQUATIONS . PART I : A MODEL PROBLEM ON THE CATENOID
Abstract. We develop a new infinite dimensional gluing method for fractional elliptic equations. In Part I, as a model problem, we construct a solution of the fractional Allen–Cahn equation vanishing on a rotationally symmetric surface which resembles a catenoid and has sub-linear growth at infinity. In Part II, we construct counterexamples to De Giorgi Conjectures to fractional Allen-Cahn equa...
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 Catenoidal End Solutions to the Allen-cahn Equation in R
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.
متن کاملEntire Solutions of the Allen-cahn Equation and Complete Embedded Minimal Surfaces of Finite Total Curvature in R
We consider minimal surfaces M which are complete, embedded and have finite total curvature in R, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation ∆u+ f(u) = 0 in R. Here f = −W ′ with W bistable and balanced, for instance W (u) = 1 4 (1 − u). We assume that M has m ≥ 2 ends, and additionally that M is non-degenerate, in the sense that its bounded Jacobi fields a...
متن کاملENTIRE SOLUTIONS OF THE ALLEN-CAHN EQUATION AND COMPLETE EMBEDDED MINIMAL SURFACES OF FINITE TOTAL CURVATURE IN R3 by
— We consider minimal surfaces M which are complete, embedded and have finite total curvature in R3, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation ∆u+f(u) = 0 in R3. Here f = −W ′ with W bistable and balanced, for instance W (u) = 1 4 (1− u2)2. We assume that M has m ≥ 2 ends, and additionally that M is non-degenerate, in the sense that its bounded Jacobi fiel...
متن کامل