Existence and instability of deformed catenoidal solutions for fractional Allen--Cahn equation

2017/11/09 by Hardy Chan, Yong Liu, Chan, Hardy +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1711.03215

This is a revised and expanded version of the previous article titled "A gluing construction for fractional elliptic equations. Part I: a model problem on the catenoid"

openalex publication_date 2017/11/09 · arxiv created 2019/09/19 · arxiv updated 2019/09/20 · openalex created_date 2019/09/26 · openalex updated_date 2026/07/28

Abstract

We develop a new infinite dimensional gluing method for fractional elliptic equations. As a model problem, we construct solutions of the fractional Allen--Cahn equation vanishing on a rotationally symmetric surface which resembles a catenoid and have sub-linear growth at infinity. Moreover, such solutions are unstable.

Citations

• On Phase-Separation Models: Asymptotics and Qualitative Properties
• Traveling wave solutions for bistable fractional Allen–Cahn equations with a pyramidal front
• Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates

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• Γ-convergence for higher order nonlocal phase transitions

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