Boundary-clustered interfaces for the Allen–Cahn equation

Andrea Malchiodi, Wei-Ming Ni and Juncheng Wei

Vol. 229 (2007), No. 2, 447–468

DOI: 10.2140/pjm.2007.229.447

Abstract

We consider the Allen–Cahn equation

where Ω = B₁(0) is the unit ball in ℝⁿ and 𝜀 > 0 is a small parameter. We prove the existence of a radial solution u_𝜀 having N interfaces {u_𝜀(r) = 0} = ⋃ _j=1^N{r = r_j^𝜀}, where 1 > r₁^𝜀 > r₂^𝜀 > ⋯ > r_N^𝜀 are such that 1 − r₁^𝜀 ∼ 𝜀log(1∕𝜀) and r_j−1^𝜀 −r_j^𝜀 ∼ 𝜀log(1∕𝜀) for j = 2,…,N. Moreover, the Morse index of u_𝜀 in H_r¹(Ω_𝜀) is exactly N.

Keywords

boundary clustered interfaces, Allen–Cahn equation

Mathematical Subject Classification 2000

Primary: 35B40, 35B45

Secondary: 35J40

Milestones

Received: 21 March 2005

Accepted: 2 June 2006

Published: 1 February 2007