Coexistence phenomenon of concentration and transition of aninhomogeneous phase transition model on surfaces

Jun Yang

doi:10.3934/dcds.2011.30.965

Article Contents

2011, Volume 30, Issue 3: 965-994. Doi: 10.3934/dcds.2011.30.965

This issue Previous Article On a generalized Poincaré-Hopf formula in infinite dimensions Next Article A Harnack inequality for fractional Laplace equations with lower order terms

Coexistence phenomenon of concentration and transition of an inhomogeneous phase transition model on surfaces

Jun Yang^1,

1.
College of Mathematics and Computational Sciences, Shenzhen University, Nanhai Ave 3688, Shenzhen 518060

Received: November 30, 2009

Revised: December 31, 2010

Published: August 2011

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Abstract

We consider the following singularly perturbed elliptic problem

ε²Δ ũ + (ũ –a(ŷ))(1- ũ²)=0 in M

&ytilde; where M is a two dimensional smooth compact Riemannian manifold associated with metric ğ, ε is a small parameter. The inhomogeneous term -1 < a(ŷ) < 1 takes maximum value b with 0 < b < 1. Assume that Γ^’ = { ŷ ∈ M : a(ŷ) = 0} is a closed, smooth curve that Γ^’ separate M into two disjoint components M⁺ and M^- and also ∂a/∂v^’ > 0 on Γ^’, where v^’ is the normal of Γ^’ pointing to the interior of M^-. Moreover the maximum value loop Γ = { ŷ ∈ M : a(ŷ) = b} is a closed, smooth geodesic contained in M in such a way and Γ separate M^- into two disjoint components. We will show the existence of solution possessing both transition and concentration phenomenon, i.e.

u_ε → + 1 in M^-\ Γ_δ, u_ε → -1 in M⁺, u_ε → 1 – C along Γ as ε → 0,

where Γ_δ is a small neighborhood of Γ and C is a fixed positive constant.

Keywords:

Inhomogeneous Allen-Cahn equation,
Coexistence of concentrations and phase transition,
Gap condition.

Mathematics Subject Classification: Primary: 35J20, 35J65, 35J40; Secondary: 35B35.

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