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

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  • 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.

    uε → + 1 in M-\ Γδ, uε → -1 in M+, uε → 1 – C along Γ as ε → 0,

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

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


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