Asymptotic concentration on the boundary for elliptic eigenvalue problems with large drift

This paper is concerned with the asymptotic concentration on the boundary for the following elliptic eigenvalue problem with a large drift: (*) { − 𝜀 ⁢ Δ ⁢ 𝜙 − 2 ⁢ 𝛼 ∇ 𝑚 ⁡ ( 𝑥 ) ⋅ ∇ 𝜙 + 𝑉 ⁡ ( 𝑥 ) ⁢ 𝜙 = 𝜆 𝛼 ⁢ 𝜙 i n Ω , 𝜙 = 0 o n ∂ Ω , where Ω ⊆ ℝ 𝑁 ⁢ ( 𝑁 > 1 ) is a bounded smooth domain, the constants 𝜀 > 0 and 𝛼 > 0 are the diffusion and advection coefficients, respectively, and 𝑚 ⁡ ( 𝑥 ) ∈ 𝐶 2 ⁡ ( ¯ Ω ) , 𝑉 ⁡ ( 𝑥 ) ∈ 𝐶 𝛾 ⁡ ( ¯ Ω ) ⁢ ( 0 < 𝛾 < 1 ) are given functions. We investigate the limiting behavior of the unique principal eigenvalue 𝜆 𝛼 for ( ⁎ ) as 𝛼 → + ∞ , which addresses some conjectures in Berestycki et al. (2005) [5] . We further analyze the asymptotic expansions of the principal eigenvalue 𝜆 𝛼 for ( ⁎ ) and the associated normalized conjugated eigenfunction 𝑢 𝛼 = 𝑒 𝛼 𝜀 ⁢ 𝑚 ⁢ 𝜙 𝛼 as 𝛼 → + ∞ , where 𝑢 𝛼 concentrates asymptotically on the boundary ∂Ω.