非均匀相变模型不稳定解的第一特征值

考虑以下奇异摄动椭圆问题ε2△u+(u-a(y))(1-u2)=0 inΩ,(e)u/(e)n=0 on (e)Ω,其中Ω是R2中一个光滑区域,-10,其中v是Ω+的外法向量.在[5]中,M.del Pino,M.Kowalczyk和J.Wei构造一族具有如下形状的解u8.当uε→1 in Ω_ 且 uε→-1 in Ω+.证明了在u8处的线性化问题的最小特征值具有渐近形式:-μ0ε+o(ε),其中μ0>0.

First Eigenvalue of Interior Unstable Interface Solutions to an Inhomogeneous Phase Transition Model

The following singularly perturbed elliptic problem ε2△u+(u-a(y))(1-u2)=0 inΩ,(e)u/(e)n=0 on (e)Ωis considered,where Ω is a bounded domain in R2 with smooth boundary,-10 on Γ,where v is the outer normal to Ω+.in [5], M.del Pino,M.Kowalczyk and J.Wei constructed transition layer solutions uε whose profile looks like uε→1 in Ω_ and uε→-1 in Ω+ as ε→0.It is shown that the smallest eigenvalue for the linearized problem at uε is-μ0ε+0(ε) for some positive constant μ0.