高维空间中一类Ginzburg-Landau型泛函的极小元的渐近分析

Asymptotic analysis for minimizers of a Ginzburg-Landau type functional in a higher dimension space

在空间H1,pg(Ω,Rn)中讨论如下一类变系数Ginzburg-Landau型泛函Eε(Ω)=∫Ωa(x)p|Δu|p+14εpb(x)(|u|2-β2(x))2dx的极小元列的渐近性质.这里2≤p<n,Ω Rn为有界光滑星形区域,g(x):Ω→Rn,且|g(x)|=β(x).存在正常数M,m>0,m≤a(x),b(x),β(x)≤M,且a(x),b(x),β(x)是光滑函数.研究了当ε→0时极小元的渐近性态,证明了极小元列在H1,pg(Ω,Rn)中强收敛于某个元素,且得到了该元素所满足的微分方程边值问题.

In this paper,the asymptotic behavior for minimizers of Ginzburg\|Landau type functional with variable coefficient Eε(u,Ω)=∫Ωa(x)p｜ Δ u｜ p+14ε pb(x)(｜u｜ 2-β 2(x)) 2 d x is discussed in H 1,p g (Ω ,R n ),where 2≤ p<n,Ω R n is a smooth,bounded and simply connected domain, g(x) : ο Ω→R n ,and ｜ g(x)｜=β(x). There are constants M,m>0,m≤a(x),b(x), β(x)≤M, and a(x),b(x),β(x) are all smooth functions. The asymptotic behavior for minimizers of functional is studied with the small parameter tending to zero.It is proved that minimizers in H 1,p g ( Ω ,R n ) strongly converge to a function. A boundary value problem of a differential equation which this function satistfies is obtained.