The Research On The Two-component BEC Passing An Obstacle

In th(?)nesis,we mainly investigate ground states of two-component BoseEinsteir(?)densates passing an obstacle in the plane,which can be described by an L2-(?)tical constraint minimization problem defined in an exterior domain Ω=R2\(?)nere ω(?)R2 denoting the region of the obstacle is a bounded smooth conveA don(?).While assuming that the intraspecies interaction of atoms inside each cc;onent is attractive,we shall investigate the interaction between two components a zero,attractive and repulsive in Chapter 2,3 and 4,respectively.Since the interaction between two component are zero,the model can be reduced to the single component BEC passing an obstacle which can be described by the following constraint minimization problem e(a):=inf {Ea(u):∫Ωu2dx=1},where Ea(u)is defined by Ea(u):=∫Ω(|▽u|2+V(x)u2)dx-a/2∫Ωu4dx,and a>0 represents that the intraspecies interaction is attractive.For the interaction between two components being attractive,we consider the following minimization problem e(a1,a2,β):=inf{Ea1,a2,β(u,v):∫Ω(u2+v2)dx=1},where energy functional Ea1,a2,β(u,v)is defined by Ea1,a2,β(u,v):=∫Ω(|▽u|2+|▽v|2)dx+∫Ω(V1(x)u2+V1(x)v2)dx-1/2∫Ω(a1u4+a2v4+2βu2v2)dx.The parameter a1,a2 and β>0 represents the intraspecies interaction of each component and the interspecies interaction are attractive.For the interaction between two components being repulsive,we consider the following minimization problem e(a1,a2):=inf{Ea1,a2,β(u,v):∫Ωu2dx=∫Ωv2dx=1},where a1,a2>0 and β<0 represents the intraspecies interaction of each component is attractive and the interspecies interaction is repulsive.The potentials 0≤V(x),V1(x),V2(x)∈C2,α(Ω)(0 ≤α<1)considered in this paper satisfy assumption(HV).(?)V(x)=∞,where V(x)>0 for x∈Ω,V(x)≡ 0 for x ∈(?)Ω.By using the energy comparing method,local Pohozaev identities,variational method,classical Gagliardo-Nirenberg inequality and Gagliardo-Nirenberge inequality with remainder,the existence and non-existence of minimizer are classified completely for each problem.By using the energy estimate,the refine blow-up analysis was done for each problem,including the limiting behavior of the minimizers,determining the location and blow-up rate.