Asymptotic Analysis Of The Radial Minimizers Of An Energy Functional

In this paper, denote B = {x∈R²;x₁² + x₂² < 1}, S² = {x∈R³;x₁²+x₂²+x₃²= 1}, g(x) = (x/|x|sin M, cos M)(0 < M <Ï€/2), and x = (cosθ,sinθ) on (?)B, we consider the radial minimizers u_εof an energy functional where u(x) is in the function classIn chapter one, we present some basic notions and facts that will be used in this paper. At the same time, we outline some fundamental results of the radial minimizers u_ε= (u_ε1, u_ε2, u_ε3) of the energy functional.In chapters two and three, we mainly prove the W^1,p convergence and C^1,α convergence of the radial minimizers u_ε= (u_ε1,u_ε2,u_ε3) of the energy functional asε→0. First, we present the W^1,p convergence of u_ε, in fact, we obtain the result thatSecond, we prove the C^1,α convergence of u_ε, namely, for any compact subset K (?) (B(0,1－σ)|—)\B(0,σ), we can prove that for allα∈(0,1/2) andσ∈(0,1/4) is an arbitrary constant. Next, we also estimate the zeros of u_ε1² + u_ε2² roughly. Finally, we present the estimates of the convergent rate of u_ε3² asε→0.