A Rigorous Derivation of a Free-Boundary Problem Arising in Superconductivity
نویسندگان
چکیده
We study the Ginzburg-Landau energy of superconductors submitted to a possibly non-uniform magnetic eld, in the limit of a large Ginzburg-Landau parameter. We prove that the induced magnetic elds associated to minimizers of the energy-functional converge as ! +1 to the solution of a free-boundary problem. This free boundary-problem has a nontrivial solution only when the applied magnetic eld is of the order of the \\rst critical eld", i.e. of the order of log. In other cases, our results are contained in those we had previously obtained ((SS2, S1, SS1]). We also derive a convergence result for the density of vortices. The Ginzburg-Landau model was introduced in the fties by Ginzburg and Landau as a phenomenological model of superconductivity. In this model, the Gibbs energy of super-conducting material, submitted to an external magnetic eld is, in a suitable normalization, 1 Here, is the domain occupied by the superconductor, is a dimensionless constant (the Ginzburg-Landau parameter) depending only on characteristic lengths of the material and of temperature. h ex is the applied magnetic eld, A : 7 ! R 3 is the vector-potential, and the induced magnetic eld in the material is h = curlA. r A = r ? iA is the associated covariant derivative. The complex-valued function u is called the \order-parameter". It is a pseudo-wave function that indicates the local state of the material. There can be essentially two phases in a superconductor : ju(x)j ' 0 is the normal phase, ju(x)j ' 1, the superconducting phase. The Ginzburg-Landau model was based on Landau's theory of phase-transitions. Since then, the model has been justiied by the microscopic theory of Bardeen-Cooper-Schrieeer (BCS theory). ju(x)j is then understood as the local density of superconducting electron pairs, called \Cooper pairs", responsible for the superconductiv-ity phenomenon. A common simpliication, that we make, is to restrict to the two-dimensional model corresponding to a innnite cylindrical domain of section R 2 (smooth and simply connected), when the applied eld is parallel to the axis of the cylinder, and all the quantities are translation-invariant. The energy-functional then reduces to
منابع مشابه
Free Vibration Analysis of a Sloping-frame: Closed-form Solution versus Finite Element Solution and Modification of the Characteristic Matrices (TECHNICAL NOTE)
This article deals with the free vibration analysis and determination of the seismic parameters of a sloping-frame which consists of three members; a horizontal, a vertical, and an inclined member. The both ends of the frame are clamped, and the members are rigidly connected at joint points. The individual members of the frame are assumed to be governed by the transverse vibration theory of an ...
متن کاملOn Approximate Stationary Radial Solutions for a Class of Boundary Value Problems Arising in Epitaxial Growth Theory
In this paper, we consider a non-self-adjoint, singular, nonlinear fourth order boundary value problem which arises in the theory of epitaxial growth. It is possible to reduce the fourth order equation to a singular boundary value problem of second order given by w''-1/r w'=w^2/(2r^2 )+1/2 λ r^2. The problem depends on the parameter λ and admits multiple solutions. Therefore, it is difficult to...
متن کاملA two-phase free boundary problem for a semilinear elliptic equation
In this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $Dsubset mathbb{R}^{n}$ with smooth boundary. We give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of Caffarelli and Friedman regarding the representation of functions whose ...
متن کاملNumerical Solution of a Free Boundary Problem from Heat Transfer by the Second Kind Chebyshev Wavelets
In this paper we reduce a free boundary problem from heat transfer to a weakly Singular Volterra integral equation of the first kind. Since the first kind integral equation is ill posed, and an appropriate method for such ill posed problems is based on wavelets, then we apply the Chebyshev wavelets to solve the integral equation. Numerical implementation of the method is illustrated by two ben...
متن کاملDerivation of Green’s Function for the Interior Region of a Closed Cylinder
The importance of constructing the appropriate Green function to solve a wide range of problems inelectromagnetics and partial differential equations is well-recognized by those dealing with classical electrodynamics and related fields. Although the subject of obtaining the Green function for certain geometries has been extensively studied and addressed in numerous sources, in this paper a syst...
متن کامل