Singular limits of anisotropic Ginzburg-Landau functional

Abstract

We derive the asymptotic behavior of the minimizers of the anisotropic Ginzburg-Landau functional of superconductivity, as the ratio between the largest and smallest effective masses is very big, hence the effective mass tensor becomes very degenerate.

Appendices

Appendix A: Proof of Lemma 6.5

Step 1. We use the notation $F^a(x')=F(x',a(x'))$, $F^b(x')=F(x',b(x'))$ and

$$\begin{aligned} L(\phi )(x')=\int _{a(x')}^{b(x')} \phi (x',x_3) dx_3. \end{aligned}$$

Let $\psi$ be an eigenfunction of $\beta =\beta ^{1,2}_3(\mathbf{A})$ and has the form of (6.4). We show that u is an eigenfunction of the following equation

$$\begin{aligned} \left\{ \begin{aligned}&\sum _{j=1}^2\lambda _j[\partial _{jj}u+(d_j-2ia_j)\partial _ju-(c_j+i\beta _j+ie_j) u]+\beta u=0\quad&\text {in }U,\\&\sum _{j=1}^2\lambda _j\nu _j(\partial _ju-ia_ju)=0\quad&\text {on }\partial U, \end{aligned}\right. \end{aligned}$$

(A.1)

where $a_j, c_j, d_j$ were given in (6.5), and

$$\begin{aligned} \begin{aligned} \beta _j(x')&={1\over b(x')-a(x')}\int _{a(x')}^{b(x')}\partial _j B_j(x',x_3)dx_3,\\ e_j(x')&={1\over b(x')-a(x')}\{B_j^b(x')\partial _j b(x')-B_j^a(x')\partial _j a(x')\}. \end{aligned} \end{aligned}$$

(A.2)

By the variational character of eigenvalue $\beta ^{1,2}_3$ we have, for any $v\in H^1(U,\mathbb C)$,

$$\begin{aligned} \begin{aligned} 0&={d\over ds}\left| _{s=0}{\int _{\Omega }\sum _{j=1}^2\lambda _j|\partial _{B_j}(u+sv)|^2dx\over \int _{\Omega }|u+sv|^2dx}\right. \\&={2\over \int _\Omega |u|^2dx}\mathrm{\;Re\;}\left\{ \int _{U}dx'\sum _{j=1}^2\lambda _j\int _{a(x')}^{b(x')} \partial _j({\bar{v}}\partial _{B_j}u)dx_3-\int _\Omega \bar{v}[\sum _{j=1}^2\lambda _j\partial _{B_j}^2u+\beta u]dx\right\} . \end{aligned} \end{aligned}$$

We compute

$$\begin{aligned} \begin{aligned}&\int _{a(x')}^{b(x')} \partial _j({\bar{v}}\partial _{B_j}u)dx_3\\&\quad = \partial _j\int _{a(x')}^{b(x')}({\bar{v}}\partial _{B_j}u) dx_3-{\bar{v}}\partial _{B_j}u\left| _{x_3=b(x')}\partial _j b(x')+{\bar{v}}\partial _{B_j}u\right| _{x_3=a(x')}\partial _j a(x') \\&\quad =\partial _j\int _{a(x')}^{b(x')}({\bar{v}}\partial _{B_j}u) dx_3-\bar{v}\partial _ju(\partial _jb-\partial _ja)+i{\bar{v}} u\left( B_j^b\partial _j b-B_j^a\partial _j a\right) . \end{aligned} \end{aligned}$$

So we get

$$\begin{aligned}\begin{aligned}&\mathrm{\;Re\;}\left\{ \int _{\partial U}{\bar{v}} \sum _{j=1}^2\lambda _j\nu _j(\int _{a(x')}^{b(x')} \partial _{B_j}u dx_3)ds\right. \\&\qquad -\int _U{\bar{v}} \sum _{j=1}^2\lambda _j\left[ \partial _ju(\partial _jb-\partial _ja)-i u\left( B_j^b\partial _j b-B_j^a\partial _j a\right) \right] dx'\\&\qquad \left. -\int _U {\bar{v}}\left[ \int _{a(x')}^{b(x')} \sum _{j=1}^2\lambda _j\partial _{B_j}^2u dx_3+\beta (b-a)u\right] dx'\right\} =0. \end{aligned} \end{aligned}$$

Since v is an arbitrary smooth complex-valued function, we get

$$\begin{aligned} \begin{aligned}&\sum _{j=1}^2\lambda _j\left\{ \int _{a(x')}^{b(x')}\partial _{B_j}^2u dx_3+\partial _jf\partial _ju-ig_j u\right\} +\beta f u=0\quad&\text {for } x'\in U,\\&\sum _{j=1}^2\lambda _j\nu _j\int _{a(x')}^{b(x')}\partial _{B_j}u dx_3=0\quad&\text {for } x'\in \partial U, \end{aligned} \end{aligned}$$

where $f=b-a$, $g_j=B_j^b\partial _j b-B_j^a\partial _j a.$ For $j=1,2$ we have

$$\begin{aligned} \begin{aligned} \int _{a(x')}^{b(x')}\partial _{B_j}u dx_3&=\int _{a(x')}^{b(x')}(\partial _j u-iB_j u)dx_3 =f\partial _ju-i{\tilde{B}}_ju,\\ \int _{a(x')}^{b(x')}\partial _{B_j}^2u dx_3&=f\partial _{jj}u-2iL(B_j)\partial _ju-iuL(\partial _j B_j)-uL(B_j^2), \end{aligned} \end{aligned}$$

hence

$$\begin{aligned} \begin{aligned}&\sum _{j=1}^2\lambda _j\left( \partial _{jj}u-2i{L(B_j)\over f}\partial _ju-iu{L(\partial _j B_j)\over f}-u{L(B_j^2)\over f}\right. \\&\quad \left. +{\partial _jf\over f}\partial _ju-i{g_j\over f}u\right) +\beta u=0&\text {in }U,\\&\sum _{j=1}^2\lambda _j\nu _j\left( \partial _ju-i{L(B_j)\over f}u\right) =0&\text {on }\partial U, \end{aligned} \end{aligned}$$

which is (A.1).

Step 2. We write (A.1) as follows:

$$\begin{aligned} \left\{ \begin{aligned}&-\sum _{j=1}^2\lambda _j\partial _{b_j}^2u-\sum _{j=1}^2\lambda _jd_j\partial _ju+\sum _{j=1}^2\lambda _j(U_j+iV_j)u=\beta u&\text {in }U,\\&\sum _{j=1}^2\lambda _j\nu _j\partial _{b_j}u=0\quad&\text {on }\partial U, \end{aligned}\right. \end{aligned}$$

(A.3)

where $U_j=c_j-b_j^2$ and $V_j=\beta _j+e_j-\partial _jb_j=d_jb_j$. (A.3) can be written as (6.6), because

$$\begin{aligned} -\sum _{j=1}^2\lambda _jd_j\partial _ju+iVu=-\sum _{j=1}^2\lambda _jd_j\partial _ju+i\sum _{j=1}^2\lambda _jd_jb_j u =-\sum _{j=1}^2\lambda _jd_j\partial _{b_j}u. \end{aligned}$$

$\square$

Appendix B: Notation

$$\begin{aligned} \begin{aligned}&d^{2,3}({\mathcal {X}}_1)&(2.6)\\&d^3({\mathcal {B}}_{1,2})&(3.9)\\&d^3({\mathcal {H}}^{1,2}),\quad {\mathcal {H}}^{1,2}&(3.4)\\&d^3({\mathcal {H}}^2_1),\quad {\mathcal {H}}^2_1&(3.16)\\&d^3({\mathcal {X}}_{1,2})&(3.14)\\&f^{1,2}_H&(3.3)\\&f^{1,2}_{3,H}&(5.1)\\&f^2_{1,H}&(3.15)\\&f^3_{1,2,A}&(4.3)\\&f^3_{1,2,{\hat{A}}}&(4.5)\\&f^3_{1,2,H}&(4.1)\\&{\mathcal {Z}}_0(\Omega )&(1.4)\\&{\mathcal {Z}}^1_1(\Omega )&(2.2)\\&{\mathcal {Z}}^1_{1,2}(\Omega )&(3.1)\\&\Psi _1(\kappa A_1),\quad \Psi _1^{(1)}(\kappa A_1)&(2.3)\\&\Psi _{1,2}(\kappa A_1,\kappa A_2),\quad \Psi _{1,2}^{(1)}(\kappa A_1,\kappa A_2)\qquad \;\;&(3.7)\\ \end{aligned} \end{aligned}$$