数理解析研究所講究録
第 891 巻 1995 年 1-9

1Signorini 型境界条件の超伝導現象への応用–ON PENETRATION PHENOMENON OF MAGNETIC
FIELD INTO SUPERCONDUCTING MATERIALS–

Hideyuki KOSHiGOE $(*)$
腰越 秀之 (千葉大・工)Hideo KAWARADA $(*)$
河原田秀夫 (千葉大・工)明 (機械技術研究所) Akira SASAMOTO $(**)$
笹本

0. Introduction

We shall deal with a behavior of penetration of magnetic field into a superconducting
material in the presence of an applied magnetic field. As well known, a superconductor
has the Meissner effect, i.e., if the magnetic field applying to a superconducting material
is less than the critical field at a sufficiently low temperature, the magnetic field does notpenetrate into the material.
This phenomenon discovered in 1933 by W.Meissner and R.Ochsenfeld, has been interpreted as follows : a superconductor has a property of perfect diamagnetism, so that
surface current occurs on it’s surface and the magnetic feild is excluded from it.
On the other hand, the superconductor can not preserve the Meissner effect under amagnetic field which is much greater than the critical field, so that a penetration ofmagnetic field into the superconductor arises and the superconductivity is lost partially orcompletely.
Our interest is to investigate the behavior of the penetration of magnetic field into thesuperconductor under the situation that a given magnetic field applying to the material isjust around the critical field.
In order to understand this penetration phenomenon, we shall present a mathematicalmodel using boundary conditions of Signorini type in section 1. In section 2, a theoremof existence and uniqueness for the model equation will be shown. Our mathematicaltechnique is to reduce the equation of the above model to certain variational inequality.
Aftcr that, we shall use the standard method to prove the existence and uniqueness theorem(for example, Duvaut-Lions [3] and Glowinski-Lions-Tr\’emoli\‘eres [5]).

$(*)$

Faculty of Engineering, Chiba University, Inageku, Chiba city 263 Japan$(**)$

Energy Division, MEL, Tsukuba city, Ibaraki 305 Japan2

1. Mathematical model for penetration of magnetic field into superconductor. Denote bya domain occupied with aLet be a bounded, connected domain in. According to the theory ofsuperconducting material which is put in and set
the following equations
[2]),
[6],
Du-Gunzburger-Petersonsuperconductivity (Tinkham:
hold in$R^{3}$$\Omega$$\Omega_{0}$$\Omega$$\Omega_{1}=\Omega\backslash \overline{\Omega_{0}}$$\Omega_{0}$

rot

(1.1)

in

$j_{0}=- \frac{c}{4\pi\lambda^{2}}B_{0}$(London equation)$\Omega_{0}$and$4\pi$inrot $B_{0}=\overline{c}j_{0},$ $divB_{0}=0$(1.2)(Maxwell equation)$\Omega_{0}$whereis a current density vector inis a magnetic induction in
represents the penetration depth.
velocity of light, and a parameter
Hence using (1.1) and (1.2), we have$j_{0}$$B_{0}$$\Omega_{0},$and$\Omega_{0}$$c$is the$\lambda$

rot rot

(1.3)

(1.4)

$divB_{0}=0$, the magnetic induction. Let us assume that:
dumping effect inIn$B_{0}+ \frac{1}{\lambda^{2}}B_{0}=0$$\Omega_{1}$$\Gamma_{1}(=\partial\Omega)$$B_{1}$$B_{1}$$in$$\Omega_{0}$in$\Omega_{0}$,.

is generated by a stational current
imposed onequation
amodifiedsatisfies
having aMaxwell’s$j_{1}$$\Omega_{1}$rot rot $B_{1}+\epsilon B_{1}=0$ ,(1.5)

and on the boundary,$\Gamma_{1}$

rot

(1.6)

$B_{1}$$divB_{1}=0$in$\Omega_{1}$satisfies$B_{1} \cross n=\frac{4\pi}{c}j_{1}$,$B_{1}\cdot n=0$

on$\Gamma_{1}$.

) andHere is the unit normal vector (directed toward the exterior of
The first condition of (1.6) represents the direction of the given current
second one also describes that the magnetic field does not exclude from.
$n$$\epsilon(>0)$$\Omega_{1}$$j_{1}$

is small.
and the$\Omega_{1}$Remark 1.1
From a physical point of view, the termin (1.5) describes
a dumping effect. In the case wherein (1.5), (1.5) together with (1.6) describes
Amp\‘ere’s law ;$\epsilon B_{1}$$\epsilon=0$(1.7)$4\pi$.

rot $B_{1}=C \int 1$ on$\Gamma_{1}$, rot $B_{1}=0$ in$\Omega_{1}$and $divB_{1}=0$ in$\Omega_{1}$.

3Finally as the transmisson condition prescribed on the surfaceof the superconductor,
we shall impose the boundary conditions of Signorini type which could be regarded as arepresentation of the Meissner effect :$\Gamma_{0}$(1.8)$B_{0}=B_{1}(\equiv B)$

(1.9)

$|$

(1.10)$|$(1.11)

$|$

rot

rot

rot

$B_{1}\cross n$$B_{1}\cross n$$B_{1}\cross n$$on$$\Gamma_{0}$–rot $B_{0}\cross n|\leq g$,on$\Gamma_{0}$,–rot $B_{0}\cross n|<g\Rightarrow B_{T}=0$ ,–rot $B_{0}\cross n|=g\Rightarrow B_{T}=0$

there exists $\mu>0$ such that

$B_{T}=- \frac{1}{\mu}$$($rot$B_{1}\cross n$or–rot $B_{0}\cross n)$ ,represents an intensity of critical current which is a inherent constant to thiswherematerial, $B_{T}=n\cross(B\cross n)$ and, i.e.,
denotes Euclidean norm in$x=(x_{1},x_{2},x_{3})$
for.
$g$

$|x|$

$R^{3}$$|x|=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}$Remark 1.2
Boundary conditions $(1.9)-(1.11)$ can be rewritten by the following
two equivalent forms (1.12) and (1.13) ;(1.12)$\{\begin{array}{l}rot B_{1}\cross n -- rot B_{0}\cross n|\leq g on \Gamma_{0},g|B_{T}|+ (rot B_{1}\cross n -- rot B_{0}\cross n)\cdot B_{T}=0 on \Gamma_{0}\end{array}$and(1.13)

Here(1.14)$-(rotB_{1}\cross n$$\partial(|x|)$–rot $B_{0}\cross n)\in g\partial(|B_{T}|)$ .

in (1.13) means the multi-valued subdifferential at $x(\in R^{3})$ , i.e.,$\partial$(国)$=\{\begin{array}{l}x/|x| if x\neq 0,\{y\in R^{3};|y|\leq 1\} if x=0 .\end{array}$

We note particularly that these conditions (1.8) -(1.11) are one of the characteristics
of our mathematical modelling (cf. Fujita-Kawarada [4], treating an example appearing
in flow problems).

4

Given a positive constantProblem I$\{B_{0}, B_{1}\}\in(H^{1}(\Omega_{0}))^{3}\cross(H^{1}(\Omega_{1}))^{3}$$g$

and a current$j_{1}\in(H^{-1/2}(\Gamma_{1}))^{3}$, findsatisfying $(1.3)-(1.4),$ $(1.5)-(1.6)$ and $(1.8)-(1.11)$ .

The physical correspondences of the conditions $(1.8)-(1.11)$ are as

Remark 1.3
follows. Set$\Gamma_{00}=\{x\in\Gamma_{0};| (rot B_{1}\cross n - rot B_{0}\cross n)(x)|<g\}$,$\Gamma_{01}=\{x\in\Gamma_{0};| (rot B_{1}\cross n - rot B_{0}\cross n)(x)|=g\}$.

implies that the magnetic field does not penetrate into(i)
superconductivity preserves. i.e., $B_{0}=0$ in( Meissner effect ).
In fact, from (1.10), we have$\Gamma_{00}=\Gamma_{0}$$\Omega_{0}$and the$\Omega_{0}$(1.15)$B_{T}=n\cross(B_{0}\cross n)=0$$on$$\Gamma_{0}$.

Note that rot $B\cross n)\cdot n=0$ . Using (1.15) together with (1.3) and Green’s formula, wethen deduce that$($$0=(rot$$=(rot$$B_{0}$$B_{0}$, rot, rot$B_{0})+ \int_{\Gamma_{0}}($$B_{0})+ \int_{\Gamma_{0}}$ $($rotrot$B_{0} \cross n)\cdot B_{0}d\sigma+\frac{1}{\lambda^{2}}(B_{0}, B_{0})$$B_{0} \cross n)\cdot(n\cross(B_{0}\cross n))d\sigma+\frac{1}{\lambda^{2}}(B_{0}, B_{0})$$=(rot$ $B$ , rot $B_{0})+ \frac{1}{\lambda^{2}}(B_{0}, B_{0})$ ,from whichin$B_{0}=0$$\Omega_{0}$.

(ii)Let be in Fol. Then the penetration of magnetic field into superconduction
arises at because of Lorentz’s force.
In fact from (1.11) 9 it follows that
$x$

$x$

rot

(1.16)$B_{1}\cross n$–rot $B_{0}\cross n=-\mu B_{T}$$=-\mu(n\cross(B\cross n))=-\mu(B-(B\cdot n)n)$Hence(1.17)$(rotB_{1}\cross n$

and$B$. (rot$($$B_{1}$–rot–rot $B_{0}\cross n)\cdot B_{T}=-\mu|B_{T}|^{2}\leq 0$$B_{0}$) $\cross n)=-\mu(|B|^{2}-(B\cdot n)^{2})=-\mu|B_{T}|^{2}$ ,5

from which(1.18)$n$. (rot$($–rot$B_{1}$$B_{0}$) $\cross B)=\mu|B_{T}|^{2}$ .

Therefore we could regard that the left hand side of (1.18) stands for the normal componentof Lorenz’ force operated to the superconducting material. And we could also deduce that
contributes to the penetration of magnetic field
the only tangential component of on
into the superconductor from (1.16) (1.18).
$B$$\Gamma_{0}$-2. Existence and uniqueness theorem for penetration of magnetic field
In this section, we shall prove a theorem of existence and uniqeness for our mathematicalmodel. Here we shall use a variational inequality formulation which is equivalent to theoriginal problem I.
Now let us introduce some notations.
. We, set
For any vector valued function
anddefined in. We then define a Hilbert
recall thatand:space$\Omega_{0}\subset\Omega,$$u_{0}=u|_{\Omega_{0}}$$\Omega$$u$$\Gamma_{1}=\partial\Omega$$\Gamma_{0}=\partial\Omega_{0},$$\Omega_{1}=\Omega\backslash \overline{\Omega_{0}}$$u_{1}=u|_{\Omega_{1}}$

$V$

(2.1)

$V\equiv\{u|u\in(L^{2}(\Omega))^{3}$, rot $u\in(L^{2}(\Omega))^{3},$ $divu=0$ inon$u\cdot n=0$$\Gamma_{1}$$\Omega$,$\}$

equipped with an inner product$(u,$ $v)\equiv(rotu$ , rot $v)_{\Omega}+(u,$$v)_{\Omega}$

for$u,$ $v\in.

V$

We remark that this norm$\Vert u\Vert=\sqrt{(u,u)}$is equivalent to the usual norm of$(H^{1}(\Omega))^{3}$(cf. Duvaut-Lions [3]).

Let $a(u, v)$ be a bilinear form on $V\cross V$ given by(2.2)$a(u, v)=(rotu$ , rot$v)_{\Omega}+ \frac{1}{\lambda^{2}}(u,$$v)_{\Omega_{0}}+\epsilon(u,$ $v)_{\Omega_{1}}$and we define a functional

$j$

(2.3)

$j(u)= \int_{\Gamma_{0}}g|u_{T}|d\sigma$:$Varrow R$for$u,$ $v\in,V$such thatfor $u\in V$.

where $u_{T}=n\cross(u\cross n)$ . Then we set up a variational inequality as follows.

Problem II Find $u\in V$ such that
(2.4)$a(u, v$

一$u)+j(v)-j(u) \geq\int_{\Gamma_{1}}\frac{4\pi}{c}j_{1}\cdot(v-u)d\sigma$

for anyHereafter let us put$B_{0}=u_{0}$and $B_{1}=u_{1}$ .

$v\in V$.

6

Problem I and Problem

Lemma 2.1.

$\Pi$are equivalent.

Let be a solution of Problem I.
Taking the scalar product of (1.3) and (1.5) with $v-u$ and using Green’s formula, wehave$v-u_{0})_{\Omega_{0}}+(rot$ rot$0=$
rot rot

Proof.

$u$$=$$u_{1}+\epsilon u_{1},$ $v-u_{1})_{\Omega_{1}}$$u_{0}+ \frac{1}{\lambda^{2}}u_{0},$$($$($rot$u_{0}$, rot$(v-u_{0}))_{\Omega_{0}}+ \int_{\Gamma_{0}}($rot $u_{0}\cross n)\cdot(v-u_{0})d\sigma$$+ \frac{1}{\lambda^{2}}(u_{0}, v-u_{0})_{\Omega_{0}}+\epsilon(u_{1},$ $v-u_{1})_{\Omega_{1}}+(rotu_{1}$$- \int_{\Gamma_{0}}($rot$u_{1} \cross n)\cdot(v-u_{1})d\sigma-\int_{\Gamma_{1}}$ $($$=a(u, v-u)- \int_{\Gamma_{0}}$$+ \int_{\Gamma_{0}}($rot$u_{1}\cross n$by (18)$($$($rot–rot$u_{1}\cross n$–rot$(v-u_{1}))_{\Omega_{1}}$rot $u_{1}\cross n)\cdot(v-u_{1})d\sigma$$u_{0}\cross n)\cdot vd\sigma$$u_{0} \cross n)\cdot u_{0}d\sigma-\int_{\Gamma_{1}}$ $($i.e., $u_{0}=u_{1}$ on, rot

rot $u_{1}\cross n)\cdot(v-u_{1})d\sigma$$\Gamma_{0})$$\leq a(u, v-u)+\int_{\Gamma_{0}}g|v\tau|d\sigma-\int_{\Gamma_{0}}g|u\tau|d\sigma$$- \int_{\Gamma_{1}}\frac{4\pi}{c}j_{1}\cdot(v-u)d\sigma$

by $(1.9)-(1.11)$ and (1.6)$=a(u, v-u)+j(v)-j(u)- \int_{\Gamma_{1}}\frac{4\pi}{c}j_{1}\cdot(v-u)d$ 伍

Hence it follows$a(u, v-u)+j(v)-j(u) \geq\int_{\Gamma_{1}}\frac{4\pi}{c}j_{1}\cdot(v-u)d$ 伍

This means that is a solution of Problem II.
Next let be a solution of Problem II. By taking $v-u=\pm\varphi(\varphi\in V\cap(\mathcal{D}(\Omega_{i}))^{3},$ $i=$$0,1)$ in (2.4), we find $u=\{u_{0}, u_{1}\}$ such that$u$$u$(2.5)

rot rot$u_{0}+ \frac{1}{\lambda^{2}}u_{0}=0$in$\Omega_{0}$7

rot rot $u_{1}+\epsilon u_{1}=0$

(2.6)

in

$\Omega_{1}$.

It is obvious that $divu=0$ in
and $u\cdot n=0$ on
becauseMoreover from (2.5), (2.6) and Green’s formula, we have$\Omega$$\Gamma_{1}$$0=a(u, v-u)- \int_{\Gamma_{0}}($$- \int_{\Gamma_{1}}$$($$u\in V$.

rot $u_{1}\cross n-rotu_{0}\cross n)\cdot(v-u)d\sigma$

rot $u_{1}\cross n)\cdot(v-u)d\sigma$ ,which, in combination with (2.4), shows that(27)

$j(v)-j(u)+ \int_{\Gamma_{0}}$$+ \int_{\Gamma_{1}}$ $($$($

rot $u_{1}\cross n-rotu_{0}\cross n)\cdot(v-u)d\sigma$

rot $u_{1} \cross n-\frac{4\pi}{c}j_{1})\cdot(v-u)d\sigma\geq 0$ .

In fact,$0=a(u, v-u)- \int_{\Gamma_{0}}$$- \int_{\Gamma_{1}}$$($$($rot$u_{1}\cross n$–rot $u_{0}\cross n)\cdot(v-u)d\sigma$

rot $u_{1}\cross n)\cdot(v-u)d\sigma$ .

$\geq-j(v)+j(u)+\int_{\Gamma_{1}}\frac{4\pi}{c}j_{1}\cdot(v-u)d\sigma$$- \int_{\Gamma_{0}}$ $($rot$u_{1}\cross n$–rot$u_{0} \cross n)\cdot(v-u)d\sigma-\int_{\Gamma_{1}}$ $($rot $u_{1}\cross n)\cdot(v-u)d\sigma$,from which we obtain (2.7). Therefore (2.7) shows that
rot

(2.8)

(2.9)

$j(v)-j(u)$ 十Substituting$\pm\lambda v,$$\lambda>0$

for$\lambda j(v)-j(u)+\int_{\Gamma_{0}}$ $($$\int_{\Gamma_{0}}($$v$$u_{1} \cross n=\frac{4\pi}{c}j_{1}$on$\Gamma_{1}$,rot $u_{1}\cross n-rotu_{0}\cross n)\cdot(v-u)d\sigma\geq 0$ .

in (2.9),

rot $u_{1}\cross n-rotu_{0}\cross n)\cdot(\pm\lambda v-u)d\sigma\geq 0,$$\forall\lambda\geq 0$

8i.e.,(2.10)$\lambda\int_{\Gamma_{0}}(g|v\tau|\pm$$- \int_{\Gamma_{0}}$As$\lambdaarrow\infty$$(g|u_{T}|+$$($$($rotrot$u_{1}\cross n$–rot$u_{0}\cross n)\cdot v)d\sigma$–rot $u_{0}\cross n)\cdot u)d\sigma\geq 0$ .

$u_{1}\cross n$in (2.10), then$| \int_{\Gamma_{0}}($rot–rot$u_{1}\cross n$$u_{0} \cross n)\cdot vd\sigma|\leq\int_{\Gamma_{0}}g|v_{T}|d\sigma$$\leq\int_{\Gamma_{0}}g$

回$d\sigma$,

which means that$($$v \mapsto\int_{\Gamma_{0}}\frac{1}{g}$is continuous on
for the norm

rot–rot $u_{0}\cross n)\cdot gvd\sigma$$u_{1}\cross n$, having the topology induced by$(H^{1/2}(\Gamma_{0}))^{3}$$(L^{1}(\Gamma_{0}))^{3}$, and has norm$\leq 1$on$(L^{1}(\Gamma_{0}))^{3}$.

is dense in$(L^{1}(\Gamma_{0}))^{3}$and the Hahn-Banach theorem,$\int_{\Gamma_{0}}g$

Using the fact that
we have$(H^{1/2}(\Gamma_{0}))^{3}$$\frac{1}{g}$with norm$\leq 1$

(2.11)

Also as$($rot(2.12)$d\sigma$-rot$u_{1}\cross n$$u_{0}\cross n)\in(L^{\infty}(\Gamma_{0}))^{3}$,. Hence$|rotu_{1}\cross n$$\lambdaarrow 0$

回–rot $u_{0}\cross n|\leq g$

a . on$e$$\Gamma_{0}$.

,$\int_{\Gamma_{0}}(g|u_{T}|+$$($rot$u_{1}\cross n$–rot $u_{0}\cross n)\cdot u)d\sigma\leq 0$ .

By use of (2.11), we get$g|u\tau|+$$($rot$u_{1}\cross n$–rot $u_{0}\cross n)\cdot u\geq 0$

a.e. on Fo,

which, in combination with (2.12), shows that$g|u_{T}|+$$($rot$u_{1}\cross n$–rot $u_{0}\cross n)\cdot u=0$a.e. on$\Gamma_{0}$a.e. on$\Gamma_{0}$i.e.,$g|u_{T}|+(rot$Therefore$u$$u_{1}\cross$n-rot

is a solution of Problem I.

$u_{0}\cross n)$$\blacksquare$$\cdot u_{T}=0$.

9We now state the existence and uniqueness theorem for our model.
There exists one and only one $u\in V$ satisfying (2.4).

Theorem 2.1.

proof. First note that the norm
and the norm associated with the inner productgiven by $((u,v))=$ a(u,v) are equivalent. Then $a(u)v)$ is a symmetric bilinear form,continuous on V and coercive on V. Also using the property of trace operator defined$\Vert\cdot\Vert$

is an l.s. . proper convex functional. Hence under these conditions,on(2.4) has a unique solution $u\in V$ (for example see Baiochi-Capel [1], Glowinski-LionsTr\’emoli\‘eres [5] .
$(H^{1}(\Omega_{0}))^{3},$$j$

$)$

$c$

$E$

References[1] C.Baiocchi and A.Capelo, Variational and Quasivariational Inequalities, A
Wiley-Interscience Publication, 1984.
[2] Q.Du, M.Gunzburger and J.S.Peterson, Analysis and Approximation ofGinzburg Landau Model of Superconductivity, SIAM REVIEW, vol.34 (1992),-pp.54-81.
[3] G.Duvaut and J.L.Lions, Inequalities in Mechanics and Physics, SpringerVerlarg, 1976.
[4] H.Fujita- H.Kawarada, Flow Problems with Unilateral (Slip-and Leak-)Boundary Conditions (to appear).
[5] R.Glowinski, J.L.Lions and R.Tk\’emoli\‘eres, Numerical Analysis of Variational
Inequalities, North-Holland, 1981.
[6] M.Tinkham, Introduction to Superconductivity, McGraw-hill, 1975.