Weierstrass-type maximum principle for microstructure in

Max-Planck-Institut

fur Mathematik in den Naturwissenschaften Leipzig Weierstrass-type maximum principle for microstructure in micromagnetics by

Martin Kruz
k and Tom
as Roub
cek Preprint no.:

40

1999

Weierstrass-type maximum principle for microstructure in micromagnetics Martin Kruz
k1 and Tom
as Roub
cek2 1 Institute of Information Theory and Automation, Academy of Sciences, Pod vod
arenskou vez
 4, CZ-182 08 Praha 8, Czech Republic. 2 Mathematical Institute, Charles University, Sokolovsk
a 83, CZ-186 75 Praha 8, Czech Republic. 1

Abstract. We derive necessary and su

cient optimality conditions for a relaxed (in terms of Young measures) variational problem governing steady states of ferromagnetic materials. Such conditions here stated in the form of a generalized Weierstrass maximum principle enable us to establish uniqueness of a solution in some specic situations and can also be used in e cient numerical algorithms solving the relaxed problems, for instance. Key words. Calculus of variations, convexication, ferromagnetism, micromagnetics, optimality conditions, relaxation, Young measures. AMS subject classi
cations: 49K20, 49K35, 49S05

1. Introduction

Steady-state con
gurations of mechanical systems are usually governed by an energyminimization type principle. In past centuries, this led to a development of the variational calculus, which resulted in formulations of optimality conditions in terms of Euler-Lagrange equations or Weierstrass maximum principle. Sometimes, the involved energy is not convex in highest derivatives, which causes \physically" a development of a microstructure and \mathematically" a failure of existence of a solution. To describe the microstructure in detail and to overcome the failure of existence, the original problem is to be extended suitably. In some situations, it may happen that the extended (relaxed) problem has a convex structure with respect to some geometry not necessarily compatible with the \natural" geometry of the original nonconvex problem. Then one can formulate the optimality conditions. For the case of the scalar variational problems this results in one half of the Euler-Lagrange equation combined with the Weierstrass maximum principle, see 28, Section 5.3]. The identi
cation of a linear structure that makes the relaxed problem convex and formulation of corresponding optimality conditions is a basis for construction of eective numerical algorithms for relaxed problems, cf. 7, 20, 28]. Let us still remark that other geometries applied to the relaxed problem may lead to other optimality conditions, cf. e.g. Chipot and Kinderlehrer 8], DeSimone 12] or Pedregal 26]. The goal of this paper is to adapt the above ideas to a steady-state micromagnetics. The variational problem, stated in Section 2, was already formulated in 3, 4, 5, 16] while its extension, stated here in Section 3, was formulated in 12, 24, 25, 27]. Our original results, i.e., the optimality conditions for the extended problem, are formulated in Sections 4 and 5 in terms of a Weierstrass-type maximum principle in the integral form (Propositions 1 and 3) and also pointwise (Propositions 2 and 4). Some consequences are mentioned in Section 6. 1

2. Steady-state model of micromagnetics

In the classical theory of rigid ferromagnetic bodies, based mainly on works of Landau and Lifshitz 22], a magnetization m :  ! IRn, describing the state of the body 
IRn, n = 2 3, depends on a position x 2  and has a given temperature-dependent magnitude

jm(x)j = const(T ) for almost all x 2  with m(x) = 0 for T  Tc the so-called Curie point. We will treat the case when the temperature is
xed below the Curie point and thus we shall assume that jmj = 1 almost everywhere in . In the so-called no-exchange formulation, the energy of a large rigid ferromagnetic body 
IRn consists of three parts and the variational principle governing

steady-state con
gurations can be stated as follows (see e.g. Brown 3, 4, 5], Choksi and Kohn 9], James and Kinderlehrer 16], James and Muller 17], Kinderlehrer and Ma 18], Tartar 29], etc.): 8 >> minimize E (m u) := Z h'(m(x)) ; H (x)  m(x)i dx + 1 Z jru(x)j2 dx e >> 2 IR < subject to jmj = 1 on  (1) > >> div(ru ; m) = 0 in IRn >: m 2 L1( IRn) u 2 W 1 2(IRn) where ' : IRn ! IR is continuous, m :  ! IRn is the magnetization, He : IRn ! IRn is a given external magnetic
eld, and u :  ! IR is a potential of the induced magnetic
eld, and  : IRn ! f0 1g denotes the characteristic function of . The
rst term in E is an anisotropy energy with a density ' which is supposed to be an even nonnegative function depending on material properties and exhibiting crystallographic symmetry. Two important cases are the uniaxial case, where ' attains its minimum along one axis, and the cubic case when it attains its minimum along three axes. The second term involving He is an interaction energy and the last term is a magnetostatic energy related with the magnetization
eld m through u = div(m ). This equation stems from the Maxwell equations (omitting constants) n

div B = 0

(2)

curl H = 0

where B is magnetic induction and H intensity of the magnetic
eld. By the de
nition, B = H + m and H = ;ru. Then u = div(m) follows immediately. Let us notice that the weak formulation of this equation reads (3)

Z h

8v 2 W 1 2(IRn) :

IR

n

In particular, putting v := u, we have

Z

ru(x) ; m(x) (x)i rv(x) dx = 0 Z

:

jru(x)j2 dx =  m(x)  ru(x) dx IR which gives krukL (IR IR )  kmkL (IR ) by the Holder inequality. It follows from the LaxMilgram lemma that (3) has for any m 2 L2( IRn) the unique solution u 2 W 1 2(IRn)

(4)

n

2

n

n

2

n

2

and that the mapping m 7! ru is linear and weakly continuous. Hence the magnetostatic R 1 energy m 7! 2 IR jru(x)j2dx is sequentially weakly lower semicontinuous. As the set of admissible magnetizations fm 2 L1( IRn) jmj = 1g is not convex, we cannot rely on direct methods (see e.g. 11]) in proving the existence of a solution to (1) cf. 16] for failure of existence of a solution in a uniaxial case. More precisely, if the weak limit of some minimizing sequence of m's in (1) lives for almost all x 2  in the unit sphere then this is the strong limit cf. 21, p. 99]. Therefore, a so-called
ne structure (or, in the \limit" we will speak about a microstructure) in m will typically develop, and we have to look for a notion of generalized solutions and to formulate a so-called relaxed problem. Let us emphasize that the
ne structure in m is actually observed in real ferromagnetic materials, see 15]. n

3. Relaxation in terms of Young measures

We need to describe suitably an oscillating character of sequences f(mk uk )gk2IN
1 L ( IRn) W 1 2(IRn) minimizing sequence (1). It is well known (see 2, 10, 30]) that we can extract a subsequence (denoted, for simplicity, by the same indices) and
nd u 2 W 1 2(IRn) and a family of probability measures  fxgx2 such that supp(x )
S n;1 := fs 2 IRn jsj = 1g which is weakly measurable in the sense that v   is Lebesgue measurable for any v 2 C (S n;1), and

w-lim uk = u & w*-lim v  mk = v   k!1 k!1 for any continuous function v : IRn ! IR, where the limits refer respectively to the weak R n 1 2 1 topology in W (IR ) and the weak* topology in L (), and v   ](x) := S ;1 v(s)x(ds) for almost all x 2 . Let us denote the set of all  fxgx2 with the above listed properties by Y ( S n;1) such  's are called Young measures. Conversely, for any  2 Y ( Sn;1) there is a sequence of measurable functions mk :  ! S n;1 such that the later convergence in (5) is ful
lled. The relaxation of the problem (1) was done by DeSimone 12], Pedregal 24, 25], Rogers 27] etc. The continuously extended problem obtained by this way looks as follows: 8> Z Z Z   1  >> minimize E ( u) :=  S ;1 '(s) ; He (x)  s x(ds) dx + 2 IR jru(x)j2 dx <  Z  Z (6) > subject to ru(x) ; S ;1(x) s x(ds) rv(x) dx = 0 8v 2 W 1 2(IRn) >> IR :  2Y ( S n;1) u 2 W 1 2(IRn) : (5)

n

n

n

n

n

The probability measure x describes in a proper (we may say \mesoscopic") way the microstructure of the \limit" magnetization at a point x. The extended problem (6) is a correct relaxation for the original problem (1). Indeed, by 12, 25], the in
mum of E is attained and it is equal to the in
mum of E . Moreover, having ( u) a solution to (6), then there is a sequence (mk uk ) 2 L1( IRn) W 1 2(IRn) satisfying uk = div(mk ) in the weak sense, jmk j = 1 a.e., minimizing E , and attaining ( u) in the sense (5). Conversely, every sequence (mk uk ) 2 L1( IRn) W 1 2(IRn) 3

satisfying uk =div(mk  ) weakly, jmk j = 1, and minimizing E , contains a subsequence attaining some ( u) 2 Y ( S n;1) W 1 2(IRn) in the sense (5), and every ( u) obtained by such way solves the relaxed problem (6). One can also think about a \coarser" relaxation in terms of the original \macroscopic" magnetization m. We denote by S ;1 the indicator function of the unit sphere, i.e. n

(

jsj = 1 S ;1 (s) = 0+1 ifotherwise : n

Furthermore, by v we denote the second Fenchel conjugate (the convex envelope of v), i.e. v = supfw convex w  vg. This can be used to pose the following relaxed problem: 8 Z Z >> minimize E~ (m u)= ' + S ;1 ](m(x)) ; He(x)  m(x) dx + 1 jru(x)j2 dx < 2 IR  (7) > subject to div(ru ; m) = 0 in IRn >: m 2 L1( IRn) u 2 W 1 2(IRn) : Note that ' + S ;1 ] equals +1 outside the unit ball in IRn so that any minimizer (m u) of (7) must satisfy jm(x)j  1 for a.a. x 2 . DeSimone 12] showed that E~ always attains its minimum on the considered admissible set, and this minimum is equal to the in
mum of (1). For any s 2 IRn, one has n

n

n

(8)

' + S ;1 ] (s) = n

inf

Z


probability on S ;1 R measure ;1 
(d)=s n

S ;1 n

'()(d) :

Sn

Note that, for jsj > 1, the set of 's considered in (8) is empty so that the in
mum in (8) is +1. It is clear that if ( u) minimizes E then (m u) with

m(x) =

Z

S ;1 n

sx(ds)

minimizes E~ . Said dierently, a unique minimizer of E implies a unique minimizer of E~ . The opposite implication does not hold because,
xing some m 2 L1( IRn) with values in the unit ball centered at the origin, we might still have many (even continuum of) minimizers of E with the
rst moment m, cf. Example 3 below. Clearly, the only term responsible for uniqueness/nonuniqueness is '. Recently, Carstensen and Prohl 6] showed that if ' corresponds to uniaxial ferromagnets, i.e. ' is nonnegative and equals zero at two points s 2 S n;1, and has a given representation, then E~ has a unique minimizer. The Euler-Weierstrass-type optimality conditions for the corresponding E will enable us to prove that under some properties of ' there also exists a unique Young measure-valued minimizer, see Examples 1{2 below. See also DeSimone 12] for earlier results on uniqueness of Young measure solutions in the uniaxial case.

4

4. Optimality conditions in terms of and u

It is usual to identify a given Young measure  2 Y ( S n;1) with the linear functional in L1( C (S n;1)) de
ned by

h hi =

(9)

Z Z

 S ;1 n

h(x s)x(ds)dx:

Thus Y ( S n;1) can be considered as a convex weakly* compact subset of L1 ( C (S n;1)), see 28, Corollary 3.1.7]. Furthermore, let us de
ne # : L1 ( C (S n;1)) W 1 2(IRn) ! W ;1 2(IRn)  = W 1 2(IRn) by the formula

Z

hv #( u)i = ;h rv  idi +  rv(x)  ru(x) dx for v 2 W 1 2(IRn), where naturally rv  id](x s) := rv(x)  s. Let us note that #( u) = 0

(10)

just means that u solves

Z 

(11)

IR

n

ru(x) ;

Z



(x) s x(ds) S ;1 n

rv(x) dx = 0 8v 2 W 1 2(IRn) :

Also note that # is (weak* weak,weak)-continuous and surjective in the sense that (12) 8f 2 W ;1 2(IRn) 9u 2 W 1 2(IRn) 9 2Y ( S n;1) : #( u) = f which follows immediately from the Lax-Milgram lemma. The relaxed problem (6) now takes the following abstract form: 8 >> minimize E ( u) < (13) >> subject to #( u) = 0 : ( u) 2 Y ( S n;1) W 1 2(IRn): Note that E is convex, # is linear, and Y ( S n;1) is convex, so that the problem (13) has a convex structure. As E is Gate^aux dierentiable and 0 2int(#(Y ( S n;1) W 1 2(IRn))) due to (12), it is known (see e.g. Aubin and Ekeland 1, p.175]) that the
rst-order optimality conditions looks as follows: E 0( u) 2 ;NKer  \ (Y (S ;1)W 1 2 (IR )) ( u) = Range # ; NY (S ;1)W 1 2(IR )( u) = Range # ; NY (S ;1)( ) f0g where E 0 = (E0 Eu0 ) : L1 ( C (S n;1)) W 1 2(IRn) ! (L1( C (S n;1)) W ;1 2(IRn))  = L1 ( C (S n;1)) W 1 2(IRn)) denotes the G^ateaux dierential of E and # = (# #u) : W 1 2(IRn) ! L1 ( C (S n;1)) W ;1 2(IRn) is the adjoint operator to #. Moreover, NY (S ;1)W 1 2(IR ) ( u) denotes the normal cone to the convex set Y ( S n;1) W 1 2(IRn) at the point ( u), and analogously NY (S ;1) ( ) is the normal cone to Y ( S n;1) at  , i.e. n o NY (S ;1) ( ) := 2 L1( C (S n;1))  8~ 2 Y ( S n;1) : h ~ ;  i  0 : n

n

n

n

n

n

n

n

5

n

Therefore, we can deduce that, if ( u) 2 L1 ( C (S n;1)) W 1 2(IRn) solves (13), then there is a Lagrange multiplier 2 W 1 2(IRn)  = W ;1 2(IRn) such that (14) #u ; Eu0 ( u) = 0 # ; E0 ( u) 2 NY (S

(15)

;1 ) ( )

n

see 28, Sec. 5.3]. As the problem (13) is convex, the conditions (14){(15) are also su%cient in the sense that, if ( u) 2 Y ( S n;1) W 1 2(IRn) satis
es #( u) = 0 and (14){(15) for some multiplier 2 W 1 2(IRn), then ( u) solves (13). The abstract conditions (14){(15) turns for the concrete data E from (6), # from (10) and Y ( S n;1) into the following integral maximum principle:

Proposition 1. Let H e n

2 L2( IRn), ' : IRn ! IR be continuous, let ( u) 2

L1 ( C (S n;1)) W 1 2(IR ) solve (13) with the data from (6) and (10). Then Z (16) h Hui = m2Lsup Hu(x m(x)) dx 1 (IR )  jmj=1 a.e. where the Hamiltonian Hu :  IRn IR is dened by n

Hu(x s) := ;ru(x)  s + He(x)  s ; '(s) : Conversely, if ( u) 2 Y ( S n;1) W 1 2(IRn) satises #( u) = 0 and if the maximum (17)

principle (16) holds, then ( u) solves (13).

Proof.n Let us evaluate the dierential of E . As for Eu0 ( u) 2 L(W 1 2(IRn) IR) =

W ;1 2(IR ), we have

Z

hEu0 ( u) vi =  ru  rv dx while for E0 ( u) 2 L1 ( C (S n;1)) = L(L1 ( C (S n;1)) IR) we have (19) E0 ( u)](~ ) = h~ 1  ' ; He  idi where naturally 1  '](x s) = '(s) and He  id](x s) = He(x)  s. (18)

The equation (14) now gives

(20)

Z

D 0 E Z  rv  r dx = h #(0 v)i = h#u vi = Eu( u) v =  ru  rv dx 

for any v 2 W 1 2(IRn), from which we get simply = u+ constant. As should live in W 1 2(IRn), this constant must vanish so that we eventually = u. The inclusion (15) results in the inequality D E 0  # ; E0 ( u) ~ ;  = h #(~ ;  0)i ; h~ ;  1  ' ; He  idi = h~ ;  ;r  id ; 1  ' + He  idi 6

for all ~ 2 Y ( S n;1). This gives h~ ;  Hi  0 with the Hamiltonian H = Hu given by (17). By (20), H = Hu. In other words, we got

h Hui = ~2Ymax h~ Hui : (S ; )

(21)

n

1

As E0 as well as # take their values in L1( C (S n;1)) rather than in L1 ( C (S n;1)), we can take into considerations only the intersection of the normal cone NY (S ;1) ( )
L1 ( C (S n;1)) with L1 ( C (S n;1)) as was already done in 28]. Hence, n



NY (S ;1)( )\L1 ( C (S n;1)) = h 2 L1 ( C (S n;1)) 8~ 2Y ( S n;1) : h~ hi  h hi Z n o 1 n ; 1 = h 2 L ( C (S )) h hi = sup h ( x m ( x )) d x m2L1 (IR )  jmj=1 a.e. which eventually gives us (16). As the problem (13) is convex, the maximum principle (16) is also su%cient in the above speci
ed sense. 2 n

n

Thanks to the special form of the set of admissible magnetizations in (1) admitting arbitrary oscillations of m, the integral maximum principle (16) can be localized into the following pointwise maximum principle, which gives a very explicit restriction on possible steady-state microstructures. Proposition 2. Let ( u) 2 Y ( S n;1) W 1 2(IRn) solve the relaxed problem (13). Then

Z

(22)

S ;1 n

Hu(x s)x(ds) = smax 2S ; Hu(x s) n

1

for a.a. x 2 

with the Hamiltonian again from (17). In other words,

supp(x)
ArgmaxHu(x ) where ArgmaxHu(x ) := fs 2 S n;1  Hu(x s) = max Hu(x S n;1)g. Conversely, if ( u) 2 Y ( S n;1) W 1 2(IRn) satises #( u) = 0 and (23) holds for a.a. x 2 , then ( u) solves the relaxed problem (13). (23)

Proof. We will show that (16) and (22) are equivalent to each other. Due to 13,

Th 1.2. Ch. VIII], there exists m~ :  ! S n;1, measurable such that maxs2S ;1 Hu(x s) for almost all x 2 . First, suppose that (16) is ful
lled. Therefore,

Hu(x m~ (x)) =

n

h Hui

=



Z Z

 S ;1 n

Z



Hu(x s)x(ds) dx =

Hu(x m~ (x)) dx =

Z

max  s2S ;1 n

7

sup

Z

m2L1 (IR ) jmj=1 a.e. n

Hu(x s) dx :



Hu(x m(x)) dx

In other words,

Z Z



Hu(x s)x(ds) ; smax 2S ; Hu(x s) dx  0 :  S ; R At the same time, S ; Hu(x s)x(ds)  maxs2S ; Hu(x s) for almost all x 2 , which n

n

1

n

1

n

1

1

shows that (22) holds. Let now (22) be satis
ed. Integrating it over  one gets

h Hui =

Z

max  s2S ;1 n

On the other hand,

Hu(x s) dx 

sup

Z

m2L1 (IR ) jmj=1 a.e. n



Z

Hu(x m(x)) dx   Hu(x m~ (x)) dx :

R max ;1 H (x s) dx = R H (x m~ (x)) dx and thus (16) holds. 2 u s2S   u n

5. Optimality conditions in terms of and m

One can also alternatively consider optimality conditions when the energy functional is supposed to depend on the \mesoscopic" Young-measure magnetization  and the \macroscopic" magnetization m. Interestingly, it turns out that such optimality conditions are the same as those derived in the previous section. In order to show this we will de
ne a new functional e : L1 ( C (S n;1)) L2( IRn) ! IR by

Z Z e( m) = h 1  'i ; He(x)  m(x) dx + 21 

IR

n

jru(x)j2 dx

with ru determined via u = div(m). Eventually, we de
ne  : L1 ( C (S n;1)) L2 ( IRn) ! L2 ( IRn) by ( m) = id   ; m and we see that  (Y ( S n;1) L2 ( IRn)) = L2 ( IRn). Thus we are concerned with

8 >> minimize e( m) < (24) >> subject to ( m) = 0 : ( m) 2 Y ( S n;1) L2 ( IRn) : Note that  is continuous and linear and e is convex. We will also show that e is Gate^aux dierentiable. The
rst order optimality conditions read in this case that, if ( m) 2 Y ( S n;1) L2( IRn) solve (24) then there is a Lagrange multiplier ` 2 L2( IRn) such that (25)

m ` ; e0m ( m) = 0

` ; e0 ( m) 2 NY (S ;1) ( ) where e0 = (e0 e0m) : L1( C (S n;1)) L2 ( IRn) ! L1 ( C (S n;1)) L2 ( IRn) and  = ( m ) : L2 ( IRn) ! L1 ( C (S n;1)) L2 ( IRn).

(26)

n

8

L2( IRn), ' : IRn ! IR continuous, let ( m) Y ( Sn;1) solve (24) and let u solve (3). Then Z (27) h h`i = m~ 2Lsup h`(x m~ (x)) dx 2 (IR )  jm~ j=1 a.e. where the Hamiltonian in now dened as

Proposition 3. Let He L2 ( IRn)

2

2

n

h` := `  id ; '

(28)

with ` = He ; ru.

Proof. First we prove that e0( m) = (' ;He + ru). The
rst component of e0,

namely e0 , is obvious because e( m) is a%ne. As to the second component, we denote by w the solution to (3) with arbitrary v 2 L2 ( IRn) instead of m. Then we have Z e0m ( m)](v) = ;He  v + 21 ddt jru(x) + trw(x)j2 dx jt=0 Z IR = ;He  v + ru(x)  rw(x) dx n

=

ZIR

n

;He  v +  ru(x)  v(x) dx = (;He + ru)  v

where we used, beside (3) with v instead of m, also the linearity of the mapping m 7! ru. Furthermore, for any ` 2 L2 ( IRn) it holds ` = (`  id ;`) because h` ( m)i = h` ( m)i = D` id  E ; h` mi = h `  idi ; h` mi : The relations (25) and (26) now turn respectively into (29)

` = He ; ru

(30)

;' + `  id 2 NY(S ; )() : n

1

Again, since e0 as well as  take their values in L1 ( C (S n;1)) rather than in L1 ( C (S n;1)) , we obtain the claimed maximum principle. 2

Proposition 4. Under the assumptions of the previous proposition (31)

h`   ](x) = smax 2S ;1 h`(x s) for a.a. x 2  : n

The proof of the above point-wise version is analogous as that one of Proposition 2. We point out that h` = Hu provided ` = He ; ru so that, in fact, Propositions 1 and 2 are equivalent with Propositions 3 and 4, respectively. 9

6. Some consequences

The following proposition gives a su%cient condition under which the relaxed problem (13) has a unique minimizer. This condition is indeed satis
ed in some physically relevant situations, see Examples 1 and 2 below, while in other situations admitting many minimizers is not satis
ed, see Example 3.

2 IRn, the function S n;1 ! IR: s 7! r  s ; '(s) attain its maximum at a nite number (r)  n + 1 (r) of points rl such that dim(spfrl gl=1 ) = (r) ; 1. Then (13) has a unique solution. Proof. The proof paraphrases that one of 28, Corollary 5.3.4]. Let (m u) 2 2 L ( IRn) W 1 2(IRn) be a unique minimizer of (7) and let ( 1 u1) and ( 2 u2) be two 1 1 2 2 1 1 Proposition 5. Let (7) possess a unique minimizer and let, for any r

dierent solutions to (13). Let us denote m = id   and m = id   . As (m u ) and (m2 u2) must solve (7), we get

m1 = m = m2 and u1 = u = u2: Then the Hamiltonian is determined uniquely, i.e. Hu1 = Hu2 . By (23), and the assumption, the probability measure xi must be supportedPat a
nite number k(x) = (x) ai (x) i

(He(x) ; ru(x)) of points sl (x) = Hl e (x);ru(x) , i.e., xi = kl=1 s (x) with al  0, l Pk(x) ai = 1 a.e. in . By (32), we have l=1 l

(32)

l

kX (x) l=1

(a1l (x) ; a2l (x))sl (x) = m1 (x) ; m2 (x) = 0 :

The full-rank condition yields a1 = a2 a.e. in .

2

Proposition 6. Let the assumptions of Proposition 5 be fullled. Then the original

problem (1) possesses a solution (being equal just to (u m), the assumed unique solution to (7)) if and only if, for a.a. x 2 , it holds (33)

(r) m(x) 2 frl gl=1

with r = He(x) ; ru(x) :

Proof. The unique solution ( u) to (13) is 1-atomic at a given x 2 , i.e. of the form

x = m(x) , if and only if (33) holds. If (33) holds a.e., (m u) then solves (1). Conversely, if (m u) solves (1), then ( u) with x = m(x) solves (13). As this solution is unique, failure of (33) for x from a positive Lebesgue measure set implies failure of existence of any solution to (1). 2

Example 1. (Uniaxial magnets I.) Let us take n = 2, ' : S 1 ! IR

'(s) = s21 + minf(s2 ; 1)2 (s2 + 1)2g such potential is related with the so-called uniaxial ferromagnet. For this type of ' there is a unique solution to (7) cf. DeSimone 12] or also Carstensen and Prohl 6]. As n = 2, 10

we have S 1 = f(cos  sin )  2 0 2]g and, for () := '(cos  sin ), it is easy to see that () = 2 ; 2j sin j. For r = (r1 r2) 2 IR2, let us further denote f ( r) = () ; r1 cos  ; r2 sin . Dierentiating f with respect to  we have

( @f ( r) = r1 sin  ; (r2 + 2) cos  if  2]0  r1 sin  ; (r2 ; 2) cos  if  2] 2 : @ We see that f ( r) has at most four local extrema at  = 0,  = ,  = 1 2 0 ] and  = 2 2  2], where ( ( r2 +2 if r 6= 0 arctan r2r;1 2 if r1 6= 0 arctan 1 r1 1 :=  and  := 2 3 if r1 = 0 if r1 = 0: 2 2

On the other hand, there is no r 2 IR2 for which f (0 r) = f ( r) = f (1 r) = f (2 r). This shows together with Proposition 3 that (r)  3, r 2 IR2, and that for ' as above (13) has a unique solution. In truth, one can even show that (r)  2, so that (13) has a solution ( u) with x = (x)s1 (x) + (1 ; (x))s2 (x) for almost all x 2 .

Example 2. (Uniaxial magnets II.) By similar arguments one obtains uniqueness also for n = 2, ' : S 1 ! IR, '(s) = '(s1 s2) = s21 as already observed in 12]. Example 3. (Cubic magnets.) Let us take n = 3, ' : S 2 ! IR, '(s) = '(s1 s2 s3) = s21s22 + s21s23 + s22s23 :

Then one can see that for He = 0 there are many solutions to (13), for example, x = 1 1 1 1 2 (0 1) + 2 (0 ;1) , x 2 , u = 0 or x = 2 (1 0) + 2 (;1 0) , x 2 , u = 0. Note that the assumptions of Proposition 5 are indeed not satis
ed because (0) = 6 > n + 1 = 4.

Acknowledgment: The research was conducted during the stay of M.K. in the Max-Planck-

Institute for Mathematics in the Sciences, Leipzig, Germany whose hospitality is gratefully acknowledged. The work was supported by the grant A 107 5707 (Grant Agency of the Academy of Sciences of the Czech Republic).

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