ON HOMOGENIZATION LATTICES: SMALL PARAMETER LIMITS, H-MEASURES, AND DISCRETE WIGNER MEASURES
NIKAN B. FIROOZYE
Abstract. We fully characterize the small-parameter limit for a class of lattice models with twoparticle long or short range interactions with no \exchange energy." One of the problems we consider is that of characterizing the continuum limit of the classical magnetostatic energy of a sequence of magnetic dipoles on a Bravais lattice, (letting the lattice parameter tend to zero). In order to describe the small-parameter limit, we use discrete Wigner transforms to transform the stored-energy which is given by the double convolution of a sequence of (dipole) functions on a Bravais lattice with a kernel, homogeneous of degree with N with the cancellation property, as the lattice parameter tends to zero. By rescaling and using Fourier methods, discrete Wigner transforms in particular, to transform the problem to one on the torus, we are able to characterize the small-parameter limit of the energy depending on whether the dipoles oscillate on the scale of the lattice, oscillate on a much longer lengthscale, or converge strongly. In the case where > N , the result is simple and can be characterized by an integral with respect to the Wigner measure limit on the torus. In the case where = N , oscillations essentially on the scale of the lattice must be separated from oscillations essentially on a much longer lengthscale in order to characterize the energy in terms of the Wigner measure limit on the torus, an H-measure limit, and the limiting magnetization. We show that the classical magnetostatic energy with added lattice-induced anisotropies corresponds to oscillations essentially on a much larger lengthscale than that of the lattice and note that this energy is nonlocal in character. We also show that if the square of a suitable extension of the dipoles to <N is precompact in L1 then the part of the limiting energy which corresponds to oscillations essentially on the scale of the lattice is local in the sense that the energy of short lengthscale oscillations generates a nitely additive nite signed measure on Borel sets. Examples are also given where the dipoles concentrate on Lebesgue null-sets and the corresponding energy of short lengthscale oscillations is not local. Several extensions are discussed as well as applications to magnetostatics. Key words. Harmonic Analysis, Pseudo-Di erential Operators, Homogenization, Magnetism, Separation of Scales, H-measures, Wigner measures, Fourier Methods, Lattice Models.
1. Introduction. Lattice models have been used extensively in the study of mechanics in elds ranging from classical magnetostatics to the quantum mechanical structures of materials, (e.g., Bloch 2]), to the modeling of coherent structures (Toda 20], among others). Lattice models are studied especially in their relation to continuum models through the use of small-parameter limits, (e.g., Vogelius 21], Fujiki et al 7], De'Bell and Whitehead 4], James-Muller 10], Khachaturyan 13], etc). Smallparameter limits have been studied in the context of -convergence, where some storedenergy is minimized for every value of the lattice parameter, or in the more general context, where compactness is assumed a priori and the limiting behavior is studied without the more restrictive assumption of energy-minimization. In both the contexts of -convergence and more general small-parameter limits, continuum limits are seen
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 U.K., and Institute for Mathematics and its Applications, University of Minnesota, 514 Vincent Hall, 206 Church St. SE, Minneapolis, MN 55455. This work was partially supported by a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship and SERC grant GR/F86427. Part of this work was completed at SFB 256, University of Bonn. 1
to be a means of extending basic principles to the modeling of macroscopic bodies, and conversely to relate experimental measurements on large bodies to the microscopic properties of underlying material lattices. The goal of this paper is to use Fourier methods and rescalings, (discrete Wigner transforms), to transform small-parameter limits of a class of lattice models with both long-range and short-range two-particle interactions with no \exchange energy," whose energies are given by convolutions, into equivalent convergence problems on tori, then to separate di ering scales of oscillation for dipole sequences so as to completely characterize the limiting energies. Our model problem involves determining the limiting magnetostatic energy of a lattice of magnetic dipoles with no exchange energy and no anisotropy energy other than that intrinsic to the lattice. Some limited -convergence results for speci c models of lattices of magnetic dipoles were found by Khachaturyan 13] where he let particles interact by both long-range interactions and short-range \exchange energies," (which penalize dipole moment oscillations on the same lengthscale as that of the lattice). Vogelius 21] described the -convergence of more general electrical networks, (rather than lattices), but only under the in uence of speci c short-range \exchange energies." Without considering small-parameter limits, Khachaturyan 14] and Chouliourous-Pouget 3] also used lattice models to describe the evolution of magnetic moments with more complicated Hamiltonians (higher-order interactions). Using nite Fourier transforms and theta-functions as a means of approximating small-parameter limits without actually re ning the lattice, Fujiki et al 7] and De'Bell-Whitehead 4] described certain lattice-induced anisotropies for dipole-dipole interactions. More recently, James and Muller 10] have used singular integrals to characterize small-parameter limits of a particular lattice energy for lattices of magnetic dipoles oscillating only on the speci c lengthscales in three dimensions with long-range interactions and no exchange energy. Our study improves upon the results of James-Muller in that we are able to completely characterize the full small-parameter limit of the \no exchange" energy (including lattice-induced anisotropies) for lattices of magnetic dipoles which are allowed to oscillate on any lengthscale in any number of dimensions. We are also able to consider the limiting energy induced by kernels in the class of all smooth functions with the cancellation property, which are homogeneous of degree where N, N being the dimension. We reposit the problem as one on a torus, relying upon the work of S. Wainger 22] on the Fourier transforms of kernels on lattices. In the process, we introduce discrete Wigner measures, the continuous analogs of which have been used by Gerard 9] and Lions-Paul 15] to describe semiclassical limits of Schrodinger equations. After transforming the problem to one on the torus, we cannot pass to the limit directly; it is rst necessary to separate di erent lengthscales of oscillation. Using essential lters to separate various lengthscales, we are able to characterize the limiting energy completely by three complementary tools: the limiting magnetization; the H-measure limit, (see Tartar 19], Gerard 8]), to represent oscillations on a much larger lengthscale than that of the lattice; and the discrete Wigner measure limit, used to represent oscillations only on the scale of the lattice.
2
The \no-exchange" energy of a sequence of dipoles d on the Bravais lattice L = L for > 0 is given by the classical formulation:
e=
X
L ;x6=y
K(x y)d (x)d (y);
where K(z) is a matrix function which has the cancellation property in z and which is homogeneous of degree for N. We show that the limiting energy can be completely characterized by the discrete homogeneous Wigner measure on the torus, the H-measure H of a ltered extension of the sequence d to <N , and the weak limit of this extended sequence d. The energy e converges to
e !e=
where
Z
T
K( )d ( ) + N
Z
SN
G(n)d H (n) + 1
X
Z
<N
^ ^ G( )d( ) d( )d ;
K( ) =
L1 nf0g
e
2 i m K(m);
is the Fourier transform of K on the torus, and in the case where = N, the correction energy G (the lattice-induced anisotropy) is given by asymptotical behavior of the Fourier transform of K near the origin, which can be calculated explicitly and is homogeneous of degree zero. The model kernel K, the Helmholtz kernel, ); K (y) = c(N) (N yiyj
ij
jyjN
jyj2
ij
which is given by Kij = rirj 1( 0), is used in magnetostatics to determine the stored energy in the eld induced by a magnetic dipole. Convolution by K on <N corresponds to projections onto gradient vector elds, (i.e., one part of the Helmholtz decomposition). James and Muller studied the homogenization of the energy given by this kernel in three dimensions. We will consider this kernel among others in any dimension. We begin our study with a discussion of preliminaries in section 2. We also give an overview of some of the complications of our approach. Additionally, we describe the lattice rescaling and the use of discrete homogeneous Wigner measures in our problem, meanwhile transforming our problem to one on the torus. In section 3, we consider sequences of dipoles which are allowed to oscillate only on a much larger lengthscale than that of the lattice (weak-long oscillations), and thus are able to characterize their limiting behavior completely in terms of double convolutions over <N . We consider, in section 4, sequences of dipoles which are allowed to oscillate only on the scale of the lattice (weak-short oscillations), and show that the limiting energy of these sequences is characterized by an integral on the torus with respect to the Wigner measure limit. Since most sequences of dipoles may oscillate on various lengthscales, in section 5 we use lters and diagonalization arguments, (essential lters), in order to separate sequences
3
of dipoles into the parts which oscillate essentially on the scale of the lattice and those parts which oscillate esentially on a much larger lengthscale. We similarly decompose the energy and are thus able to fully describe limiting energies. In sections 6 and 7, we describe the unusual behavior of limiting energies of weakshort oscillations. In section 6, we use the theory of measure lters developed by Firoozye and Sverak 5] to give an example of a sequence which oscillates on the scale of the lattice and to describe the limiting energy of a localization of this sequence to a given domain. We show that the limiting energy acts as a measure on this given domain. In section 7, we show that this example is not unique and that if the square of an extension of the sequence of dipoles to <N does not concentrate on a Lebesgue nullset and no mass escapes to in nity, the limiting energy induces a nitely additive nite signed measure on <N (i.e., there is no long-range interaction energy). By inducing a measure, we mean that if d is a sequence of dipoles oscillating only on the scale of the lattice, then localizing the sequence by multiplying by a characteristic function, d gives a limiting energy e( ), parameterized by the set . This energy is then a nitely additive measure on . Also, in section 7 we give examples of sequences which do concentrate and whose corresponding energies are not additive. In section 8, we apply our results to the case considered by James-Muller 10], Khachaturyan 13], De'Bell and Whitehead 4], and Fujiki et al 7], of magnetostatics in <3. We analyze lattice-induced anisotropies for several types of lattices and explain the signi cance of the weak-long part of the energy. We then describe the importance of the locality of the weak-short part of the energy and its implications. Our analysis of separation of scales may be extended easily to other small-parameter limits in <N where there is a preferred lengthscale, such the characterization of limits of the sort: N K( x y )f (x)f (y)dxdy;
ZZ
^ where jjf jjL2 C, and K is not of any particular homogeneity, but K 2 C0(<N nf0g), N )), and the asymptotics of K at the origin are known. Most of our ^ ^ (or K 2 C0(< results for convolutions on lattices are immediately applicable to problems of this type, and in fact the analogous statements are much simpler in the context of convolutions over <N . Our characterization of the limiting energy is complete because of our ability to separate scales of oscillation. Our techniques of rescaling, using discrete Wigner transforms, essential lters, and blowup techniques, are very general and can be applied to a variety of problems of convolutions on lattices. We easily see that our analysis also applies to a large variety of short-range interactions as well. The considerations of \exchange energies" is equally simple within this framework. 2. Preliminaries and Rescalings. As a starting point, we will introduce the discrete Wigner transform and show its use in the context of convolutions on lattices. Let L be a Bravais Lattice in <N (L = f N riei : ri 2 Zg, where feigN is an orthogonal i=1 1 basis for <N normalized so that the unit cell, U = f siei : si 2 0; 1]g has vol(U) = 1).
P P
4
Let d be a sequence of functions in l2(L ; <M ), where L = L is the rescaled lattice with parameter . The energy of a sequence of dipoles d on the Bravais lattice L for > 0 is given by the classical formulation: e= K( x + y ; x y)d (x)d (y); 2 L ;x6=y
X
where K(x; z) is a matrix function which is smooth in the inhomogeneity x and has the cancellation property in z and which is homogeneous of degree for N in z. Introducing new variables,
e=
= = where K(r; ) =
P
X
r2L =2 ;z2Lnf0g
X
K(r; z)d (r + z=2)d (r K(r; z)d (r + z=2)d (r U (r; )d ;
and
z=2) z=2)
r2L =2 ;z2Lnf0g
X Z
r2L
=2
TN
K(r; )
2iz
z2Lnf0g K(r; z)e
U (r; ) =
X
z2L
d (r + z=2)d (r
z=2)e
2 iz
;
for (r; ) 2 L =2 T N is the discrete Wigner transform of the sequence d and has the property that U 2 l1(L =2) L1(T N ). It also satis es the property
X
r2L
P
=2
U (r; ) = jd j2( );
~ if d 2 l1(L ), where d ( ) = L d ( n)e in , (see Folland 6] for other properties of the continuous Wigner transform). Markowich recently used the Wigner transform for converting Schrodinger equations into Vlasov-Liouville equations 16]. Gerard 9] and Lions-Paul 15] have expanded upon this use, introducing the corresponding continuous Wigner measure solutions to Vlasov-Liouville equations as a tool for obtaining semiclassical limits for Schrodinger equations. We will be concerned instead with spatially homogeneous kernels. Let K be z 1 a kernel homogeneous of degree with N, K(z) = jzj K 0( jzj ), with K 0 2 C 1(S N 1; <M M ) having the cancellation property:
Z
SN
1
K 0(s)ds = 0:
We are interested in the energy, (1)
e=
X
n;m2L n6=m
K(n m)d (n)d (m);
5
and its small parameter limit, where we will assume a priori that a suitable extension of d to <N : (N+ )=2; ~= d (x) def (2) m+ U (x)d (m)
X
m2L
~ is uniformly bounded (weakly compact), i.e., jjd jjL2(<N ) C, where U is the unit cell of the lattice L1 = L, normalized so that jUj = 1, and m+ U (x) is the characteristic ~ function of a translated and scaled cell. The choice of scaling in d is exactly that which insures that e will remain bounded in the limit. As in all homogenization problems, we are interested primarily in the relationship between the limiting energy, e ! e, and ~ the limiting magnetization, d * d, where the convergence is weak-L2(<N ). Let us begin by rescaling:
e=
= = (3) where h (n) = =
=2 d
X
n;m2L n6=m
X
K(n m)d (n)d (m) K (n m) d ( n)d ( m) K(n m)
=2 d
n;m2L1
X
n;m2L1
X
( n)
=2 d
( m)
n;m2L
K(n m)h (n)h (m);
( n) is de ned on L1 and we note that ~ jjh jjl2(L) = jjd jjL2(<N ):
Using Plancherel's theorem, (3) can be written as (4) where
e=
Z
Q
K( ) h ( ) h ( ) d ;
X
K( ) =
P
n6=0;n2L
K(n)e
2 in
;
and h ( ) = n2L1 h (n)e 2 in are functions periodic on the unit cell Q of the reciprocal lattice. We will not make any distinction between the unit cell Q and the N-torus T N below, and will use Q to represent both. We have chosen the notation f to denote the Fourier transform of functions de ned on the lattice L, and we will reserve the notation f^ for the Fourier transform of functions de ned on <N . The relationship ~ between h ( ) and the Fourier transform over <N of d (x) will be useful to us at a later point: (5) ~ h ( )^U ( ) = S 1 d ]( );
c
6
where (6)
U
is the characteristic function of the unit cell U of the lattice L, and
x); ^ is an isometry on L2(<N ). We note that S f] = S 1 f]. ~L To show existence of the limit, we note that jj jh j2jjL1(Q) = jjd jj2 2(<N ) C 2 and that, due to the homogeneity and cancellation property, K 2 L1 (Q). In fact we can say more about K following Wainger, ( 22], see also Stein-Weiss 18], ch. 7, x6.1): Theorem 1 (Wainger,1965). Let K(x) = jxj K 0 (x=jxj) for N, with K 0 2 C 1(S N 1) and
d Z
S f](x) =
N=2f(
S
N1
K 0(s)ds = 0;
and, without loss of generality, assume that
1 0 (s) = X X a Y (s); K l;m l;m l=1 m
< N + , where where Yl;m is a spherical harmonic of degree l and N smallest integer for which l < implies that al;m = 0 for all m. Let
1 (t) = 0 for t 2 ; 1 for t 1 with 2 C 1( 1; 1) and 0 (t) 1 for all t. Let F ( ) = (jxj)e jxjjxj K 0 x
(
is the
Then lim !0 F ( ) exists for all 6= 0. Letting F( ) = lim !0 F ( ), we nd that F is in nitely di erentiable for all 6= 0. Also, for any integer r, jF ( )j = O(j j r ) uniformly in 0 as ! 1. Moreover, we have that F is bounded at the origin and at = 0,
jxj
_
:
F( ) = j j
where
NK ~
;N
j j + E( );
1 ( 2 (N + l )) 1 ( 2 (l + )) Yl;m(s);
~ K ;k (s) =
N
XX
1
l= m
( i)lal;m
is smooth on S N 1 , with E( ) = O(j j ) + o(1) as j j ! 0. Also, F is dominated by an L1 function and
x ^ F(x) = (jxj)jxj K 0 jxj :
7
An easy application of Poisson's summation formula to Theorem 1 gives the following theorem, which we will make extensive use of below. Theorem 2 (Wainger,1965). Let K(x) be as in Theorem 1. Let f ( ) = e 2 im e jmj jmj K 0 m :
X
m6=0
jmj
Then lim !0 f ( ) exists for all which are not lattice points. Letting K( ) = lim !0 f ( ), we nd that K 2 Cper (Q n f0g) \ L1 (Q) and K is bounded at the origin and K( ) = F( ) + L( ); where F is as in Theorem 1 and L 2 C 1 (Q), with X L( ) = F( + n):
n6=0
Moreover, K has the following asymptotics at the origin:
K( ) = j j
~ where K
;N
NK ~
;N
0 j j + S + L ( );
is as de ned in Theorem 1 and
S = lim !0
X
is a lattice dependent constant, ) Sj = o(1) as j j ! 0. Furthermore c K(n) = K(n). From this point on, we will refer to the asymptotics at the origin for K( ) as G( ), i.e.,
L1 nf0g and jL0 ( )j = jL(
e
jmj jmj
m K 0 jmj ;
(7)
G( ) def j j =
NK ~
;N
~ where K ;N and S are as de ned in Theorem 1 and 2, respectively. A simple corollary to Theorem 2 is that K 2 U(Q), the subset of the bidual of Cper (Q), C (Q), of universally integrable functions, since it is easy to see that it can be represented both as K = inf sup G ; and as K = sup inf L ; , where G ; ; L ; 2 Cper (Q), (see Kaplan 12]). Thus K can be integrated against any nite Radon measure 2 M(Q) = Cper (Q). If > N then K 2 Cper (Q). In order to represent the limiting energy, we rst rewrite (4) as
j j + S;
e = K( )d ( );
where the measure d ( ) = h ( ) h ( )d is the discrete homogeneous Wigner measure induced on the torus. The mass of the Wigner measure is uniformly bounded since ~L jj jjM(Q) = jjh jj2 2(Q) = jjd jj2 2(<N ) C 2: L
8
Z
Thus, we can use the compactness of the unit ball in the vague topology on measures to extract a subsequence, which we call such that * . We call the discrete Wigner measure limit of h . Note that is an M M Hermitian matrix of measures, i.e., ij = ji which is positive de nite in the sense that Q M ij i j 0 for i;j i 2 Cper (Q), i = 1; . . . ; M. In the case where > N, the Wigner measure limit tells us everything we need to know about the limiting energy; since K 2 Cper (Q) in this case, we will be able to pass to the vague limit directly. Some properties of the limiting energy will be described in sections 3 and 4. If we also know that no mass is \lost at in nity," condition (32), ~ and that the sequence jd j2 is strongly precompact in L1, we will also show in section 7 that the limiting energy generates a nitely additive nite signed measure on <N in the ~ generates the limiting energy e( ) which is a measure on sense that the sequence d the Borel set . The situation is much more complicated in the case when = N. It is clear that we cannot just represent the limiting energy e by the limiting Wigner measure alone, since K 62 Cper (Q), so we cannot pass to the vague limit under the integral. But since K is well behaved everywhere except at the origin, all is not lost, and a more re ned ~ analysis of the sequence d is necessary. We will see in section 4 that the Wigner measure limit is su cient to describe the limiting energy only in the case when the ~ oscillations of d are essentially on the scale of the lattice. If the sequence d oscillates only on a much larger scale or converges strongly, the Wigner measure limit will be a point mass at the origin|exactly where K fails to be continuous. Therefore, in this case the Wigner measure limit will tell us nothing of the limiting energy. Conversely, we will nd that the part of the Wigner measure limit singular with respect to a point mass at the origin depends only on those oscillations essentially on the scale of the lattice. The complications involved with characterizing the limiting energy in the weak-long case and the blowup techniques used to determine this energy will be described in section 3. We remark that in the case where < N (when K 62 L1 (Q)), boundedness of the ~ energy requires much more complicated bounds on d and on the growth (or decay) of ~ d^ ( ) as j j ! 0. We will not be discussing these cases of homogenization on lattices in the paper because the nature of these bounds is entirely di erent from those bounds we have stated above. 3. Long Waves and Strong Convergence. Following James and Muller ( 10]), ~ ~ we say that d * d converges weak-long if the L2-modulus of continuity, 2 d ]( ), converges to zero, i.e., ~ ~ ~ (8) 2 d ]( ) = sup jjd ( + ) d ( )jjL2 (<N ) ! 0
R P
as ! 0, where is large enough so that B \ L1 contains N independent vectors (i.e, ~ U B ). If d ! d strongly in L2(<N ), then d * d weak-long as well. We note the following properties of the modulus of continuity (see, for example 18], x1.1): (9a) 2 f](c) 2jjfjjL2 (<N ) ;
9
2B
for any c > 0,
f fjjL2 2 f]( ); by Minkowski's inequality, where (x) = (1= N ) (x= ) is an approximation of the identity with supported in the unit ball (i.e., 2 C0(B1), (x)dx = 1), (9c) 2 f](c ) c 2 f]( ); where c is a positive integer, and (9d) 2 f](a) 2 f](b); when a b. Note too that 2 S f]](1) = 2 f]( ), where S is the isometry on L2(<N ) de ned in (6). We also de ne the modulus of continuity for functions de ned on the lattice L, i.e., 2 h](1) = sup jjh( + ) h( )jjl2 (L):
(9b)
R
jj
2L\B
Note that properties (9a{9d) are also true of the modulus of continuity on the lattice, so we can prove the following lemma: ~ Lemma 1. The moduli of continuity 2 h ](1) and 2 d ]( ) are equivalent, i.e., ~ 2 h ](1) 2 d ]( ) c 2 h ]( );
where c 1. Proof. RecallP S f](x) = N=2 f(x= ). Since that ~ ](x) = L h (m) m+U (x), we nd that since S d
2 2
~ d ]( ) =
2
~ S d ]](1), and
h ](1) = sup
2B
2
X
sup j = =
2
2B \L
Z
jh (m + ) h (m)j2
h (m)
m+U (x +
X
)
X
h (m)
m+U (x)j
2dx
2 ~ S d ]](1) 2 ~ 2 d ]( ) : To prove the opposite inequality, we rst x x and . Then j h (m) m+U (x + ) h (m) m+U (x)j2 0 if x and x + are in the same cell, = jh (m0) h (m)j2 otherwise, with m0 a neighboring lattice point to m. Thus, j h (m) m+U (x + ) h (m) m+U (x)j2 jh (m0) h (m)j2;
X X ( X X X
10
m0 =m+ 2B L
giving us,
2
~ S d ](1)
2
X
X
m 2B \L1
jh (m + ) h (m)j2 jUj jh (m + ) h (m)j2 jUj #fB L1g;
where # is the cardinality function. We remark that, although h is uniformly bounded in l2(L1), and thus has a weakly convergent subsequence, (h * h weak-l2(L1)), in the case of weak-long oscillations this weak limit contains no information; rescaling gives us that 2 h ](1) ! 0; and by lower semi-continuity of the norm, we see that 2 h](1) = 0, thus h must be a constant and since it is also in l2(L), the constant must be zero. The measures , being \concentration" measures, may contain more information. Heuristically, that h converges to a constant implies that h h converges to a constant and thus that converges to a point-mass at the origin. Let us make this rigorous: ~ Lemma 2. If d * d weak-long or strong, then the Wigner measure limit has the property that 0 62 supp j j. Proof. By properties (9b) and (9c), 2 h ](1) ! 0 as ! 0 implies that
X
2B \L1 m
sup
X
jh
hj =
2
Z
= (1 ( ))2trfd g( ) Q !0 for any 2 l1(L1 ) with nite support and L = 1 (i.e., for any trigonometric poynomial with (0) = 1), where trfd g( ) is the trace of the matrix of measures ( ). Thus, by passing to the limit,
P Z
Z
jh ( )j2(1 Q
( ))2d
for all such trigonometric polynomials. But, since the trigonometric polynomials are dense in the uniform topology in Cper (Q),
Z
Q
(1
( ))2d ( ) = 0;
for any 2 C0(B2 ) with 0 1, B2 Q, and 1 in B , i.e., supp trf g\(Qn B ) = ; for all > 0, which by the positive de niteness of implies that suppj ij j \ (Q n B ) = ; for all i; j = 1; . . . ; M, where j ij j is the total variation of ij .
Remark 1. Using similar methods to the above, we can also show that
Q
(1
( ))trfd g( ) = 0;
lim !0
Z
(1 <N
=
~ ~ )( )d ( ) d ( )d = 0;
c c
11
for any > 0 where = ( ) = (( = ) ) with 2 C0 (B2), 0 1 and 1 inside B1 . We can use the last lemma to prove the following: ~ Lemma 3. If d * d weak-long or strong, then the Wigner measure limit is a point mass at the origin, i.e., Z ~ ~ * = 0 lim d (x) d (x)dx: Conversely, if * c 0, then condition (8) is satis ed. Proof. Let f be a continuous periodic function on Q. Then
!0 <N
(10)
Z
Let 2 C0(B2 ) with 0 1 and ( ) 1 for 2 B , where > 0 is xed. Since f is continuous, lim !0 sup 2B jf( ) f(0)j = 0, and then
Z
Q
f( )d ( ) = f( ) h ( ) h ( ) d :
Z
Q
f( ) f(0) fd g( )
Z
jf( ) f(0)j ( )trfd g( ) + jf( ) f(0)j(1 )( )trfd g( ) sup jf( ) f(0)j jh j2( )d
Z Z
+2jjfjjC(Q) (1 sup jf( ) f(0)j C 2 (11)
B
B
Z
)( )jh j2d )( )jh j2d ;
+2jjfjjC(Q) (1
Z
since jjh jjL2 C. Applying Lemma 2, we see that the second term on the right of (11) tends to zero for arbitrary . Letting ! 0 we see that the rst term on the right-hand side of (11) also tends to zero. Thus,
Z
f( )d ( ) ! f(0) lim d ( )
= f(0) lim h ( ) h ( )d Q ~ ~ = f(0) lim <N d (x) d (x)dx;
Z
Z
Q Z
where we have used the fact that 1 2 Cper (Q) to show that the limit in fact exists and Plancherel's theorem and the de nition of h to show the connection between h and ~ d. To prove the converse, rst let us transform the weak-long condition on the lattice to a condition on the torus, i.e., we wish to show that if * c 0, weak-? in the sense of measures, then lim !0
2
h ](1) = lim 2B \L1 (e max !0
12
Q
Z
i
1)2d ( ) = 0:
First note that the maximum is over nitely many points 2 B \ L1, and we can change the order of the maximum and the limit. Since * c 0 weak-? in the sense of measures, g ( ) = (e i 1)2 is continuous and g (0) = 0, Q(e i 1)2d ! 0. Thus, trivially, 2 h ](1) ! 0. So, when = N in the cases of weak-long or strong convergence, the Wigner measure limit is concentrated exactly where K is discontinuous, and thus o ers no useful information for characterizing the limiting energy. Fortunately, by a blowup argument, we are able to say more: ~ Theorem 3. Suppose that d * d weak-long or strong L2 (<N ). Then, if > N, ~ ~ e = K( )d ( ) ! lim hS d (x); d (x)idx:
R Z Z
If = N and G( ) = K ;N ( =j j) + S (as de ned in equation 7),
Q
(12)
e=
Z
Q
K( )d ( ) ! lim !0
Z
where the right-hand side of (12) can be represented by an H-measure limit, i.e., Z Z ^ ^ e= G(s)d H (s) + G( )d( ) d( )d :
SN
1
<
N
~ ~ G( ) d ( ) d ( ) d ;
c c
H
and the weak
<N
that (13)
Proof. Before proceeding, note that jjGjjC0 (<N )
Z
jjKjjC (Qnf0g). First, let us show
0
converges to zero. We can bound (13) above by
Z
Q
K( ) h ( ) h ( )d
Z
Q
G( )h ( ) h ( )d
= (14)
Q Z Q
jK( ) G( )jjh ( )j2d
( )jK( ) G( )j jh ( )j2d + (1
Z Z
sup jK( ) G(
2B
)jC 2 + 2jjKjjL1
(1
Q
)( )jK( ) G( )j jh ( )j2 d )( )jh j2d ;
where 2 C0(B2 ), 0 1 and 1 inside B . By Lemma 3, for any > 0, the second term on the right of (14) tends to zero as ! 0. Since was arbitrary, we let it tend toward zero, making the rst term on the right of (14) go to zero too by Theorem 2. In the case where > N, we nd that G S + O(j j N ), thus
Z
Q
(G( ) S) h ( ) h ( )d
Z
Q
( )jG( )j jh ( )j2d
Z
+ (1
Q
)( )jG( )j jh ( )j2 d
Z
(15)
13
N C 2 + jjGjj 1 L
(1
)( )jh j2d ;
where is as before. Again we note that the second term on the right of (15) tends to zero as ! 0 for any > 0 by Lemma 3. Letting ! 0, we nd that the rst term on the right of (15) also tends to zero and thus, e ! lim hh ; h id , which by the de nition of h and the Plancherel's theorem, gives the stated result. In the case where = N, the result is slightly more di cult. We must use equation (5) and the fact that ^U is twice di erentiable at the origin, (in fact it is analytic by Paley-Wiener), ^U ( ) = 1 + o(j j) as ! 0 since jUj = 1 and U is symmetric about the origin. Then, by using the homogeneity of G and changing variables,
R Z
Incorporating this,
Z
Q
G( ) ^U (
)]2h
( ) h ( )d =
c
Z
Q (1= )(
~ ~ )G( )d ( ) d ( ) d :
c c
<N
Q (1= )(
~ ~ ) 1 G( ) d ( ) d ( )d
c Z
=
Z
()
Q (1= )( =
+ (1
~ ) 1 jd j2 jG( )jd ~ )( ) Q (1= )( ) 1 jd j2 jG( )jd
c c
~ (16) + 2 d ]( 1) : By choosing small enough and then letting ! 0, we see that the right of (16) tends to zero. By the continuity of ^U at the origin we can see that
Z
jjGjjL1 C 2 sup 2B
2
Q(
)1
also tends to zero. Thus we have proved the result. Note that, because of the homogeneity of G, the limit (17) can be represented by an integral with respect to a homogeneous H-measure induced ~ by (d d) * 0 in L2, (see Tartar 19], and Gerard 8]) and by an integral of the weak-limit, i.e., ^ ^ e= G(n)d H (n) + G( )d( ) d( )d :
Z Z
Q
jG( )( ^U ( )]2 1)h ( ) h ( )jd
Z
lim !0
<
~ ~ G( )d ( ) d ( ) d N
c c
~ In the case where d ! d strongly in L2(<N ), (17) becomes ^ ^ G( ) d( ) d( ) d :
Z
SN
1
<N
<N
Thus, in the case of weak-long or strong convergence, we retrieve the classical continuum energy with some extra lattice-induced anisotropy terms given by S.
14
verges weak-short if (18)
4. Weak-Short Oscillations. We say that a sequence d~ * d weak-L2(<N ) conlim lim !0 !0
X
m2L
j
1=
h j2 = 0;
where 1= (x) = ( )N ( x), with (x) = ( + 1)jxj N=2 JN=2+ (2 jxj) with JN=2+ a Bessel function of degree N=2 + for > (N 1)=2 xed. Note that is the kernel used for taking Bochner-Riesz means, (see Stein and Weiss 18], Chapter 4, Theorem 4.15), and that (1 j j2 ) if j j 1 ( ) = ^( ) = (19) 0 if j j 1. We will see below that our choice of is one of convenience; any other (x) with nice decay at in nity which has ( ) = ^ ( ) monotone decreasing in j j, and continuous with bounded support and with (0) = 1 would su ce in our de nition, and the conditions of bounded support and monotonicity can also be relaxed. We will see in Remark ~ 3 that in order for (18) to be satis ed, oscillations in d must be essentially only on the scale of the lattice and that this condition is in direct contrast to condition (8). Although our terminology is borrowed from James and Muller 10], our de nition of weak-short is di erent from theirs; they ne de it to be not weak-long, which would thus allow oscillations on the scale of the lattice as well as on much longer lengthscales. The utility of our de nition will become apparent from the results presented in this section, and those in section 5 where we demonstrate that any sequence can be separated into its weak-short and weak-long parts. We will see in Remark 2 that condition (18) is equivalent to the following:
(
lim lim !0 !0
Z
<N
j
=
~ d j2dx = 0;
where = (x) = ( = )n ( x= ) is as de ned above. We claim that under the above conditions, the Wigner measure limit of the ~ sequence d is singular with respect to a point mass at the origin. Let us make this rigorous: ~ Lemma 4. Let d * d converge weak-short, i.e., let h satisfy (18), then = h h d * where ? 0, i.e., ( = )d ( ) ! 0 as ! 0 for 2 C0(B1). Proof. Because of nice decay at in nity, satis es the hypotheses for Poisson's summation formula, thus, taking the Fourier transform of condition (18),
R
(20) since
lim lim Q j ( )h ( !0 !0
X
Z
)j2
= lim lim Q j ( )j2trfd g( ) = 0; !0 !0 =
X
Z
N
L
( m)e
2m
15
L
+ m def ( ); =
where L is the reciprocal lattice to L and where has support in the unit ball. For small enough we see that ( ) = ( = ) for 2 Q. Because * and 2 Cper (Q), the limit in in (20) exists and we have that lim j ( = )j2trfd g( ) = 0:
Q
Z
Due to monotonicity in , the limit in exists. We also see that for any 2 C0(B1), lim Q ( = )trfd g( ) = 0, and due to positive de niteness, this is also true for ij and j ij j for i; j = 1; M, where j ij j is the total variation of ij . Thus j ij j ? 0. Remark 2. We can now easily see that condition (18) is equivalent to the following condition: ~ (21) lim lim j = d j2dx = 0;
R Z
!0 !0
where = (x) = ( = )N ( x= ). By rescaling (21) and using Plancherel's theorem, we see that Z 1Z j c ~ ]j2dx = 1 ~ (1 j = j2)2 jS 1 d ]j2d 1= S d 2 2 jj Z X (1 j = j2 )2 jh j2d = j 1= h j2
jj
Z
2
jj
(1 j = j
2 )2
~ jS d
1
c
]j2d
L
=2 j
Z
1=
~ S d ]j2dx;
for small, where we have used (5) and the fact that ^U ( ) is continuously di erentiable at the origin, and ^U ( ) = 1 + o(j j) for j j ! 0. Passing to the limits we see the equivalence immediately. Remark 3. We note that the much more easily veri able condition, X 1 X h 2 = 0; (22) lim sup lim N !0
!1
m2L k2B (m)\L
implies (18). Note too that in order for condition (22) to hold, oscillations must be only on the scale of the lattice and that this is in direct contrast to condition (8). To show that condition (22) implies (18), we take the Fourier transform of (22): X 1 X h (k) 2 = Z jf ( )h ( ) j2d ; N
m2L k2B (m) L
where f ( ) = Pn2B \L (1= N )e 2 in is a sequence of trigonometric polynomials which have f (0) = 1 and f 0 (0) = 0, with jjf jjL2 (Q) ! 0 as ! 1. We claim that there exists a constant M > 0 such that for all > 0 there exists an 0 and for any 0, we have the following inequality:
(23)
1
2
16
Mjf ( )j;
for all j j
. If we choose
1=2
= 1 and rescale, we obtain the inequality
2
(24)
1
p
M
X
N=2
X
B Lp
e
2 im
;
for all j j 1=2. The right hand side of (24) converges pointwise to j^B1 j. Since ^B1 is bounded away from zero in a neighborhood of the origin, and because of the monotonicity of the left-hand side of (23) in we see that inequality (23) must be true as according to the claim. Thus, condition (22), a much easier condition is seen to imply (18). We use methods similar to those used to prove Lemma 4 to show that in the case of weak-short convergence the Wigner measure gives all the information necessary to nd the limiting energy. ~ Theorem 4. If d * d weak-short, then
(25a) (25b) (25c)
e ! K( )d ( ) = lim (1 !0
Z Z
Z
Z
( = ))K( )d ( )
= lim K ( )d ( ) !0 = lim f ( )d ( ) !0 1,
Z
where ( ) 2 C0(B1 ) with (0) = 1, 0
and where f is the regularization de ned in Theorem 2. Proof. First, let us start with the easiest regularization of K in (25a), which e ectively carves out the origin, i.e.,
Z
K ( ) = jB1jN
B()
K(y)dy;
K( )d ( )
Z
K( )(1
( ))d ( )
Z
(1 (1
Z
+ j j jKjtrfd g( ) )K(d ( ) d ( )) +jjKjjL1 j jtrfd g( ):
Z
Z
)K( )(d ( ) d ( ))
(26)
Note that K(1 ) 2 Cper (Q) for all > 0 and, thus by the vague convergence of to , for small, the rst term on the right-hand side of (26) is arbitrarily small. Condition (18) implies that the second term on the right-hand side of (26) is arbitrarily small for and small. Thus, the right hand side of (26) tends to zero. Continuing, we now turn to the justi cation of (25b). We see that K ( ) ! K( ) pointwise on Q, and thus K * K weak-? in Cper (Q), (see Kaplan 12], x54.2), and also uniformly on Q n B for > 0. Using this, we estimate the following:
Z
K( )(1
( = ))d ( )
Z
Kd
Z
Z
Q Q
(K K )d +
Z
K ( = )d
Z
(K K )d + jj
KjjL1 (d )
( = )d :
17
We rst claim that jjKjjL1(d ) < 1, since jjKjjCper < 1 (i.e., it is bounded in the sup norm). We know that (K K)d ! 0 due to weak-? convergence in Cper (Q). We also know that ( = )d tends to zero from the fact that is bounded with compact support and from condition (18). Thus, we have justi ed (25b). Finally, the justi cation of (25c) is quite similar to that of (25b). Theorem 2 states that f ! K pointwise on Q n f0g, and thus that f (1 ) * K(1 ) weak-? in Cper (Q) for all > 0. Thus,
R R Z
K(1
R
( = ))d
Z
fd
Z
(27)
Z
(K f )(1 (K f )(1
( = ))d + j j jf jd ( = ))d + jjf jjL1 (d
)
Z
Z
j jd :
Again (K f )(1 ( = ))d tends to zero as ! 0 for all > 0 by weak-? convergence. We also claim that jjf jjL1(d ) < 1 independent of because it is uniformly bounded in the sup norm and thus, by condition (18), that the right-hand side of (27) tends to zero, completing the proof. Concluding, we see that the Wigner measure limit on the torus fully describes the limiting energy. Moreover, the Wigner measure limit gives us the limiting energy explicitly if we have complete knowledge of the pointwise behavior of K. In section 7 we will describe some further properties of the energy of weak-short oscillations. 5. Separation of Scales. We are able to show that general sequences can be decomposed or \ ltered" into their weak-long and weak-short parts and the energy can thus be decomposed into the constituent parts. This is important since most sequences ~ d neither oscillate on the scale of the lattice alone nor on a much longer lengthscale alone and thus, neither the analysis for weak-short nor for weak-long oscillations applies in the form given in sections 3 or 4. Our method for decomposing the sequence of dipoles into its constituent parts is to nd an \essential" lter, one which lters out the oscillations which are essentially on the scale of the lattice and calls them the weak-short part, and also lters out the oscillations essentially on a much larger scale and calls these the weak-long part. We will prove the following: ~ Theorem 5. Let d * d weakly in L2 and let f g >0 , (with * weak-? in the sense of measures), be the induced sequence of Wigner measures on the torus Q. Then there is a decomposition =
L + S;
where L * a0 0 weak-? in the sense of measures and S * S weak-? where S ? 0 and = a0 0 + S . Moreover, L (or the corresponding ltered sequence of dipoles, ~ dL * d) induces an H-measure H with total mass equal to a0 so that the total energy converges to
e!
Z
SN
1
G(s)d H (s) +
Z
Q
K( )
18
S(
)+
Z
<N
^ ^ G( )d( ) d( )d :
Proof. Let ( ) = ( = ) be as de ned in equation (19). This will be our lter. Since we are unable to decide a priori which frequencies to place into the weak-short part and which into the weak-long part, we use the intermediary step:
= S; + L; = (1 j j2) + j j2 : We will show that S; satis es the weak-short condition (18) and L; satis es the weak-long condition (8) if we choose = ( ) appropriately. The rest of the proof will be devoted to this choice of ( ). First let us check that L; satis es (8): xing > 0, So we must choose = ( ) ! 0 as ! 0, which gives us that lim (1 j j2)d !0
Z
lim (1 j j !0
Z
2)d L;
= lim (1 j j2)j j2d = O( j = j2jj jj): !0
L; ( ) = 0;
Z
for all > 0, verifying (8). We remark that (8) is even more easily veri able if we test against trigonometric polynomials 2 C0(B2 ) with 1 inside B . Second, let
a0 = lim lim j ( )j2 d ( ); !0 !0
(we note that the limit in exists because ( ) = L ( + m) since supp Q implying that 2 Cper (Q) and because the limit in exists from the monotonicity of ). From a diagonalization argument of Attouch ( 1], Lemma 1.15 and Corollary 1.16), we can choose = ( ) ! 0 as ! 0 such that ( ) and
P
Z
a0 = lim Q j !0
lim lim j j2d !0 !0
Z
Z
()
j2 d :
()
Letting ( ) = ( ) as de ned above, then the weak-short condition (18), can be tested:
S; ( )
= lim lim j j2d !0 !0 = a0 a0 = 0;
= lim lim j j2(1 j !0 !0
Z
Z
Z
j2)d j ( ) j2 j j2 d
since for j j lim lim 1
( )< , ( )2 2
Z
j
2 ( )j d = lim lim 1
( )2 1 ()
2
Z
1 1
lim lim
19
Z
22
()
22
d
22
d
= lim lim j lim lim (28)
Z
Z
()
j2 j j2 d
22
1
(
= lim lim j
Z
() 2 )j d
d
and since ( ) ! 0 as ! 0, ( ( ) ), we see that both upper and lower bounds in (28) converge to a0. Thus, for ( ) as chosen above, S; ( ) satis es condition (18) and L; ( ) satis es condition (8). Letting S = S; ( ) and L = L; ( ), we obtain the desired decomposition. We can trivially decompose the energy:
e = Kd + Kd L ;
S
Z
Z
The weak-short part is easily dealt with by the methods of section 4. The weak-long part is also easily dealt with once we realize that
j^U j2 L = j^U j2j ( )j2h h ~ ~ = j ( )j2S 1 ^ ] S 1 ^ ] d d
~ ~ = S 1 d^L ] S 1 d^L ] ~ ~ d where d^L ( ) = ( = ( ))^ ( ). We also know that ( )= (x) = ( ( )= )N ( ( )x= ) = L is a sequence of Wigner measures generated by the sequence ( )= (x) Thus, L= ~ ~ ~L d ( )= d . Since jjd jjL2 < 1 by Minkowski's inequality, a subsequence is weakly ~ ~d ~ convergent. In fact, dL * d; it is easy to see that because ( ) , d^L ^ * 0 weak-? ~ in the sense of measures, and thus weakly in L2. Thus, the sequence dL d * 0 gener~ ates an H-measure, L , (generally not equal to the H-measure generated by d d * 0), H L = a0 . We can see that the theory of section 3 then applies and we get with SN 1 d H the desired energy decomposition,
R
e!
Z
Q
R
K( )d
S(
)+
Z
SN
1
G(n)d
L (n) + H <N
Z
^ ^ G( )d( ) d( )d :
We note that the weak-short part of the energy can also be represented by the singular integral: lim !0 K d , where 2 C0(B ) and ( ) 1 for 2 B =2. 6. An Example of Weak-Short Oscillations. We already know that the energy associated with weak-long oscillations converges to a double convolution over <N ~ and that this is nonlocal by nature, i.e., if d converges weak-long then the limiting ~ ~ energy associated with 1 d and that associated with 2 d do not sum to give the ~ energy associated with ( 1 + 2 )d , where 1 and 2 are disjoint sets. There is an interaction energy for assembling multi-bodied systems of weak-long dipoles. We can also
20
characterize this weak-long energy explicitly with the use of H-measures. The weakshort part of the energy is not so easily characterized, since we would need to know K completely on Q, (whereas Wainger's results 22] only give us complete knowledge of its asymptotics around the origin). However, we are able to characterize the behavior of the weak-short part of the energy as being essentially local in nature when the sequence ~ jd j2 is strongly precompact in L1 and no mass escapes to in nity, (which we will do in section 7). We provide this example, a generalization of example 8.4 from James and Muller ( 10]), as a motivation for this approach. Let (x) satisfy the hypotheses of Poisson's summation formula, i.e., j (x)j A(1 + jxj) n and ^ = with j ( )j A0(1 + j j) n , and let be a measure on the torus, Q, with Fourier series: ( )=
X
a(m)e
2 im
:
= N and the sequence
We will consider a kernel K which is homogeneous of degree of dipoles,
d (m) =
N
(m)a(m= ):
The corresponding rescaled function h is given by h (m) = N=2 ( m)a(m). Note ~ that d is a weak-short sequence. By Poisson's summation formula, + m def L ( ); 1 N=2 ( m)e 2 im = = N=2
X X
L
L
where L is the reciprocal lattice to L. Then h = L . In a similar fashion to Firoozye and Sverak, ( 5]), we prove the following: Theorem 6. Under the above hypotheses, where also can be decomposed into its atomic and di use parts, i.e., = a + d , where a = ai xi, the Wigner measure sequence = h h d converges weak-? in the sense of measures to j j2dx a2 xi . i Proof. We note rst that h = L = am N=2 ( m)e 2 im x and that since decreases rapidly at in nity and am 2 l1(L), the sum converges absolutely. We can also bound L uniformly in L2, i.e.,
P R P P Z
jL j2dx = Q
=
X
LX N j L
j
N=2
( m)j2 (m)j2;
and, since 2 L2 \C 0, this Riemann sum converges to jj jj2 2 , thus the Riemann sums L are uniformly bounded (and also will be bounded if = with a bounded domain) Using Jensen's inequality, we nd that
jj jh j2jjL (Q) = jjL
1
jj2 (Q) jjL jj2 (Q) jj jj2 : L L M(Q)
2
21
2
So we know that, after passing to a subsequence, jh j2 * weakly in the sense of measures. Also it is simple to show that h * 0 weakly in l2. First, we show that jL j2 * 0 j j2dx weak-? in the sense of measures. Let f 2 Cper (Q). Then N=2 x + m j2f(x)dx jL j2f(x)dx = j
R Z Z X
Q
= (29) since
P
Z
QL
= (y + m= ) = (y) +
X P
Z
j (y) + O( Q (1= )
X
j Q (1= )
X
(y + m= )j2 f( y)dy
N+
)j2f( y)dy;
m6=0
(y + m= ) and A (y + m= ) N+ m6=0 (1 + jy + m= j)
m6=0
C
N+
R
;
for y 2 Q1= . Finally, the right hand side of (29) converges to j j2dx f(0) by the dominated convergence theorem. We are now prepared to prove the theorem. Let have the decomposition into di use and atomic parts, = a + d where a = ai xi . For the time being, suppose that d = 0. Then
P Z
jh j2f(x)dx = Q jL Q
=
Z Z
Z
j2f(x)dx
N=2 N=2
= = (30)
Z
Q Q
(
((x y)= ) + O(
N=2+
))
X
ai xi (y)dy f(x)dx
2
X
((x xi)= )ai + O( ((x xi)= )j2f(x)
N=2+
X
jaij) 2f(x)dx
Z
X
Q
a2 i
N
Z
Nj
X
+
Q i6=j
aiaj ((x xi)= ) ((x xj )= )f(x)dx + O( 2 );
P
and we see that the rst term on the right of equation (30) converges to a2f(xi). The i second term on the right of (30) converges to zero because of the growth conditions at in nity, (see Firoozye-Sverak, 5], Theorem 1). If d = 0 then we have two other terms: 6 (31) jL ( a + d )j2f(x)dx= (jL
ZZZ Z Z
aj
2+
jL
dj
2 + 2hL
a; L
d i)f(x)dx:
Following Firoozye-Sverak, we use Fubini's theorem to expand the second term thus:
L (x y)L (x z)f(x)dxd d (y)d d (z) = F (y; z)d(
R
Z
d
d )(y; z);
where F (y; z) = L (x y)L (x z)f(x)dx. It is quite simple to see that F (y; z) concentrates on the diagonal = f(y; z) : y = zg. We also know that ( d d) ] 6= 0
22
is equivalent to d(fxg) 6= 0 for some point x 2 Q, which is clearly impossible. Thus the second term in equation (31) converges to zero. We deal with the third term in equation (31) similarly, noting that ( a d) ] = 0 as well. Thus, we have proven the theorem.
a characteristic function as well, since jL j is still uniformly bounded in for 2 L2 (<N ), and all other results are easily seen to be extendable to the general case. This gives us the \locality" of the energy. In other words, we have proved the following: Corollary 7. Let ( ) be a nite measure on the torus, with = a + d , where a = P ai xi , with Fourier coe cients a(m). Let the dipole moments d (m) = N (m)a(m= ), where is the characteristic function of a nite volume set, . Then the stored-energy associated with the sequence of dipoles fd g is
2
Remark 4. We note that the above result holds for
L1(Q)
e( ) =
Z
Q
K( )d a ( ) vol( );
a - nite measure. Of course, this example of \locality" of the weak-short part of the energy leads one to suspect that locality is a general quality of weak-short oscillations. We will see below that this is indeed true but only under special circumstances. 7. The Locality of the Weak-Short Energy. The weak-long part of the energy is represented via limits of convolutions over <N and is non-local in that if we were to assemble a system of two magnetic dipoles each with weak-long internal oscillations, the energy of the system is di erent from the energy of the separate components. The example of the preceeding section leads one naturally to the question of whether the energy given by weak-short oscillations is local, i.e., additive. We will see in this section that under the condition of \no mass-loss at in nity," (a condition that the example of the preceding section does not satisfy),
(32)
lim sup !1
Z
jxj
~ jd j2dx = 0;
~ and that if the sequence jd j2 is strongly precompact in L1, (a condition which the example of the preceding section satis es), the energy of weak-short oscillations is indeed local. In fact, it will generate a measure. Before we proceed, we must phrase the concept of \locality" in a mathematically ~ rigorous language. Suppose d is a sequence of dipoles converging weak-short in L2. Then, as we have seen above, the energy e ! Q K( )d ( ), where is the Wigner measure limit of the sequence d . Let be the characteristic function of a set <N . ~ Then d converges weak-short again and induces a Wigner measure and the corresponding energies converge e ( ) ! Q K( )d ( ) = e( ). We will show that the energy is \local" in the sense that the energy parameterized by , e( ) is in fact a measure on . We will prove the following:
R R
23
~ to in nity, condition (32), and = N. Let <N . Then d also converges weakshort and induces a Wigner measure limit on the torus, (where is a parameter). R The limiting energy e( ) is given by e( ) = Q K( )d ( ). This energy e( ) is a nitely additive nite signed measure on Borel sets in <N . The same result holds in the ~ case where > N when d is a sequence of dipoles converging weak-short, weak-long, or strong satisfying condition (32). Before proceeding to the proof of this theorem, we need several approximation lemmas. First we will show that our kernel can be approximated by functions of bounded support, i.e., kernels whose Fourier transform on the torus are trigonometric polynomials. ~ Lemma 5. Let d be a sequence of dipoles. Then the energy associated with a kernel of degree where > N can be approximated arbitrarily closely by kernels ~ of nite support. If d converges weak-short and = N then the same approximation result holds. Proof. Consider the energy associated with the kernel f with nite support, i.e., where f is a trigonometric polynomial:
~ Theorem 8. Let d converge weak-short in L2 (<N ), assume that no mass escapes
ef =
= =
X
X
L
f( x y )h (x= )h (y= )
L Z
Z
f(x y)h (x)h (y) f( )h ( ) h (x)d
where ( ) 2 C0(B1) with (0) = 1 and 0 1. The function (1 ( = ))K( ) 2 Cper (Q) and thus can be approximated arbitrarily closely in the uniform norm by trigonometric polynomials. Thus, eK can be approximated arbitrarily closely by ef where f is a kernel of nite support when the sequence of dipoles converges weak-short. We will also show that the domains can be approximated: ~ in L1. Then the limiting energy ef (B) associated with B (x)d (x),...