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u* - 0,
Lemma 2 can be proved using Tartar's method described in
2. HOMOGENIZATION OF STRATIFIED STRUCTURES
Of great importance for applications in technology is the study of processes in bodies made of stratified elastic materials.
The
elastic properties of such bodies vary rapidly in only one direction. In
this
case one can obtain of
coefficients
the
explicit
homogenized
formulae which
elasticity
express through
system
the the
coefficients of the initial system for a stratified elastic material. Thus
it
becomes possible to
find
in
explicit
characteristics of a stratified elastic body.
form the subject of a vast literature
Stratified composites
in mechanics
Composites with plane layers are considered in
many effective
form
(see e.g.[111).
We study here
[36].
stratified structures without any assumptions on their periodicity (see [37]).
Consider a sequence
ZE
of differential operators of linear
elasticity
8au I
XE (u) = axi (CEj ('p(x),x...... xn) x J j where
CEJ(cp(x),xl,.... xn)
CE1h(t,y), y - (y5,...,yn)
functions of derivatives components
t E R', in
smooth domain in
p(x)
and
Rn,
(16) whose
(n x n) - matrices
are bounded
y e Rn
y...... ynr
u5,...,uns
are
- f,
(uniformly
with bounded
u - (u1,...un)
is
Txiwxi
e)
measurable
in
e)
first
a column-vector with
is a scalar function In 0 t T(x) c 1,
in
(uniformly
elements
C°(li),
12
is a
3 const > 0.
Here and in what follows we assume summation over repeated indices from
1
to
n.
60
Oleinik: Spectra of singularly perturbed operators
The elements of matrices
CEi(t,y)
are required to satisfy
the following conditions
K nlnl t Cij (t,y)nin.i c K nini, E,hl h1 2 1 I I I 1
CEJlh(t,y) - CEihl(t,y) -
where
n - Nnii
(17)
ih(t,y),
CE,J
an arbitrary symmetric matrix and
is
positive constants which do not depend on
K1,K2
are
E,t,y,n.
Let us also consider another system of equations of linear elasticity r(u) =
(Cij((p(x),x1,...,xn)8xu
l
8xi
- f' 1)
whose coefficients satisfy conditions similar to (17) but with constants A
and are smooth functions of
K1,K2
t
and
We obtain an estimate
y.
for the difference between a solution of the Dirichlet problem for operator
in a smooth domain
ZE
0 and a solution of the corresponding
Dirichlet problem for operator L This result is used here to estimate the difference between the respective energies contained in a subdomain
w
of a stratified elastic body
f1
and also between the respective
eigenvalues and eigenfunctions of the operators Dirichlet problem.
for the strong as
to the operator
j ID7vI2dx, km
space
components in (u,v)m
H-1(11).
of
Z
Z.
(Ho(t)) we denote the completion in the norm
NvNm - [h/1 E
by
ZE
e - 0 and give explicit formulae for the coefficients of
the
for the
Z
We also obtain necessary and sufficient conditions
G-convergence of the sequence
By Hm(n),
of
and
X.
7 - (1.....,yn),171 - 71+...+yn,
n
vector-valued
C-(f1) (C-o(s1).
functions
The scalar product in
while the Hilbert space dual
We assume that
f1
v - (vl,...,vn)
to
Hm(fl)
HI(fl)
Is a bounded smooth domain in
with
is denoted
Is denoted by An.
Consider the following Dirichlet problems
YE (uE) - f,
uE a Ho(n),
f e L2 (fi),
(18)
Oleinik: Spectra of singularly perturbed operators
61
A
X(U) - f,
Define matrices
Ns(t,y) - I
f E L2(fl).
u E Ho (f2),
t -1
0
by the formulae
MIs(t,y)
and
NS(t,y)
(19)
,pp(y)[CPS(T,y)
-
CpS(T,y))dT, (20)
JOIyj(y)CEj(T,y)[`pl(y)`pk(y)CE1(T,y)]-1 9p(y)(Cps(T,y)
Mis(t,y) -
- Cps(T,y), + CES(T,y) - C1s(T,y)}dT,
where
inverse
(T1(y)...... fn(y)) -
matrix
to
[
grad T
ax n
sax 1
i,s - 1,...,n,
and A 1
is the
It can be proved that the matrix exists and its elements are functions bounded
A.
uniformly in e. Set
sE - xmax0 [IMIj
(,(x),x)I, INi (,(x),x)I, IaylMij (w(x),x)I, Iay1NJ (9(x),xI}.
l,i,j -1,...,n
For a matrix
we denote
A - lakil
H0'((2)
which satisfy (18), Theorem 2.
u
aklakl
by W. Solutions of
are taken to be vector valued functions from
problems (18) and (19)
the sense of distributions.
(19) in
Let
f E L2(fl).
Then for the solutions
WE
and
of problems (18) and (19) respectively, the following estimates hold
luE - u - NS(cp(x),x)au 11 4 Cos% 1flo,
(22)
s
Clu
10 5 Cls
is(`p(x),x)
at
1f1 o,
(23)
s
ax
IEW(uE) - Ew(u)I
4 C2(w)s
lflo,
(24)
62
Oleinik: Spectra of singularly perturbed operators
e
where
yE
CEj ax
depend on
es
w
,
7i
Cij ax
and the constants
is a smooth subdomain of
vector-valued functions
We
lw(CEj
describe
ax
j
e
do not
(u,v) = uj )
f2,
^u ax j
briefly
F, (u) = j (Cij
,dx,
the
main
ideas
of
for any
and
v = (v1,...,vn), u - (u1,.... un)
E
EL (uE)
Co, C1, C2
j
LU, au
l axi,dX.
axj
proof
the
of
Theorem 2.
Define a function
in il,
YE (vE) - 0
as a solution of the problem
vE(x)
vE = NJi(,f(x),x)axj
on
an,
vE E H' (n).
It is easy to prove that
reWE - u - Ns((p(X),x)u
+ VE)
S
((Cis ax1 L l
- Cis - Cij E
E
This equation is satisfied
BNsJauj ax j
ax
-
8
asu
i
E sax ax ax (CiiNE
j=F
in the sense of distributions.
-
E.
(25)
One can
easily get that
at(Mis(f(x),x))
= fklgrad fl-2 ax (Mls(v(x),x), + sis(x,E),
(26)
k
axi
(Mis(f(x),x),
_
S(X" ),
CiS(f(x),X) - CES(f(x),x)-CEj(f(x),x)aX Ns(f(x),x)
am
atis + ais(x,E),
J
where Isis(X,E)I < C95E,
and the constants
C9,C41Cs
Ias(X,E)I
do not depend on
right-hand side of (25) in the norm of
Iais(X,E))I t C56E,
6 C466,
H-11(n).
e.
Let us estimate the
Using (25) and (26) we
63
Oleinik: Spectra of singularly perturbed operators
obtain
IFEIH-1(0) t C66.IuI2,
where the constant coefficients of
(27)
does not depend on
C6
and the boundary of
p
Nu12 F C71flo,
and
a
Since the
u.
a are smooth we have
C7 - const.
From (27) and (25) we get
lue - u - NS(,p(x),x)X + VEIL t C66ENf1,
(28)
s
where the constant
does not depend on
C.
Now we have to estimate 9E - Nj(,p(x),x)au 46
,
- 1
4
In the
and f.
e
IvEI1.
Set
we - ve - 9.,
Be - neighbourhood of
&R,
where yE - 0
j
outside the
26. - neighbourhood of
the integral identity for
ve'
an,
0e E C"(Q),
0
>i
0e i 1. Using
conditions (17) and Korn's inequality,
we obtain IwEN1 6 CS No ell.
To estimate
we consider
19.11
89
axk
-
8N au axk ax j
4'e + Nj
a2u
au
a+y
(29)
ax j axk
ax j
It follows from (29) that
lep2
4
c10 61c Nu12 + C11 Igrad UIIL2(w 1
where
w1
is the
26E - neighbourhood of
an.
),
It is easy to prove that
C1266Iu12,
since according to the imbedding theorem we have
Iax N2(r) L 4 C Iu12 j
for
any
u e H2(0)
and
any
smooth
surface
r
belonging
to
a
Oleinik: Spectra of singularly perturbed operators
neighbourhood of
64
Therefore
an.
IVC11 i C13s54 Iflo.
(30)
Estimates (30) and (28) imply (22).
From (22) it follows that au
ONE au
au
qL,
(31)
axj ' ax + axj axs+
where
Iqf1o f C146
From (26) and (31) we get
Iflo.
Cijau - Cijau - -MIS au - u e axj
axs
at
axj
au + C.. ` isOx s e q
The estimate (23) follows easily from (32). proved using estimates (22)
(32)
The estimate (24) can be
and (23).
Now we consider the strong G-convergence of operators
to
t A
A sequence of operators
to operator
if for any
i
is called strongly G-convergent
Z.
the sequence
f e H-1(n)
of problem (18) converges to a solution Ho(se)
as
and
0
e
7E
as
L2(si)
e
-
axi
1 = 1,...,n
0,
Denote by
C°,S
of solutions
uE
of problem (19) weakly in
u
converges to
Ci i Lue 11
in
X.
a - 0.
as
weakly
y1
(see [4],[9]).
the space of functions
t e [0,1],
g(t,y),
y E n with the norm Ig(t,y')-g(t,y")I
Igho,s - sup Ig(t,y)I + sup t,y t,Y'Y"
t E [0,11,
Iy'-y"18
Y,Y',Y" E (1,
y' m y",
0 < 8< 1.
It is easy to prove the following lemmas.
Lemma 3.
norms in e' - 0 in
Co,8
Suppose that a family of functions
uniformly bounded in
and a function
L2(0,1)
for every
y e si.
e.
If
has
Then there is a subsequence
such that
m E C0,8
0e(t,y)
O(t,y)
0,- (t,y)
cy6(t,y) -. 0
as
e
0
weakly
weakly in
65
Oleinik: Spectra of singularly perturbed operators
t
for every
L2(0,1)
C([0,1] X 0),
in the norm
0
OE((p(x),x)
Lemma x e fl,
0E(t,y) - J 4e(T,y)dT - 0
then
y E fl,
Suppose
4.
such that for
weakly In
0
e
-
that CE]
and
0
as
H1(Q)
there - CEl
exist
as 'J
0.
a
matrices + CEj
NE 5(x),
we have
1
1)
NE s E H1(fl), NE s
2)
Eli
3)
X (CEj - Eli)
weakly in
0
weakly in
Eli
LZ(fl),
H1(fl), s - 1...,n,
(33)
i,j - 1,...,n,
(34)
a
0
in the norm of H 1(fl),
i - 1,...,n.
(35)
j
Then the sequence
X.
Is strongly G-convergent to an operator
the matrix of coefficients
Cij(x)
X
with
satisfying conditions similar to
(17). Conditions [9],[30]).
(33) -
(35) are known as the N-condition
(see
The proof of Lemma 4 is given in [30].
We introduce the following notation 1
BE(t,y) _ [TI(y)(pk(y)CEl(t,y)]
BE(t,Y) = [wl(Y)wk(Y)CEl(t,Y)]
wj(y)Cps(t,Y),
(t,y) = ,j (y)CEj (t,y)[Ipl(y),k(y)CEl(t,y)] pp(y)Cps(t,y) - CES(t,y), B°(t,y) =-
Bs(t,Y) =- [1 (Y)'Pk(Y)CkI(t,Y)]
Tp(Y)Cps(t,Y),
i
Bls(t,y) ° Yj(y)C1)(t,y)[Tl(Y)yk(y)Ckl(t,y)I Yp(y)Cps(t,y) - CES(t,y).
66
Oleinik: Spectra of singularly perturbed operators
The following theorem gives the necessary and sufficient A
conditions for the strong G-convergence of operators E
X
to
XE
as
'' 0.
Suppose that the elements of matrices
Theorem 3.
uniformly bounded in
C0,8
have norms in
CEi(t,y)
Then the sequence
E.
is
XE
A
strongly G-convergent to
if and only if the following conditions are
X
satisfied BS(t,Y),
Bis(t,y),
BES(t y) 8
(36)
8
a ayl B°(t,y),
BE(t,y) ayl
BE(t,y)
a BS(t,Y), ayl
8y1
8
BES(t,y) ayl
weakly in
BE(t,y)
B°(t,Y),
BE(t,y)
l,s,l = 1,...,n,
Bis(t,y), ay l
L2(0,1)
as
0
e
for any
y E H.
We outline here the proof of Theorem 3. that conditions (36) are sufficient for to
X
as
e
-
X.
Let us show that in this case
0.
Let us prove first
to be strongly G-convergent 0
S.
as
0.
e
One can easily check that
Ns(t,y) - jt [BE(r,y)(B°(r,y))-'B(t,y) - BE(r,y)1dr,
(37)
O
M1s(t,y) a jt [(BE(r,y))*(B°(r,y))-iBS(r,y) - BES(r,y) 0
- (Bi(r,y))*(B°(r,y))-'B5(r,y) + Bis(r,y)Jdr.
Therefore by virtue of Lemma 3 and conditions (36) we get that as
e - 0.
From
f E L2(0),
uE
N'S(v(x),x) ax - 0
Theorem
u
(estimate
2
weakly in
Ho(t)
in
weakly H'(f1)
(22))
as
c - 0
it
as
follows E
S. -- 0
that
0
for
since
according to Lemma 3.
s
The estimate (23) implies that
yE
yi
,
I - 1,...,n,
weakly in L2(f)
67
Oleinik: Spectra of singularly perturbed operators
as e
since
0
(37).
Approximating any
easily obtain that L2(n)
as a- 0
weakly in
0
at Mis(T(x),x)
by functions from
f e H-1(1)
weakly in
u
uE
and for any
due to (36),
L2(n)
Ho(n),
f e H-1
for
one can
L2(n)
yE
weakly in
y1
(Cl).
The proof of the necessity of conditions (36) for the strong A
G-convergence of
to
gE
Y
as
is based on Lemmas 3,4 and the
c - 0
theorem about the uniqueness of the strong
G-limit (see [9],[30]).
It
is easy to prove that if conditions (36) are valid for some matrices B1J(t,y), then for these matrices the condition
Bs(t,y),
9°(t,y),
N
is satisfied.
Let the elements of matrices
Theorem 4. norms
in
matrices
uniformly bounded
Co,S
B°(t,y),
e
0
Then
and the coefficients of C1s
s,i,
that
such
1,...,n,
is strongly G-convergent to
t
as
are given by the formulae
Y
(B1)*(Bo)-1Bs
=
Z.
have
Suppose that there are
e.
B1s(t,y),
Bs(t,y),
conditions are satisfied.
in
CEJ(t,y)
_ Bis
(38)
Let us consider some examples when conditions
(36)
are
satisfied.
Suppose that the elements of matrices
Theorem 5.
the form k E 11R1
Chile
x)
have
CEJ
are almost periodic functions of
and
satisfying conditions (17).
G-convergent to the operator
i
Then the sequence
is strongly
X.
whose matrices of coefficients are
given by the formulae C1j a - +
1
T
where
>
are matrices with elements
Moreover estimates (22),
If CO)
hiare
where the constant
(23),
(24) are valid with
1-periodic functions C
= lim hl Tim
{,{ E R1,
in
does not depend on
e.
Cij(i;)dt. J-T hl B.
0
then
as 6E
e
0.
4 Ce,
68
Oleinik: Spectra of singularly perturbed operators
We introduce a class such that for some
Cf(y),
consisting of functions
A.
f(t,y)
g(t,y) one has
t
f(s,y)ds - Cf(y)t - g(t,y). j0
of
The functions
ag
are supposed to be
Cf, g,
f,
Holder continuous
ayl in
y E fi
uniformly in
t;
1
- 1,
n,
Ig(t,y)I + Ia g(t,y)) i Co(1 + itl1-Q, yl
where the constants
Co,o
Obviously, for
f E A.
THEOREM 6.
-
do not depend on
lim
and
o e (0,11.
Set
T
1
T- 2T J-T we have
t
f(s,y)ds.
- Cf(y).
In (16) let
CEj(t,y) = IClJih(t,y)I - ICih(E,y)1,
t - p(x), y - (y..... ,yn).
Suppose that the elemeents of matrices BE(t,y) = Bo(E,Y) = B°(T,Y), Bis(t,y)
BE(t,y) = B°(E,Y) a B°(T,Y),
Bis(E,Y) = Bis(r,y), i,s - 1,...,n,
defined above are functions of class
A°,
in
r
and
y
and
, , are matrices whose elements are smooth Then the sequence of operators
functions.
- ICih
the matrices
CiJ
operator
with the coefficients
i
(x),x)I
Z.
corrresponding to
is strongly G-convergent to the
Cis(x) - *-1 - .
Moreover,
in Theorem 1 satisfies the inequality
BE
the constant
C
does not depend on
BE i Ce°,
where
E.
Theorems 5 and 6 are obtained as a consequence of Theorems 2
69
Oleinik: Spectra of singularly perturbed operators
and 3.
Now we consider eigenvalue problems related to the operators
Z.
and
Z: ZE(ue,k) + akpE(x)uE.k - 0 ue,k a Ho(n),
0 < ai
S a2
S
(uE,k, uE.mpE)o
... t ak t ..
Z(uk) + akP(x)uk - 0
0< )` 1 4 X2 4 5km
times
as
(39)
5km,
,
in Cl,
(uk umP)o = 5km
uk a Ho(n),
where
in Cl,
...
..
OXkf
(40) ,
is the Kronecker symbol, each eigenvalue is counted as many its
multiplicity,
and
p
pE,
are
bounded
measurable
A
functions such that
PE 3 const > 0,
p i const > 0.
We also consider an auxiliary eigenvalue problem ZE(vE,k) + akP(x)ve,k = 0 ve,k E Ho(n),
(ve,k, ve,k
in Cl, np)o
- 6km,
(41)
0< ae 4 ;6 4 ... 4 ;k 4 ... Let
p(x)
be a bounded measurable function which can be represented in
the form
p - fo + Oxj,
fo, fj a LP (0),
p ; n.
(42)
Set
-
1p1-1 'p
inf {1fo1Lp fo,fI
where the infimum
Ifi1Lp(n)}'
+ (n)
i-1
is taken over all representations of
p
in the form
(42).
Lemma 5.
Let 4 and
ak
be eigenvalues of problems (39)
and (41) respectively. Then I(ak)-1 - (ae)-11 where
C
4 CIIp, - P1-11p,
is a constant which does not depend on
p ; n, a
and
k.
The proof of this lemma is based on the variational theory
70
Oleinik: Spectra of singularly perturbed operators
of elgenvalues, Korn's inequality and the Imbedding theorem: IuHtp(n) t Colull'
1 - 2+ n 3 0.
if
p
Theorem 7.
Suppose that
and matrices
p E C1(fi)
CEJ
in
Then
(16) satisfy the conditions of Theorem 2. I(4)-1
C1(HpE - Pl-1,p + 6
- (ak)-11
where
ak,
),
(43)
p a n,
are eigenvalues of problems (39), (40) respectively and
Xk
C1 is a constant which does not depend on
The inequality
(43)
E.
is proved on the basis of H. Weyl's
theorem (see [4Q]).
Theorem satisfied.
Let
8.
conditions
the
all
r(uE,k, N(ak,Y)) 4 C06% + up, -
where the constant eigenfunction of formed
space
eigenvalue
of
Theorem
7
be
Then
'k
does not depend on
Ck
(39) corresponding to
by
eigenfunctions
all
and
r(uE.k,N(ak,))
N(ak,Y) in the space
p1,p
L2(fl)
e;
ak,
of
is the linear
corresponding to uE,k
(40)
is the distance between
with scalar product
Note that in the case
Is the k-th
uE,k
N(ak,X)
pe(x)= p r-!
the and
(u,v) _ (u,vp)o.
,x), where
p(t,y) E A0,
c
we have
Up(x)1,
when
p(x)
R,
- p(X)l-1
.
The constant
C Ear
C
does not depend on
E.
Proofs of Theorems 7 and 8 are given in [37].
3 HOMOGENIZATION OF EIGENVALUES AND EIGENFUNCTIONS OF BOUNDARY
VALUE PROBLEMS IN PERFORATED DOMAINS FOR ELLIPTIC EQUATIONS
AND THE ELASTICITY SYSTEM WITH NON-UNIFORMLY OSCILLATING COEFFICIENTS
Boundary value problems for partial differential equations describing processes in strongly non-homogeneous media are a subject of intensive
research
(see
[1] - (38]).
In
many
applications
it
is
71
Oleinik: Spectra of singularly perturbed operators
important to estimate various characteristics of these processes composites
elastic
order
displacements,
as
stres
tensors,
in
energy,
First we consider a mixed boundary value problem for
eigenvalues, etc. second
such
elliptic
equations
non-uniformly
with
oscillating
coefficients in a perforated domain with a periodic structure of period e
and then a similar problem for the system of linear elasticity. estimates
obtain
eigenfunctions
of
difference
the
for
problem
this
eigenvalues
between
eigenvalues
the
and
and
We
and the
In the case of
eigenfunctions of the related homogenized problem.
ellpitic second order equations these differences are of order
(see
a
[171) and in the case of the elasticity system these differences are estimated by Cu ,
C - const,
non-intersecting
smooth
0 < xj < 1,
j- 1,...,n}.
by a vector
z
G1 -
U
[25].
be a set which consists of a finite number of
G°
Let
domains
belonging
Denote by
and denote by
eX
to
a
cube
0 - {x a Rn,
a shift of the set
X + z
the set {x a An, e-lx a X}.
X E An Let
where Z is the set of all vectors z - (z1,...,zn)
(G° + z),
zE7ln with integer components, GE - eG1, where
n
ne - n\Ge,
is a smooth domain in
are connected open sets in
SE - anE\an, An.
re - &I n aft"
We assume that
R0.
The domain
consider
second
ne
,
and
Rn\G1
is called a perforated
nE
domain.
In
we
RE
order elliptic differential
operators
1'E
where
ai
(aiJ (x,xe) ax J
aij(x,k) are functions of class
(1,...{n
with period 1 (1-periodic in
C°°
in
(x,t),
periodic
in
i;),
aij(x,f) - aji(E,x), x11n12 d aij(x,i;)ninj
for any
n E Rn,
eigenvalue problem
f e Rn,
4 x,101,
x E n.
In
nE
x1,x2 - const > 0,
(44)
we consider the following
72
Oleinik: Spectra of singularly perturbed operators
LE(ue'k) + akp(x,E)UE'k - 0
in f2E,
(45)
aE(uE'k) - 0
where
on
on
k - 1,...,
rE,
BuE,k
x
a(ue'k)
uE'k = 0
Se'
aij(x,E)
vi, v = (vt,...,vn)
is a unit outward
8x J
normal to W. We assume that
is a function of class
p(x,t)
C0
in
(x,t) E fl x An, p(x,t) 3 po - const > 0,
0< X!
t
...
4
4
X11
and each eigenvalue is counted as many times as its multiplicity. consider such functions
which belong to
u6'k
H'((1E)
uE'k- 0
We
on rE
and satisfy the integral identity E,k
(- aij(x,E) JaE
for any function
axv
+ ak p(x,E) uE'kv)dx - 0
axj
v e H'(fE)
i
and
v
on
0
re,
and
x p(X, E) dx - skm. f2
eUe,kue'm
Let us define functions
as 1-periodic
Ns(x,{), s - 1,...,n,
in
t
solutions of the following boundary value problems
8N (x,t))
a
8al
(x,{) in
sadJ
at
stn\G1
(46)
1 = ati
on
a(Ns) - - aisvi
J
8G1,
0,
Q\G°
where
(v1,.... vn)
the
is
62n\-G,, a(N) a aij(x,t) aN vi;
outward
unit
normal
to
the
domain
is considered as a parameter. Set
x c- 11
J
1
hpq(x) = Lrmeas (Q\G°)]
J
Q\G°
LLlapq(x,{) + ajq(x,t) aEd{ (47)
73
Oleinik: Spectra of singularly perturbed operators
and consider the differential operator
e
(hpq(x) axq) p
which is called a homogenized operator for
It is proved in [1] that
LE.
satisfies conditions (44) with
L
some positive constants
X1, X2. Consider the eigenvalue problem for the homogenized operator
L(uk) + Xkp(x)uk - 0
in fi,
uk-0
(48)
on8fl,
where
P(x) - fineas (Q\G°)] l
and
ak
p(x,{)d r; 1
L
Q\G°
are also enumerated according to their multiplicity:
0 < al S a2 4
...
4 ak 4 ..
,
r uk(x)um(x)p (x)dx - skm J
n
Theorem 9. (45),
Let
(48) respectively.
ak
ak
be eigenvalues of problems
Then
lxk - akl 4 C(k)e,
where the constant
and
C(k)
k - 1,2,...,
does not depend on
e.
Assume that for some
j>0
aj-1 < aj - ... - aj+r-s < aj+r, r a 1, a° - 0. Then for sufficiently small
e
Huj+t - vE+t1L2(fE) b Cj+tE,
r - 1,
t
+t where
vE+t
is an eigenfunction of problem (48),
from the linear space formed by eigenfunctions problem (45) and
Cj+t
is a function uE,j
is a constant independent of
uE,j+r-1
of
e.
To prove Theorem 9 we use Lemma 1 and a generalization of
74
Oleinik: Spectra of singularly perturbed operators
the maximum principle for solutions of the mixed boundary value problem for elliptic second order operators.
Let w be a bounded domain in J. Assume that the boundary aw
of w
has the form v e Lip(w)
function
aw -
and
where
ro u rl,
v - 0
on
is such that for any
ro
the Friedrichs inequality
ro
av
n
CjIi
IVIL2(w)
Iaxj1L2(w)
¢
holds with constant
Independent of
C
By
v.
Lip(w)
we denote a
w with the finite norm
space of continuous functions in
Iv(x) - v(y)I Ivl -
sup
Iv(x) +
XEW By
RI(w)
sup
Ix - yl
X,yEW
we denote the completion of
IF
HI(w): NviI - INvIL2() + jE By
RI(w,ro)
Lip(w)
in the norm
Lip(w)
(49) j
IL2(w)1
we denote the completion in the norm (49) of a subspace in
consisting of the functions equal to zero on
ro.
Consider the
boundary value problem
j
au
Aij Here
in w,
lAji(x) a , - 0
-
vi - 0
ax
rI,
on
u - pp
are bounded measurable functions,
Aji(x)
r
av
Aii(x) - Aji(x)
and
We set
the condition similar to (44) is satisfied.
a(v,w) =
(50)
ro s 0.
ro,
on
aw
Ali(W )
axidx,
ax
j J
j
and let T E RIM. The function (50),
if
a(u,v) - 0
u E It' (w),
u(x)
is called a weak solution of problem
u - N E RI(w,ro)
is satisfied for any
and
the
integral
identity
v c let(w,ro).
The existence and uniqueness of a weak solution of problem (50) can be proved using the F.Rlesz theorem. Lemma
weak solution
6.
u(x)
(Maximum principle).
Let
W E C' M.
of problem (50) satisfies the inequality
Then any
75
Oleinik: Spectra of singularly perturbed operators
min (p
( u(x) 4 max'p
r°
for almost all
(51)
r°
x e w.
To prove estimates (51) we use the variational method for the solution of problem (50). In order to prove Theorem 9 we use the following lemmas.
be a weak solution of the
Let 0 e L2 (SE ), v(x)
Lemma 7. problem
v - 0 on rE, aE (v) - 0
in flE,
LE (v) - 0
on
SE.
Then
NvOH1(nE) ¢ C1e
RONL2(S );
C1 - const.
E
Let
Lemma B. function defined in
f(x,t)
fl x An
be a measurable, 1-periodic in
If(x,t) - f(x°,t)I 4 Clx - x°I
where the constant
N E L1 (fl),
C
t
and such that for all
does not depend on
x,x° E fl,
x,x°,t.
t E IRn,
Then
for any
we have I
'p(x)f(x,x)dx -, J cp(x)f(x)dx
as
e
-
0,
E
where f(x) = JQ f(x,t)dt.
We construct an approximate solution of the problem (45) in
order to use Lemma 1 and to estimate the eigenvalues of the problem (45).
Let us define functions
in
solutions of the following boundary value problems
t
aNpga(x,t)
at (aij (x,t) 1
Npq(x,t), p,q - 1,...,n,
as 1-periodic
_ hpq(x) - apq(x,t)
)
(52)
- aaiq(x,t)
aNp(x,t)
N (x,t) - 2ai (x,t) ati
p
a(Npq) - - aiq(x,t)Np(x,t)vi
q
on
in
ati
aG1,
pn\G1,
76
Oleinik: Spectra of singularly perturbed operators
Npq(x.t)dt - 0. t
0\G° Set
l1
hs(x) - [meas (Q\G°)]
Lax(aij(x.a J
)+
8aia(xt)ldt
Q\G°
JJ
and define
Ns, N
as 1-periodic in
(aij(x,t)
t
solutions of the problems
aN({.t ))
- P(x) - p(x.t)
in
Rn\G1.
at J
J
(53)
o(N) - 0
N(x,t)dt - 0,
on 8G,,
Q\G° BNSa{.t
-I[aij(x,t)
- hs(x) - axi (aiJ(x.t) j
1
J
(54) 82Ns(x.t) - aais (x,t - aaij(x,t) aN5(x,t) -
a iJ (x,t)
axi
atiaxj
axJ
ati
in
a(NS) - ajJ(x.t)
Vi
on
Rn\G1
Ns(x,t)dt - 0.
aG1, 1
J
Q\G°
Existence of solutions of problems (46),
(52),
(53),
(54) can be proved
by the methods of functional analysis.
In these problems
xl,...,xn
are considered as parameters,
and due to the smoothness of the boundary and of the coefficients of the equations the functions in
11 x (Rn\G1).
N5,Npq,N,Ns
are smooth with respect to
(x,t)
77
Oleinik: Spectra of singularly perturbed operators
Let us define
uE(x)
by the formula
k
+ E2pq(x,
£E(x) _- uk(x) + ENp(x,E) CIX
IN
P
)
82k 8xP 8x+ q
k Np(x, E )
8x
p
+ akN(x,0uk],
where
is an eigenfunction of the problem (48) corresponding to
uk(x)
the eigenvalue
as a solution of the boundary
We define
ak.
value problem
in nE,
LE (vE) - 0 vE - UE
Denote by
on rE,
the space
XE
on S.
aE(vOE ) -
La(nE)
(u,v)gE -
with scalar product
u(x)v(x)p(x,E-lx)dx. J
nE
We introduce an operator
which maps a function
AE:1E - xE
f E X.
of the problem
into a solution LOE We ) - f
in nE,
aE(UOE ) - 0
on SE,
Define an operator
(55)
AE:ICE -9E
uE - 0
on r.
by the formula
AEw =- AEp(x,X)w. E
The operator
- AE
is positive, self-adjoint and compact In 1t E.
The proof of Theorem 9 is based on the following lemma. Lemma 9.
Let
Then for sufficiently small
we - uE - VE.
e
we have
IwE + XkAEwEIg
(56)
t CE, E
wE(x) - uk(x) + EWE,
IWEIg
i K,
(57)
E
K2 ) Iwelx
) Ki > 0, E
(58)
78
Oleinik: Spectra of singularly perturbed operators
where constants
do not depend on
C,K,KI,K2
e.
The proof of Theorem 9 Is given In [17].
A theorem similar to Theorem
also proved
is
9
for the
elasticity system with rapidly oscillating coefficients in a perforated domain.
Consider the system of elasticity given by
where are
Ch.k(x,t)
f,
8xh(Chk(x E-ix) auks -
xE(u) =
are
matrices with elements
(n x n)
n x An,
C°° - functions in
1-periodic in
a column-vector with components
Cij (x, t)
t - (ti,..-,td,
X1nini t Cij(x,t)ninj c XZnini,
Denote by
H1(w)
is
(59)
and for any symmetric matrix
(x,t) a ft x Rn,
u
ul,.... un. Assume that
Cij(x,t) - Cji(x,t) - Chj(x,t)
for any
which
n - Intel,
X1'X: - const > 0.
the space of vector valued functions
(60)
u - (ul,.... un)
with norm
IuI HI In the domain
CIE
8u i 8ui)dx1
rr rulus
(w)
- Uw l
8xl
+ 8x]
]
we consider the following eigenvalue problem
2E(Ue,k) + 4p(x,E-lx)uE.k - 0
oE(ut,k) - 0
0 < ai t X6
where
on
SE,
...
i ak
S
oe(u) a Chk ax vh,
in
on
0
(61)
s2E,
r6, k - 1,2,...,
...,
(v...... vn)
(62)
is a unit outward normal to W.
k
We
assume
that
p(x,t)
p(x,t) 3 po - const > 0
times as its multiplicity.
such that
uE,k E H1(SiE),
is
a
function
of
class
C-(-(l x Rn),
and each eigenvalue in (62) is counted as many We consider eigenfunctions of problem (61) u6,k - 0
on
re
and
ue,k
satisfies the
79
Oleinik: Spectra of singularly perturbed operators
integral identity E,k
l
_
r aXh)dx
,
+ ak
J.E
p(x,E-1X)(UE-k,v)dx - 0
LE(Ckh(x,E-ix)a-k for any vector valued function
on
v e H1(f1E), v - 0
TE,
(v,w) - vjwj.
We assume that e, k, UE,m)p(x,E-1x)dx = 6km.
L
OE
We We introduce matrices
Ns(x,t),
s - 1,...,n,
which are solutions of
the boundary value problems
ahlchk(x,t)aNS(x,t))
- - atkCks(x,t)
in
82n\G1,
on
aG,,
k
a(Ns) = Chk(x,t)at vh - - Cks(x,t)vk k
Ns(x,t)dt - 0,
(63)
J
Q\G°
Ns
Variables
is 1-periodic in
xl,.... xn
t - (t1,...,tn).
are considered in problems (63) as parameters.
We set hPq - [meas [(Q\G°)]1
lICpq(x,t) + Cpj(x,t)aa
ldt.
,
j
Q\G° The operator A
r(u) =
Xp(hpg(x)axq),
u - (ul,.... un
is called a homogenized operator for hpq(x) - lhhq(x)N [12]).
ZE.
The matrices
satisfy conditions similar to (59),
(60).
(See [2],
Oleinik: Spectra of singularly perturbed operators
80
Consider the eigenvalue problem for the homogenized operator A
Y(uk) + akp(x)uk - 0
in
P(x) - [meas (Q\G°)]
uk - 0
n,
on
(64)
all,
p(x,k)d{,
f
Q1\G°
0 < al 4 aZ
i ...
i Xk H . ..
(65)
.
We assume
Each eigenvalue is counted as many times as its multiplicity. that (uk,um)p(x)dx - skm f ft
Theorem 10.
Let
(61) and (64) respectively. lak - Xkl
where the constant
be eigenvalues of problems
ak
Then
t C(k)e%,
C(k)
and
ak
k - 1,2,...,
does not depend on
(66) E.
Suppose that for some
j>0 Xj_1 < aj - ... - Xj+r-i < aj+r, r ) 1, ao - 0. Then for sufficiently small
e
luj+t - of +t IL2(IIE) 4
Cj+t e, t - 0,1,...,r - 1,
(67) jE+t
where
uj+t
is the eigenfunction of the problem (64),
(61), and
Cj+t
,ue,j+r-i
ue,j,
linear combination of the eigenfunctions
v
is a constant which does not depend on
is a of problem
Moreover,
e.
+t
luj+t - vE+t + eNs(x,C-1x)au
IHt(fE) t Cj+t E%. S
where the constant
Cj+t
does not depend on
e
and
Ns
are solutions
of problems (63).
The proof of Theorem 10 follows the same scheme as the proof
of Theorem 10.
However Lemma 6 used in the proof of Theorem 9 is not
known for the system of elasticity.
Instead of Lemma 6 we use here the
Oleinik: Spectra of singularly perturbed operators
81
following lemma which gives Integral estimates for solutions of the That is why we have
elasticity system.
10.
Lemma
suppose that
Let
f e L2 (fiC ),
belongs to
we
Eq
in (66),
(67) instead of e.
p a L2 (SE ),
m e H%(8n)
and
and Is a weak solution of the
Ht(fiE)
problem
Ze(wE) - f
in ne,
we - 0 on
re ,
on
QE (wE) - T
SE .
Then
Iwe I
Hi
(fe) t C[IfIL2 (nE) +
C - const does not depend on
where
IOIH%(8n) + E
ITIL2
(SE )1
e.
The proof of Theorem 10 is given in [25]. 4 ON VIBRATION OF SYSTEMS WITH CONCENTRATED MASSES.
We consider an eigenvalue problem in a bounded domain for
Laplace operator with Dirichlet boundary
the
density
assumed
is
to
be
constant
E-neighbourhood of the origin In nue,k +
xkpe(x)uE,k
everywhere
ue,k - 0
except
in
The the
The problem Is given by
0 e n. - 0
conditions.
n
in
n,
(1)
on
on,
(2)
+ E-m rtE(x)
for Ix1 t e, he(x) - rl(E),
where
PC - 1
for 1X1 > e, h({) - 0
j
pE(x)
10 > 1, h({) ) 0
for
for
1{I
t 1,
M - const > 0,
ri({)d{ - M,
I{1 0
An, x - (x1,...,xn),
M
is a small parameter.
In this section we number the formulae, theorems and lemmas starting again from 1. As
is known,
the problem
positive discrete spectrum.
(1),
(2),
for any
e > 0 has a
We consider weak eigenfunctions.
82
Oleinik: Spectra of singularly perturbed operators
ue,k function ue,k E Ho(n) (2) if
The
problem (1),
called a weak eigenfunction of
is
and for some
ak
it satisfies the
integral identity
r_ n aue , k av J
for any function
v
from
As before,
-0
v) dx axj+ 1`k Pe uE,k
L. j_laxj
Ho(n).
the space
Is the completion of
Ho (n)
Co(n)
(the set of infinitely differentiable functions with compact support in n)
in the norm Ur
MuN
1 - Un
We assume that 10,k.1
0<x"
ni
uj
2
E I ) dx)
j-
x
1 and that
... t kki ..
a2
,
where each elgenvalue is counted as many times as its multiplicity.
We study the behaviour of n - 3
as
1'k
e
0.
The case
m - 3,
A similar problem for the system of
has been considered in [41].
elasticity was studied in [42]. Let
n ) 3.
Consider the elgenvalue problem
Auk + xkuk - 0 uk - 0
on
(3)
n,
in
n,
0 < a1 i a2 4 ... ( 1`k t .... where each elgenvalue is also counted as many times as its multiplicity. Set au av
r
(u,v)W - J uvdx,
[u,v]W -
axjdx.
axj
I
Theorem 1. of problem (1),
Let
n 3 3,
(2) and eigenvalues
m < 2.
Then for eigenvalues
Xk
estimate is valid
I(ak)-1 - "k)-'l 4
where the constant
C
C62-m
ak
of problem (3) the following
k - 1,2,...,
does not depend on
a
and k.
83
Oleinik: Spectra of singularly perturbed operators
According to the variational principle (the minimax
Proof.
principle), we have ((P,, - 1)u,U)n
I(ak)-1 - (ak)-11
sup
i
(4)
(u,u]n
uEHo(n) It follows from the imbedding theorem that
lulLp(n)4 C1 l u N1,
if
1 - 2 +
C1 - const,
u E Ho(n),
(5)
From (4) and (5) we obtain
3 0.
p 1
I(aE)-1 - (ak)-11
E[u,u]n'
sup
t
UEH0(n)
j
u2Pdxlpl
C',2--m,
J `IXI<e
IXI<E
rigdx(1 t 1
1
n
where the constant 1
2
1
2
p
n
q
n
Lemma 1. problem (1),
does not depend on
C.
Let
n > 3,
m is
2.
and
e
n
1 - 2 + - 0, 2p
Then for the eigenvalues of
(2) we have Xk 3
CoEm-2
where the constant Proof.
Co
(6)
does not depend on
From (1),
and
e
k.
(2) we get
- [UE,k,UE,k]n +
E- mak(hEUE,k,UE,k) - 0.
According to the Friedrichs inequality we have
(UE,k,UE,k)n 6 C1[Ue,k,uE,k]n,
Let
akC1 < 2.
Then for
q - 2,
1 + 1 - 1 P
q
C1 - const.
we obtain from (7) that
(7)
84
Oleinik: Spectra of singularly perturbed operators
2 4 akE-m(,iEUE.k,ue.k )1t 1l1
l1
r
akE-m(r (ue.k)2pdx]p
4
akCze2-m,
,igdx}q 4
I
n 1x1<E
where
akC3< 2.
[uE,k,uE,k]n - 1,
since (7) and (5) are valid and
C2 - const,
Therefore
The lemma is proved.
ak > C3Em-2.
Suppose that
m ) 2
n 3 3,
and consider the eigenvalue
problem in An AOuk + h(x)uk - 0 in the space H
with the norm
H
as the completion of
[u,ul%
is the norm in
(8)
Iu1H - ([u,u]pn).
in the norm
Co (An) H
We obtain
the space
The fact that
IuIH.
is due to the Hardy inequality [43].
We can
rewrite (8) in the form - Ak[uk,v]pn + (ftuk'v)jRn - 0,
We can also write
(9)
uk, v E H.
in the form
uk, v E H.
- Ak[uk,v],Rn + [Auk,vlpn - 0,
It
is not difficult to prove that
compact operator.
(9)
In fact for any
I [Au,v]AnI = I (liu,v)pnl
4
A
is a positive,
u, v E H
l
1
self-adjoint,
we have
u21x1-2dx}
I
j
lS4
ti2v21xl2dxJ
.
llxl 0.
Theorem 2.
Let
(11)
Then for any eigenvalue
n i 3.
m > 2,
(11) of problem (9) there exists a sequence of eigenvalues (2) such that
problem (1),
Em-2(ap(,6))-'
Ak
as
i'p(E)
of
and
0
E
em-2
- AkI t CIE%, I
C1 - const.
(12)
XE
P(C)
Proof.
We have from (1),
(2) that
(XP(E)) 1(UE.P((),V)n + (UE.P(E),V)n + E-m(ft UE.P(e) V)n - 0, (13) uE,P(E) E H.
v E H,
be a function from
Let
4(x)
for
IxI > 1.
Co(Rn), O(x) - 1
We assume that the ball
for
IxI < 1
IxI
2,
as
e
0.
is an elgenvalue of problem (9). Proof.
Let us change the independent variables in (1), (2)
x
by setting
E
and suppose that
We obtain
1. E
AE,p(E)AuE,p(E) + (Em +
ue , P (1
=0
on
0
in
LIE,
af2E ,
AE.P(E) = Em-2 (AE
)-1
p(E )
Let
-
v E Co(I2n).
Then for sufficiently small
AE.P(E)[UE.P(E) v]jl + E
It
E
is easy to see that
we have
- 0,
+ (AuE.p(E) v)(1
Em(uE'p(E),v)n
[UC,p(E) UE.P(e))(I
c
(17)
E
- 1.
Em(UE,P(E),v)
0,
since according to the
Friedrichs inequality
4 COE-1. E)
(18)
87
Oleinik: Spectra of singularly perturbed operators
We set
v - ue,P(E)
-
Then we obtain
(17).
in
AE.P(E)[UE.P(E),UE.P(E)]nl + Em(UE.P(E) UE.p(E)) E
- 0.
+ (RUE.P(E) UE.p(E))n E
Em(ue,p(E) uE,p(E))Ci4
It follows from (18) that
C1Em-2
and since
e
Em(u6,p(E),uE.P(E))a
m > 2,
as
0
Thus we have
0.
E
E
M1 i
( UE.P(E) UE.P(E))il
( M2,
(19)
E
[uEP(E),uE,P(E)]SiE
since
not depend on u6.P(x)
and is weakly compact in
L2(w)
Let
w.
Then
e - 0.
M1, M2 do
According to the Hardy Inequality (10), the sequence
e.
is compact in
bounded domain
where the positive constants
- 1,
be a
u
u6.P(x)
limit function for
since due to
u A 0,
(19) we have
eigenvalue of (9).
A
Ciem-2 f ak S
C2Em-2
where the positive constants
m 3 2,
k,
k - 1,2,...,
C1,C2
do not depend on
n 3 3,
m - 2
but do depend
e
k.
Theorem 4. Then
Is an
The theorem is proved.
It follows from Lemma 1 and Theorem 1 that for any
on
as
where
(u,u)B > 0,
On passing to the limit in (17) we get that
B - (x:lxl < 1).
for any
H1(w)
either
is
x
Let
eigenvalue of problem (9). values
Akt,
Ak
where
All eigenvalues
)'p(E)
Assume that
A - X-1
the problem (1),
as
as
a
as
c
(9),
Is
an
are limit
(2).
0 and
e - 0.
(20)
is an eigenvalue of problem (9).
(2) in the form (17) and take
0.
e
k-1
of problem (3) and all
kk
of problem (1),
1p (E) - )
(UE.P(X)UE,p(E))1 - 0 In this case
ap(E) (3) or
is an eigenvalue of problem
points for some sequences of Proof.
and
an elgenvalue of problem
Let us rewrite
[uE,p(E),uE,p(E)]
- 1
x
in the new variables
t -
Because of the last condition and (20) we
88
Oleinik: Spectra of singularly perturbed operators
have
as e - 0.
0
On passing to the limit in
E
(17) as
we get a 1
0,
e
is an eigenvalue of (9).
Theorem 2 that for every elgenvalue limit point for a sequence
e - 0. as
e
i'p(E)
It is proved in
At
of problem (9),
Ak
is a
of eigenvalues of problem (1),(2) as
Suppose that i'p(E) - a and (ue'P(E),uE,P(E ))n it a - const > 0 0.
Then
a
is an eigenvalue of problem (3).
- Hue'p(E)'V]n +
XP(E)E-2(jjEUE,p(E).V)n
+
Xp(E)(UE.p(E) V)n
In fact, we have
- 0,
[(UE-P(E) UE,P(E)]n - 1,
for any function the sequence Ho((1).
According to the Friedrichs inequality,
v e Ho(n).
uE,P(E)
is compact in
Let us prove that
and is weakly compact In
L2(n)
E-2(REuEP(E),v)n
0
as
c
-. 0.
We have 1
V)
I (E-2/t E
n
I
t E-2NUE.P(E
)N
(
HgvgdX
LP(n)[J
E
J
Ixl<E
4 C1E-2NUE'P(()N
max IvIE(n+2)/2 LP(n) II
C2E(n-2)/2
i
if
1s1-1 1-n+2 p
2
n'
q
2n
since for such
p
the inequality
(5)
is
valid.
We prove now that every elgenvalue
limit point for some sequence
as
xp(E)
of problem (3) Is a
Xk
In order to apply
e - 0.
Lemma 1 of Section 1 we consider the equality -ak1[uk,v]n + (uk,v)n + E-2(riEuk,v)n _ E_2(h' Uk,v)n.
It is easy to see that 1
IE-2(AE uk,v)S2 14 E-21VI
r
LP(n)l
f IXI<E
rl2(Uk)2dX q t C! E(n-2)/2NvI E
]
1
89
Oleinik: Spectra of singularly perturbed operators
if
1-1
1
p- 2
1-1
1
y+
q
n
since
n,
is a smooth function.
uk
Using the methods applied above for the case
study problem
(1),
In the case
(2)
n i 3, one can
We get the following
n - 2.
theorems.
Let
Theorem 5.
Ilak)-1
where the constant
f'k
does not depend on
C(y)
of
and
e
and
k,
7
is
2 - m.
Let us consider now the case H - [u.[u,u]R2 < -,
ak
of problem (3) we have k - 1,2,...,
- (ak)-1I c CE/,
any constant less than
Then for elgenvalues
n - 2, m < 2.
(2), and eigenvalues
problem (1),
n - 2, m i 2.
In the space
we consider the eigenvalue problem
hudx - 0} J
Ixl 1.
n - 1,
(2) such that
sequence of eigenfunctions of problem (1),
uniformly on
as
(a,b)
e
the inequality
ap(E) i Co
depend on
Then
e.
Lemma problem (1),
ue,p(e) is a ue,p(e) -. u(x)
-. 0, and for the related eigenvalues
where the constant
is valid,
ap(e)
Co does not
u(0) - 0.
Let
3.
m > 1.
n - 1,
Then
eigenvalues
for
of
(2) the estimate Cem-i
ak i
is valid, where the constant
C
does not depend on
Consider the elgenvalue problem in
Ak
d2uk + Msuk - 0,
a
and k.
Ho(fl):
uk(a) - uk(b) - 0.
(25)
dX1
It is easy to see that the eigenvalue problem (25) has only one positive eigenvalue
A0.
Theorem em-'(a1 )-1
A0
as
Suppose
11. a
0,
Xk
that
ak-i
as
n - 1, e
0
1 < m < 2.
and
k > 1,
Then where
1'k is an eigenvalue of the problem
d2uk + kkuk - 0
on
(a,0)
(o,b),
dx2i
uk(a) - uk(b) - u(0) - 0,
0< k1 4
1.2
4 ...
4
),k 4
(26)
...
92
Oleinik: Spectra of singularly perturbed operators
In order to study the case
m) 2,
n - 1,
consider the
eigenvalue problem
__
duk(_1) - duuk(1)
A k dxuk+ iuk - 0
on
(-1,1),
- 0
(27)
J_ hudx - 0). i
The problem (27) in the space
Al i A2
ii
...i Ak it ... > 0.
Theorem 12. For any eigenvalue
such that e
A
then
-. 0,
Ak
Let
is
as
(28)
m > 2.
n - 1,
Then
ak - 0
as
an 0,
e
e - 0.
as
If
0.
f
of problem (27) there exists a sequence
Em_a(ap(E))-t -. Ak
Em-'(a1)-' - A0
has a positive discrete spectrum
if
ap{E)
Em-2(ap(E))-' - A
as
Moreover
eigenvalue
of
where
is the eigenvalue of problem
A0
problem
(27).
(25). Theorem 13. where
e - 0,
as
e
n - 1,
m - 2.
Then
Ao
is the eigenvalue of problem (25).
Ao
then
-. 0,
Let
k
ap(E) - X s 0
If
is either an eigenvalue of problem (26) or
is an eigenvalue of problem (27). is a limit point for some sequence
Every eigenvalue ap(E)
as
Ak
as
a-' - A
of problem (27)
e - 0.
The above methods can be applied to study similar problems
in the case of higher order elliptic equations, conditions,
boundary r,
1. 2.
several concentrated masses,
point,
as
well
as
masses
some other boundary
or a mass concentrated at a
concentrated
on
a
submanifold
r c n..
Benoussan A., Lions J.-L., Papanicolaou G. Asymptotic Analysis for Periodic Structures. North Holland Publ. Co. (1978). Sanchez-Palencia E. Non-Homogeneous Media and Vibration Theory. Lect. Notes in Phys. 17. Springer Verlag. (1980).
93
Oleinik: Spectra of singularly perturbed operators
3.
4.
5.
6.
7.
8.
Bakhvalov N.S., Panasenko G.P., Homogenization of Processes in Periodic Media. Moscow: Nauka. (1984). Trends and Applications Oleinik O.A. On homogenization problems. of Pure Mathematics to Mechanics. Lecture Notes In Phys. 195, pp. 248-272. Springer Verlag. (1984). Oleinik O.A., Shamaev A.S., Yosifian G.A. On the convergence of the energy, stress tensors and eigenvalues in homogenization problems of elasticity. Z. angew. Math. Mech., 65, no. 1, 13-17. (1985). Tartar L. Etude des oscillations dans les equations aux derivees partielles non lineaires. In Trends and Applications of Pure Mathematics to Mechanics. Lecture Notes in Phys., 15, pp. 384-412. Springer Verlag. (1984). Duvaut G. Homogeneisation et Materiaux Composites. In Trends and Applications of Pure Mathematics to Mechanics. Lecture Notes In Phys., 195, pp. 35-62. (1984). Tartar L. Homogeneisation. Cours Peccot au College de France. Paris. (1977).
9.
10.
11.
12.
13. 14.
15. 16.
Kozlov S.M., Oleinik O.A. Zhikov V.V., Kha T'en Ngoan. Averaging and G-convergence of differential operators. Russian Math. Surveys, 34, no. 5 (1979). Lions J.-L. Remarques sur l'homogeneisation. Computing Methods in Applied Sciences and Engineering VI. Ed. by R. Glowinsky and J.-L. Lions, INRIA, North Holland, pp. 299-315, (1984). Pobedria B.E. Mechanics of Composite Materials. Moscow: Moscow University. (1984). Oleinik O.A., Yosifian G.A. On homogenization of the elasticity system with rapidly oscillating coefficients in perforated domains. N.E. Kochin and Advances in Mechanics, pp. In 237-249. Moscow: Nauka. (1984). Cioranescu D., Paulin S.J. Homogenization in open sets with holes. J. Math. Anal. and Appl., 71, no. 2 pp.590-607. (1979). Oleinik O.A., Shamaev A.S., Yosifian G.A. Problemes d'homogeneisation pour le systeme de 1'elasticite lineaire a coefficients oscillant non-uniformement. C.R. Acad. Sci. A. 298, pp. 273-276. (1984) Vanninathan M. Homogeneisation des valeurs propres dans les milieux perfores. C.R. Acad. Sci. A, 287, pp. 823-825. (1978). Kesavan S. Homogenization of elliptic eigenvalue problems. Appl. Math. Optim. Part I, 5, pp. 153-167, Part II, 5 pp. 197-216. (1979).
17. Oleinik O.A., Shamaev A.S., Yosifian G.A. Homogenization of eigenvalues and eigenfunctions of the boundary value problems in perforated domains for elliptic equations with
non-uniformly oscillating coefficients. In Current Topics in Partial Differential Equations, pp. Kinokuniya Co. Ltd. (1986). 18. Oleinik O.A., Shamaev A.S., Yosifian G.A.
187-216.
Tokyo:
Homogenization of eigenvalues and eigenfunctions of the boundary value problem of elasticity in a perforated domain. Vestnik Mosc. Univ., ser. I., Math., Mech., no. 4, pp.53-63. (1983). 19. Marchenko V.A., Khruslov E.Y. Boundary Value Problems in Domains with a Finely Granulated Boundary. Kiev: Naukova Dumka. (1974).
Oleinik: Spectra of singularly perturbed operators
20. Marcellini P. Convergence of second order linear elliptic operators Boll. Un. Mat. Ital., B(5), 15. (1979). 21. Spagnolo S. Convergence in energy for elliptic operators. Proc.
Third Sympos. Numer. Solut. Partial Differential Equations, College Park, Md., pp. 469-498. (1976). L.Un terme etranger venu d'ailleurs. In 22. Cioranescu D., Murat F. Non-linear Partial Differential Equations and Their Applications. College de France Seminar, vol.Il, pp. 98-138, Vol.III, pp. 154-178. Pitman. (1981,83). 23. Oleinik O.A., Shamaev A.S., Yosifian G.A. On the asymptotic expansion of solutions of the Dirichlet problem for elliptic equations and the elasticity system in a perforated domain.
Dokl. AN SSSR,, no.5, pp. 1062-1066. (1985). 24. Oleinik O.A., Shamaev A.S., Yosifian G.A. Asymptotic expansion of eigenvalues and eigenfunctions of the Sturm-Liouville problem with rapidly oscillating coefficients. Vestnik Moscow Univ., ser.I, Math., Mech., no. 6, pp.37-46. (1985). 25. Oleinik O.A., Shamaev A.S., Yosifian G.A. On eigenvalues of boundary value problems for the elasticity system with rapidly oscillating coefficients in a perforated domain. Matem. Sbornik, (to appear). (1987). 26. Kozlov S.M., Oleinik O.A., Zhikov V.V. On G-convergence of parabolic operators. Russian Math. Surveys, 36, no.1, 27. De
28.
29.
30.
31.
32.
33.
34.
35.
(1981). E.,
Spagnolo S. Sulla convergenza degli integrali dell'energia per operatori ellittici del 2 ordine. Boll. Un. Mat. Ital., (4), 8, pp. 391-411. (1973). Oleinik O.A., Panasenko G.P., Yosifian G.A. Homogenization and asymptotic expansions for solutions of the elasticity system with rapidly oscillating periodic coefficients. Applicable Analysis. 15, no. 1-4, pp. 15-32. (1983). Oleinik O.A., Panasenko G.P., Yosifian G.A. An asymptotic expansion for solutions of the elasticity system in perforated domains. Matem. Sbornik. 120, no. 1, pp. 22-41. (1983). Shaposhnikova T.A. On strong G-convergence for a sequence of the elasticity systems. Vestnik Moscow Univ., ser. I, Math., Mech., no. 5, pp. 29-34. (1984). Oleinik O.A., Zhikov V.V. On homogenization of the elasticity Vestnik Moscow system with almost-periodic coefficients. Univ. ser. I, Math., Mech., no. 6, pp.62-70. (1982). Oleinik O.A., Zhikov V.V. On the homogenization of elliptic operators with almost periodic coefficients. Rend. Semin. Mat. e Fis. Milano, 52, pp. 149-166. (1982). Oleinik O.A., Shamoaev A.S., Yosifian G.A. On homogenization problems for the elasticity system with non-uniformly oscillating coefficients. In Mathematical Analysis B.79, Teubner-Texte zur Mathematik, pp.192-202. (1985). Lions J.-L. Some Methods in the Mathematical Analysis of Systems and Their Control. Beijing, China: Science Press. New York: Gordon & Breach. (1981). Asymptotic expansions in perforated media with a Lions J.-L. periodic structure. The Rocky Mountain J. of Math., 10, no. 1, pp. 125-140. (1980). Giorgi
94
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36. Oleinik O.A., Shamaev A.S., Yosifian G.A. On homogenization of elliptic operators describing processes in stratified media. Uspekhi Mat. Nauk, Al, no. 3, pp. 185-186. (1986). 37. Oleinik O.A., Shamaev A.S., Yosifian G.A. On the homogenization of stratified structures. Volume dedicated to the 60th birthday of J.-L. Lions (to appear). (1987). 38. Oleinik O.A., Shamaev A.S., Yosifian G.A. On asymptotic expansions of solutions of the Dirichlet problem for elliptic equations in perforated domains. In Non-linear partial differential equations and their applications. College de France Seminar, 8 (to appear). (1987). 39. Vishik M.I., Liusternik L.A. Regular degeneration and boundary layer for differential equations with a small parameter. Uspekhi Mat. Nauk, 12, no. 5, pp. 3-122. (1957). 40. Riesz F., Sz.-Nagy B. Lecons d'Analyse Fonctionnelle. Budapest: Academiai Kiado. (1952). 41. Sanchez-Palencia E. Perturbation of eigenvalues in thermoelasticity and vibration of systems with concentrated masses. In
Trends and Applications of Pure Mathematics to Mechanics.
Lecture Notes in Phys. L, pp.
346-368.
Springer Verlag.
(1984).
42. Sanchez-Palencia E., Tchatat H. Vibration de systems elastiques avec des masses concentrees. Rend. Sem. Mat. Univ. Politech. Torino. 4, no. 3, pp. 43-63. (1984). 43. Kondratiev V.A., Oleinik O.A. Sur un probleme de Sanchez-Palencia. C.R. Acad. Sci. A., 299, no. 15, pp. 745-748. (1984).
Conservation Laws in Continuum Mechanics Peter J. Olver School of Mathematics, University of Minnesota, Minneapolis, MN, 55455
USA
Preface. This article reviews some recent work on the conservation laws of
the equations of continuum mechanics, with especial emphasis on planar elasticity. The basic material on conservation laws and symmetry groups of systems of partial differential equations is given an extensive treatment in the author's book, [6], so this paper will only give a brief overview of the basic theory.
Some of the applications appear in the published papers cited in the references, while others are more recent. This research was supported in part by NSF Grant DMS 86-02004.
1. Conservation Laws of Partial Differential Equations. The equations of non-dissipative equilibrium continuum mechanics come from minimizing the energy functional W[u] = f UWx,ulnl) dx
(1)
a
Here the independent variables x=(xl,...,xP) a S2 represent the material
coordinates in the body, and the dependent variables u=(ul,...,uP) the deformation, where p=2 for planar theories, while p=3 for fully three-dimensional bodies. In the absence of body forces, the stored energy W will usually depend just on x and the deformation gradient Vu, but may, in a theory of higher grade
Olver: Conservation laws in continuum mechanics
97
material, depend on derivatives of u up to order n, denoted u(n). Smooth minimizers will satisfy the Euler-Lagrange equations E"(W) = 0,
V= 1'... ,P,
(2)
which, in the case of continuum mechanics, form a strongly elliptic system of partial
differential equations of order 2n. Strong ellipticity implies that this system is totally nondegenerate (in the sense of [6; Definition 2.83]). Given the system of partial differential equations (2), a conservation law is a divergence expression
DivP=iDiPi=0
(3)
i-1
which vanishes on all solutions to (1), where the p-tuple P(x,u(m)) can depend on x, u and the derivatives of u. For static problems, conservation laws provide path-
independent integrals, which are of use in determining the behavior at singularities such as cracks or dislocations. For dynamic problems, conservation laws provide constants of the motion, such as conservation of mass or energy. Two conservation laws are equivalent if they differ by a sum of trivial
conservation laws, of which there are two types. In the first type of triviality, the p-
tuple P itself vanishes on all solutions to (2), while the second type are the null divergences, where the identity (3) holds for all functions u=f(x) (not just solutions to the system). As trivial laws provide no new information about the solutions, we are only interested in equivalence classes of nontrivial conservation laws. An elementary integration by parts shows that any conservation law for the nondegenerate system (2) is always equivalent to a conservation law in characteristic form
DivP=O.E(W)=Q vE (W) v V-1
(4)
Olver: Conservation laws in continuum mechanics
98
where the p-tuple Q=(Q1,...,Qp) is the characteristic of the conservation law. A characteristic is called trivial if it vanishes on solutions to (2), and two characteristics
are equivalent if they differ by a trivial characteristic. For nondegenerate systems of partial differential equations, each conservation law is uniquely determined by its characteristic, up to equivalence.
Theorem. If the system (2) is nondegenerate, then there is a oneto-one correspondence between (equivalence classes of) nontrivial conservation laws and (equivalence classes of) nontrivial characteristics.
2. Symmetries and Noether's Theorem. A generalized vector field is a first order differential operator a
V = L. (X,U(m)) a + 61
ax
cp(X,U(m))
a
a,-1
aua
If the coefficients ti and cpa, depend only on x and u, then v generates a oneparameter group of geometrical transformations, which solve the system of ordinary differential equations dx'
i
-= a (X,u), de
dua de
= cp (x,u). a
For general v, the group transformations are nonlocal, and determined as solutions of a corresponding system of evolution equations. The vector field v is a symmetry of the system (2) if and only if the infinitesimal invariance condition prv [Ev(W)] = 0,
v =1, 1'.,p,
holds on all solutions to (2). Here pr v denotes the prolongation of v, which determines how v acts on the derivatives of u. An elementary lemma says that we can always replace v by the simpler evolutionary vector field
Olver: Conservation laws in continuum mechanics
VQ =
± Q (X,U(m)) a.1 °
99
a
aua
where the characteristic Q = (Q1, ... (:Q) of v is defined by Q
i
Q
i-1
aU
a
ax
(See [6; Chapter 5] for the explicit formulas.) The infinitesimal invariance condition pr va [Ev(W)] = 0, whenever E(W) = 0,
v = 1,... ,p,
(5)
constitutes a large system of elementary partial differential equations for the components of the characteristic Q. Fixing the order of Q, the defining equations (5) can be systematically solved so as to determine the most general symmetry of the given order of the system.
An evolutionary vector field vq is a trivial symmetry of (2) if the
characteristic Q(x,u(m)) vanishes on all solutions to (2). Two symmetries are equivalent if they differ by a trivial symmetry. Clearly we are only interested in determining classes of inequivalent symmetries of a given system of partial differential equations.
More restrictively, the evolutionary vector field va is called a variational symmetry of the variational problem (1) if the infinitesimal invariance condition
prva(W)=DivB
(6)
holds for some p-tuple B(x,u(k)). Every variational symmetry of a variational integral
(1) is a symmetry of the associated Euler-Lagrange equations (2), but the converse is not always true. (The most common counterexamples are scaling symmetry groups.) It is easy to check which of the symmetries of the EulerLagrange equations satisfy the additional variational criterion (6); see also [6; Proposition 5.39].
Olver: Conservation laws in continuum mechanics
100
Noether's Theorem provides the connection between variational symmetries of a variational integral and conservation laws of the associated EulerLagrange equations E(W) = 0.
Theorem. Suppose we have a variational integral (1) with non degenerate Euler-Lagrange equations (2). Then a p-tuple Q(x,u(m)) is the characteristic of a conservation law for the Euler-Lagrange equations (2) if and only
if it is the characteristic of a variational symmetry of (1). Moreover, equivalent conservation laws correspond to equivalent variational symmetries and vice versa.
Thus there is a one-to-one correspondence between equivalence classes of nontrivial variational symmetries and equivalence classes of nontrivial conservation laws. The proof rests on the elementary integration by parts formula
prv0(W)=Q.E(W)+DivA,
(7)
for some p-tuple A = (A,, ... Ap) depending on 0 and W. (There is an explicit formula for A, but it is a bit complicated; see [6; Proposition 5.74].) Comparing (7)
and the symmetry condition (6), we see that Div(B - A) = 0. E(W),
and hence P = B - A constitutes a conservation law of (2) with characteristic Q. The nontriviality follows from the theorem of section 1.
3. Finite Elasticity. As an application of the general theory, we consider the case of an elastic material, so the stored energy function W(x,Vu) depends only on the
deformation gradient. We show how simple symmetries lead to well-known conservation laws. Material frame indifference implies that W is invariant under the
Euclidean group
u-Ru+a of rotations R and translations a. The translational invariance is already implied by the fact that W does not depend explicitly on u, while rotational invariance requires
Olver: Conservation laws in continuum mechanics
101
that W(x,R.Vu) = W(x,Vu) for all rotations R. The conservation laws coming from
translational invariance are just the Euler-Lagrange equations themselves
ii-1Di{ aW
J = 0,
au"
written in divergence form. The rotational invariance provides p(p-1)/2 further conservation laws
i Di( ua i-1
aW
- u- aW
aup
aua
_ 0.
If the material is homogeneous, then W does not depend on x, and we have the additional symmetry group of translations
x-4x+b in the material coordinates. There are thus p additional conservation laws
D1{±ua aW -8!W}=0, a-1 i au' J
whose entries form the components of Eshelby's celebrated energy-momentum tensor. If the material is isotropic, then W is invariant under the group of rotations in the material coordinates, so W(Vu.R) = W(Vu) for all rotations R. There are an additional p(p-1)/2 conservation laws
ail
± [XiUk - Xkuiaj
aW
- [&ixk - SkxJ)W } = 0.
Scaling symmetries can produce conservation laws under the assumption that W is a homogeneous function of the deformation gradient W(X.Vu) = Xn.W(Vu),
? > 0.
Olver: Conservation laws in continuum mechanics
The scaling group (x,u) -+ (X x,
102
is a variational symmetry group, leading to
A(n-P)/nu)
the conservation law
,
Di{
,I «-1
-ua n
, x'U. I I=1
1
aW + X'W) = 0. aua
Of course, stored energy functions which are invariant under the scaling symmetry group are rather special. If one writes out the above divergence
in the more general case, then we obtain the divergence identity
I aW+zW)=pW. i-1
a=1
j.1
DUa
This was used by Knops and Stuart, [3], to prove the uniqueness of the solutions
corresponding to homogeneous deformations. This latter identity is closely related to the general dentities determined by Pucci and Serrin, [10]. Indeed the general formula used by Pucci and Serrin to determine their identities is a special case of the integration by parts formula (7) in the case that the characteristic 0
comes from a geometrical vector field. Particular choices of the coefficient functions i;i and cpa lead to the particular identities that are used to study eigenvalue problems and uniqueness of solutions, generalizing earlier ideas of Rellich and Pohozaev.
4. Linear Planar Elasticity. Although the general structure of symmetries and conservation laws for many of the variational problems of continuum mechanics remains an open problem, the case of linear planar elasticity, both isotropic and anisotropic, is now well understood. In this case, the stored energy function W(Vu) is a quadratic function of the deformation gradient, which is usually written in terms of the strain tensor e=(Vu+VuT)/2. We have W(Vu) = E qkl eii ew,
(8)
Olver: Conservation laws in continuum mechanics
103
where the constants gjkl are the elastic moduli which describe the physical properties of the elastic material of which the body is composed. The elastic moduli must satisfy certain inequalities stemming from the Legendre-Hadamard strong ellipticity condtion. This states that the quadratic stored energy function W(Vu) must be positive definite whenever the deformation gradient is a rank one tensor, i.e. Vu = a®b for vectors a, b. Following [7], we define the symbol of the
quadratic variational problem with stored energy (8) to be the biquadratic polynomial Q(x,u) = W(x®u) obtained by replacing Vu by the rank one tensor x®u. In this case, the Legendre-Hadamard strong ellipticity condtion requires that Q(x,u) > 0 whenever x * 0, and u * 0.
(9)
The symmetry of the stress tensor and the variational structure of the equations impose the symmetry conditions COd = 91d = qjk'
c ld = Cidy.
on the elastic moduli, which are equivalent to the symmetry condition Q(x,u) = Q(u,x) on the symbol.
For each fixed u, Q(x,u) is a homogeneous quadratic polynomial in x,
and so we can form its discriminant A ,(u) (i.e. b2-4ac), which will be a homogeneous quartic polynomial in u. The nature of the roots of A ,(u) provides the key to the structure of the problem. First, the Legendre-Hadamard condition (9) requires that A,(u) has all complex roots. There are then only two distinct
cases. Theorem. Let W(Vu) be a strongly elliptic quadratic planar variational problem, and let e,(u) be the discriminant of its symbol. Then exactly one of the following possibilities holds. 1. The Isotroicr Case. If z(u) has a complex conjugate pair of double
roots, then there exists a linear change of variables
Olver: Conservation laws in continuum mechanics
x -4 Ax,
u - Bu,
104
A, B invertible 2x2 matrices
which changes W into an isotropic stored energy function.
2. The Orthotropic Case. If it (u) has two complex conjugate pairs of simple roots, then there exists a linear change of variables x -+ Ax,
u -+ Bu,
A, B invertible 2x2 matrices
which changes W into an orthotropic (but not isotropic) stored energy function.
(Recall, [2], that an orthotropic elastic material is one which has three
orthogonal planes of symmetry. Thus, this theorem states that any planar elastic material is equivalent to an orthotropic (possibly isotropic) material, and so has three (not necessarily orthogonal) planes of symmetry. The analogous result is not true in three dimensions, cf. [1].) This theorem is a special case of a general classification of quadratic
variational problems in the plane, [7], and results in the construction of "canonical elastic moduli" for two-dimensional elastic media, [8]. One consequence is that in planar linear elasticity, there are, in reality, only two independent elastic moduli,
since one can rescale any orthotropic stored energy to one whose elastic moduli have the "canonical form" 01111 =62222 = 1,
C1122=C2211=(X, C1212=0,
C11112=01222=0-
Thus the constants a and R play the role of canonical elastic moduli, with the special case 2a + R = 1 corresponding to an isotropic material. This confirms a conjecture made in [5]. Extensions to three-dimensional materials are currently under investigation.
Although isotropic and more general orthotropic materials have similar looking Lagrangians, the structure of their associated conservation laws is quite dissimilar. (For simplicity of notation, we write (x,y) for the independent variables and (u,v) for the dependent variables from now on.)
Olver: Conservation laws in continuum mechanics
105
Theorem. Let W[u] be a strongly elliptic quadratic planar variational problem, with corresponding Euler-Lagrange equations E(W) = 0.
1. The Isotropic Case. If W is equivalent to an isotropic material, then there exists a complex linear combination z of the variables (x, y), a complex linear combination w of the variables (u, v), and two complex linear combinations!;, rl of the components of the deformation gradient (ux,uy,v,,vy) with the properties:
a) The two Euler-Lagrange equations can be written as a single complex differential equation in form Dzrl=O.
(Recall that if z = x + iy, then the complex derivative C is defined as (DX - iDy).)
b) Any conservation law is a real linear combination of the Betti reciprocity relations, the complex conservation laws Re[DZF]=0, and
Re[Dz{(4+z)Gin + G}] = 0,
where F(z,rt) and G(z,rl) are arbitrary complex analytic functions of their two arguments, and the extra conservation law Re[ Dz{wn - izn2} ] = 0.
2. The Orthotropic Case. If W is equivalent to an orthotropic, nohisotropic material, then there exist two complex linear combinations z, of the variables (x, y), and two corresponding complex linear combinations r;, rl of the with the properties: components of the deformation gradient
a) The two Euler-Lagrange equations can be written as a single complex differential equation in either of the two forms Dz 4=0,
Olver: Conservation laws in continuum mechanics
106
or
Dc11=0.
b) Any conservation law is a real linear combination of the Betti reciprocity relations, and the complex conservation laws Re[DZF]=0,
and
Re[cG]=0,
where F(z,i;) and G(t ,1t) are arbitrary complex analytic functions of their two arguments. Thus one has the striking result that in both isotropic and anisotropic planar elasticity, there are three infinite families of conservation laws. One family is
the well-known Betti reciprocity relations. The other two are determined by two arbitrary analytic functions of two complex variables. However, the detailed
structure of these latter two families is markedly different depending upon whether one is in the isotropic or truly anisotropic (orthotropic) case. The two orthotropic families degenerate to a single isotropic family, but a second family makes its appearance in the isotropic case. In addition, the isotropic case is distinguished by the existence of one extra anomalous conservation law, the significance of which is not fully understood. The details of the proof of this theorem in the isotropic case have appeared in [4; Theorem 4.2] (although there is a misprint, corrected in an Errata to [4] appearing recently in the same journal); the anisotropic extension will appear in [9].
I suspect that a similar result even holds in the case of nonlinear planar elasticity, but have not managed to handle the associated "vector conformal
equations", cf. [5]. Extensions to three-dimensional elasticity have only been done in the isotropic case; see [4] for a complete classification of the conservation laws there. In this case, beyond Betti reciprocity, there are just a finite number of conservation laws, some of which were new.
Olver: Conservation laws in continuum mechanics
107
References [1]
Cowin, S.C. and Mehrabadi, M.M., On the identification of material symmetry for anisotropic elastic materials, IMA preprint series #204, Institute for Mathematics and Its Applications, University of
Minnesota, 1985. [2]
[3]
Green, A.E. and Zema, W., Theoretical Elasticity, The Clarendon Press, Oxford, 1954.
Knops, R.J. and Stuart, C.A. Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity, Arch. Rat Mech. Anal. 86 (1984), 234-249.
[4]
Olver, P.J., Conservation laws in elasticity. II. Linear homogeneous isotropic elastostatics, Arch Rat. Mech. Anal. 85 (1984), 131-160.
[5]
Olver, P.J., Symmetry groups and path-independent integrals, in: Fundamentals of Deformation and Fracture, B.A. Bilby, K.J. Miller and J.R. Willis, eds., Cambridge Univ. Press, New York, 1985, pp.57-71.
[6]
Olver, P.J., Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986.
[7]
Olver, P.J., The equivalence problem and canonical forms for quadratic Lagrangians, Adv. in Math., to appear
[8]
Olver, P.J., Canonical elastic moduli, preprint.
[9]
Olver, P.J., Conservation laws in elasticity. Ill. Linear planar anisotropic elastostatics, preprint.
[10]
Pucci, P. and Serrin, J., A general variational identity, Indiana Univ. Math. J.
35 (1986), 681-703.
ON GEOMETRIC AND MODELING PERTURBATIONS DIFFERENTIAL EQUATIONS PARTIAL
IN
L E Payne Department of Mathematics Cornell University Ithaca New York 14853, U S A
I. INTRODUCTION
This paper discusses certain types of stability questions
that have been
largely
ignored
in
the
literature,
i.e.
continuous
dependence on geometry and continuous dependence on modeling. Although we shall consider these questions primarily In the context of ill-posed
problems we shall briefly indicate some difficulties that might arise under geometric and/or modeling perturbations in well posed problems.
In setting up and analyzing a mathematical model of any physical process it is inevitable that a number of different types of errors will
be introduced e.g.
determining coefficients,
etc.
errors There
in measuring data, will
errors
in
also be errors made
in
characterizing the geometry and in formulating the mathematical model.
In most standard problems the errors made will induce little error in the solution itself, but for ill-posed problems in partial differential equations this is no longer true.
Throughout this paper we shall assume that a "solution" to the problem under consideration exists in some accepted sense, but in the case of ill-posed problems such a "solution" will invariably fail to
depend continuously on the data and geometry.
We must appropriately
constrain the solution in order to recover the continuous dependence (see
[81)1
determine.
however, In
the
appropriate restrictions are often difficult to first
place
any
such
mathematically and physically realizable.
constraint
must
be
both
At the same time a given
constraint must simultaneously stabilize against all
possible errors
that may be made in setting up the mathematical model of the physical problem.
Since a constraint restriction has the effect of making an
otherwise linear problem nonlinear, one must use care in treating the various errors separately and superposing the effects.
In any case a
Payne: Geometric and modelling perturbations
109
constraint restriction which stabilizes the problem against errors in
one type of data might not at the same time stabilize the problem against errors in other types of data, errors in geometry or errors made in setting up the model equation.
Although we are concerned here primarily with the question of
errors in geometry we must point out that modeling errors are likely to be
more
serious,
simply
modeling errors precisely.
because
it
is
impossible
characterize
to
Hence it is never certain that a constraint
requirement which stabilizes the problem against other sources of error will
also
stabilize
against
it
modeling
errors.
This
means
that
predictions based on results for the model problem will have to be verified in the physical context which is being modeled before these predictions can be considered reliable.
In order to keep the arguments from becoming excessively involved we
shall
not
in our study of the question of continuous
dependence on geometry simultaneously try to deal with all other types of errors that may have been introduced in setting up our mathematical
models.
Except for modeling errors we could In fact deal with these
other error sources, but In the interests of simplicity we consider only
As reference to previous work on continuous
special types of errors.
dependence on geometry we mention the paper of Crooke and Payne [6] and the recent results of Persens [121.
Papers that might be thought of as
investigations of continuous dependence on modeling include those of Payne and Sather [111, Adelson (1,21, Ames [3,41 and Bennett [5].
These
latter papers are merely illustrative examples which indicate that In some
cases
the types of constraint restriction
used
to
stabilize
problems against errors in Cauchy data may not in fact stabilize against errors in modeling.
Since there
is
no general
theory
for handling
ill-posed
problems for partial differential equations we consider here a number of relatively simple special examples.
By way of comparison we also give a
few specific related examples of well posed problems subject to errors in modeling and geometry.
Also since we are interested primarily in
techniques, we shall assume throughout that classical "solutions" exist. Extensions of these results to weak solutions will be obvious.
In the subsequent text we shall
adopt the convention of
summing over repeated indices, and a comma will denote differentiation.
Payne: Geometric and modelling perturbations
110
II GEOMETRIC AND MODELING ERRORS IN STANDARD PROBLEMS
In this section a few
illustrative examples are given
which demonstrate the effects of geometric and modeling errors
in a
number of well-posed problems.
A. Modeling errors for the Navier-Stokes equations
Let us suppose that the problem we wish to solve is the following:
We seek a vector
ui,
i - 1,2,3, which satisfies
vnui + p,i 22-L at + uj ui,j -
in
ui - 0
(2.1)
on aDx[O,T),
ui(x,O) - Egi(x),
where
v
pressure),
is
gi
smooth boundary
a
physical
If
x e D,
constant,
is prescribed, and aD.
Dx(O,T),
=0
uj j
E < < < 1
p
is
an
unknown
scalar
is a bounded region in
D
R3
(the
with
we would like to know whether the
solution of (2.1) Is approximated well enough by
where
Evi,
vi
is a
solution of avi
- vAvI + q,i at
in Dx(O,T),
vj j
- 0
Vi
- 0
(2.2)
on 8Dx[0,T),
vi(x,0) - gi(x), x e D.
As mentioned earlier it is assumed that classical solutions of both problems exist.
[Ig12 + JDl Iog12) dx E M2.
To compare
ui and
g,v, and
D
are such that
We make the assumption that
(2.3)
evi set
wI - ui - Evi,
(2.4)
O(t) - JD Iw(x,t)IZdx =_ Uw(t)92.
(2.5)
and let
Payne: Geometric and modelling perturbations
111
Clearly
dt -
2 lw' atJ 2 Jwi{vOwi + (p - Eq),i - ujui,j}dx
- - 2vJDwi jwi jdx - 2EJDwiujvi jdx
- - 2vJDwi jwi jdx - 2EJDwiwjvi jdx - 2EZJDwivjvi,jdx. (2.6)
Here
we
have
carried
out
obvious
the
integration
by
parts
and
application of differential equations and boundary conditions.
constant y defined as
We now introduce the Sobolev
inf
1V4)I
1+y1
(2.7)
01 E HO(D) JD[OiOiJZdx
which, upon use of Schwarz's inequality in (2.6), yields 1vws1//2
dt 6 - 2vIVw12 + 2Ey
1'2
1w1 2IVvl
IvvI'4
+ 2E 27 To bound
1vv1
(2.8)
note that
Jpvi,jvi jdx = 2 f Dvi jvi jtdx
dt
2
-
JDvi tvi tdx.
V
(2.9)
An application of Schwarz's inequality on the right then leads to
[Dvivi,tdx] d
2
JDvi jvi jdx dt
- V 1v(t)I2
21
JDvi jvi jdx
(2.10)
Payne: Geometric and modelling perturbations
where
al
112
is the first eigenvalue of
nu+1,u-0
inD (2.11)
u-0
on 8D.
An integration of (2.10) yields 2)1t
Ivv12 4 1vgl2e
(2.12)
In a similar way it can be shown that zalt 1v12 t
Ig12e
(2.13)
Returning to (2.8), we now have
i
Ivw12 + 2E7'2 X1 4 Ivgl Ivw12e)'1t dt 4 - 2v
+ 2E271/2X1/8 1g11/4
Ivw12e 2X1t,
1vg17/*
(2.14)
E7i/2X1
or
41vg1e X1t-S) Ivw12
dL 4 - 2(v -
1/2
+
E4
a'1/2
28
for any positive constant
e
we choose
4
1 g 11/2 1 Vg 17/2
8.
e
-4a 1 t (2.15)
Now assuming that
a144v/(y1/2M)
(2.16)
8 as S - 2 (v-E71/2MX1_1/4)
(2.17)
and obtain
E
dt
4
20X o + 1
411t
M4e 1 LVX/4
_
(2.18)
113
Payne: Geometric and modelling perturbations
An integration yields -4X t 0(t) c
1-e
le
e4H4
I.
-20X t
vX/4 - e-/
0#2
I
1
2X110-21
2M]
(2.19)
4X11 to
which shows that exponential
in
0(t)
is the product of a term Inequality
time.
,
0 - 2
0(e4)
and a decaying
is of course a Reynold's
(2.16)
Using the triangle inequality it follows that
Number hypothesis.
1v(t)1 - 10(t)I i 1u(t)1 t Iv(t)1 + 10(t)l,
and an application of and
lower bounds
on the left and right thus leads to upper
(2.19)
for
(2.20)
One could have obtained a slightly
Iu(t)I.
different result for (2.19) by first integrating by parts the last term of (2.6).
However, the order of
a
would have been the same.
B. Errors in initial geometry for the Heat Equation
In this problem we assume that not all Cauchy data were measured precisely at time measured
along
a
but that
t - 0,
surface
t - ef(x)
in
in
fact the data were
(x,t)-space.
for
If,
instance,
If(x)I < 1,
(2.21)
Is it possible to solve the problem with the measured data prescribed on t - 0
and thus obtain a close approximation to the actual physical
problem being modeled?
To make this precise let
u
be a solution of
8u
- Au - 0
in Dx(0,T)
8t
u - 0 u(x,0) where
D
- uo(x)
on 8Dx [0,T) in D,
is a bounded region in
IRN
with smooth boundary
problem we should be solving asks for the solution
8v
at
(2.22)
v
The
of
- AV - 0
f(x) < t < T,
xeD
v - 0
f(x) i t< T,
x e 8D
v(x,ef(x)) - uo(x),
8D.
x e D.
(2.23)
Payne: Geometric and modelling perturbations
114
The first potential difficulty arises from the fact that
t - ef(x)
not a characteristic surface and thus the problem (2.23) for
a standard well-posed problem. solution
v
of
v
Nevertheless we wish to compare the
(assumed to exist) with the solution
(2.23)
is
Is not
u
of
(2.22).
Again set w = (u - v)
observe
and
that
(2.24)
standard
from
properties
of
solutions
of
initial-boundary value problems for the heat equation it follows that
for
e
: t < T, a-Zal(t -E)
1w(t)12 i 1w(E)12
(2.25)
We, therefore, need to bound
Iw(E)l
in terms of the data.
Now define
u(x,t) - (u(x,t) [uo(x)
v(x,t) - (v(x,t)
,
t > 0
,
t 4 0
I
t > ef(x)
(2.26)
uo(x)
(2.27)
t t Ef(x).
Then
Iw(E )12 = 2
SE f E
ww,ndxdn 4 ED
17
1E 1w,n12dn.
(2.28)
-E
Using the arithmetic-geometric mean Inequality we have E r1u
16e 1
r
n12dn +
Iw(e)I2 t
I
rE I
[v n]Zdndxj.
(2.29)
0
D ef(x)
Now rOE
E
J0lu n12dn - j0 (u nnu)dn
(2.30)
Ivuo(x)12dx,
S 2 J
D
115
Payne: Geometric and modelling perturbations
and E
E
r
(v n)2dndx -
J
v n ov dndx
J
I
(2.31)
f
D ef(x)
D Ef(x)
t
J[v tv ini - 2 v iv intids, E`ll
where
is the surface
E
t - ef(x), x E D.
E
J
We may rewrite (2.31) as
rr
(v n)2dndx i Jv l[v tni - V int]ds + 2 Jv iv intds.
J
E
D Ef(x)
E
(2.32)
Now assume that
nti
M2E,
(2.35)
is the indicated bound on the data.
M2
Thus for
t i E
it
follows that 2ai(t -E)
1w(t)1 t ilee
(2.36)
a result which clearly implies continuous dependence on the initial geometry.
No analogous result for the Navier-Stokes equations has been derived.
116
Payne: Geometric and modelling perturbations
C. Errors in spatial geometry for the Heat Equation
In this example It is assumed that some error was made in
characterizing the spatial geometry and as a result one looks for the solution
of
u
au
- nu
in
Dx(O,T)
on
8D1x[O,T)
at
u - 0
u(x,0) - g(x)
of the solution
instead
(2.37)
in D1,
of the actual model problem
v
av
- nv
in D2x(O,T)
at
v - 0
on 8D2x[O,T)
(2.38)
v2(x,0) - g(x) in D2.
For simplicity of presentation we assume that identical on
g(x)
and
q(x)
are
D - D1 o D2.
Setting w - u1 - u2
(2.39)
and restricting the problem to
we have from standard a priori
D
inequalities that Iw(t)12 < kJtpLlw(x,n)IZdsdn,
(2.40)
JJ
O8D
for some computable constant
depending only on the geometry of D.
k
Here the norm i s the ordinary
norm over the domain D. Now Crooke
L2
and Payne [6) showed that
Jf [w(x,n)]2dxdn O8D
4
k16[J J
L
UU
for a computable constant
distance to the boundary
Here
k1. in
(2.41) Ioul2dxdn + jO JDIoul2dxdn] 2
11
is essentially the maximum
(D1 u D2) n Dc.
computations (2.12) we now know that if D1 U D.
6
%1
But from our previous
is the first eigenvalue for
then 21, 1t
1w(t)12 t kk16M2e
where
M
represents the Indicated bound on the initial data.
(2.42)
Payne: Geometric and modelling perturbations
117
We have presented three illustrative examples of the effects
of modeling errors and geometric errors on the solutions of well-posed problems.
In the next section we shall
see that the behavior
is
radically different for analagous ill-posed Cauchy problems.
AGAINST
III STABILIZING
GEOMETRICAL
ERRORS
IN
POSED
ILL
PROBLEMS
In the previous section we gave examples of two different types of errors in geometry, i.e. errors in the time geometry and errors
in the spatial geometry.
For ill-posed problems very little work has
appeared in the literature.
stabilizing an
The first paper to study the question of
ill-posed problem against errors
in the initial
time
geometry was that of Knops and Payne [7] who investigated the question in the context of classical elastodynamics.
Analogous results for the
backward heat equation were announced in [10].
The first paper dealing with the stabilization of ill-posed problems against errors in the spatial geometry was that of Crooke and Payne [6] who derived the appropriate stabilization inequalities for the
initial-boundary value problem for the backward heat equation Dirichlet boundary conditions.
with
Other problems have subsequently been
investigated by Persens [12].
We first present a sharpened version of the result of Knops and Payne in a special case.
Continuous Dependence on Initial Geometry
A.
As in Problem B of Section 2 we consider a problem in which initial data were actually taken on the surface 1f(x)1 < 1.
t = ef(x) where
The problems whose solutions we now wish to compare are z zl
P(x)
= (ci]kl uk,l),j
in Dx(0,T)
at
ui = 0 ui(x,0) = gi(x),
on ODx(O,T)
as (x) = hi(x),
x E D
(3.1)
118
Payne: Geometric and modelling perturbations
and
p(x)
til. -
for ef(x) < t < T, x e D
(cijkl (x)vk, 1)
for Ef(x) 4 t 0
x E D,
and
Cijkl - cklij.
I)
ii)
(3.3)
Cijkl tt'ij'lkl 4 0,
for all tensors
Oij
i.e. symmetry and negative semidefiniteness of the
As the constraint assumption we prescribe that
strain energy. T
T
f
10J[Pui,tui,t]dxdt +
[Pvi,tvi,t]dxdt 4 M2.
(3.4)
J
D ef
Using (3.4) we now derive a continuous dependence inequality for the L2
integral of A
method (see
wi =- u1 - vi
straight
.
forward
[8]) leads, for
1D pwi(x,t)wi(x,t)dx 4 2
application
t < 2,
of
the Lagrange
identity
to
JD p[wi(x,E)wi(x,E)
+ wi(x,E)wi(x,2t -e)]dx
2t-e +
2 J
(3.5) J
pw1 n(x,n)wi(x,n)dxdn.
D
E
Now observe that
JDp[wi(x,2t-E)wi(x,2t-E)]dx -
pwi(x,E)wi(x,E)dx JD
(3.6)
2t -e + 2
I
pwi(x,n) at wi(x,n)dxdn.
1 E JD
Using the fact that
w1
vanishes on
8D,
the Poincare inequality for
such functions and (3.4), we easily conclude that the last integral in
Payne: Geometric and modelling perturbations
(3.6) is bounded by a multiple of
119
Similarly
M.
2t-E J pwi(w,n)wi(x,n)dxdn t CM2 JE
(3.7)
D
Thus from (3.5) it follows after a use of Schwarz's inequality that i
rr JD pwi(x,E)wi(x,E)dx
pwi(x,t)wi(x,t)dx t
+ CM
Dpwi(x,E)wI(x,E)dx}'2 uu
"
JD
i
t(x,E)dxl2
CZMt{Ifr
+
pwi t(x,E)wi
(3.8)
.
D
Let us now continue (ui(x,t),
as follows:
and vi
ui
0 < t < T,
x e D,
t
x e D,
(3.9)
ui
gi(x),
Then for at
t - e,
C i(x),
fixed,
x
wl a w1.
0,
Ef(x) < t < T, x e D,
i(x,t),
vi -
>e
t 4 Ef(x).
wi - ui -vi
0.10)
x e D.
Is a continuous function of
t
and
Thus
E
p(x)wi(x,E)wi(x,E)dx - 2J-E
JD pwi(x,n)ui,n(x,n)dxdn
JD
84E
E
(3.11)
pwi,7(x,n)wi 7(x,n)dxdn J-E JD
E 16E UH0JDpui,nuindxdn
n
E
+ J J
1
Pvi,nvi,ndxdnJJJl.
D ef(x) Here the arguments are the same as in (2.28).
Clearly, the integrals in braces may be bounded in terms of M2
so 16EM2
p(x)wi(x,E)wi(x,E)dx 4 JD
(3.12)
.
IT
We must now bound the last term
in
(3.8).
before, defining continuations as in (3.9) and (3.10) for we obtain
Proceeding as ul t and vi ,t
Payne: Geometric and modelling perturbations
E
16E
JDpwi,t(x,E)wi t(x,E)dx F
120
E
tr
IJOJDpui,nnui,nndxdn +
n
JJ pvi,nnvi,nndxdnl. J
D of (3.13)
To bound the second integral on the right observe that
IfT
E
n z
=EJ [pvi,nnvi,nn - cijklvl,jnvk,ln]dxdn
JJ pvi,nnvi,nndxdn =
i k1M 191 -g21D + k2Mr f 0 I f - f218Didn] + k3M2s + k4M2a, UU
(3.24)
for
0
c t c T/4.
the term
a
Here the notation is as before with the addition of
which Is an explicit measure of the maximum deviation
between the outward normal directions at points on points
on
8D2.
deviation Is small.
The quantity
a
will
be
8D1
small
if
and associated
this maximum
Payne: Geometric and modelling perturbations
123
iii) The Cauchv problem for the Poisson equation Here the comparison was between solutions of eu'X
- F.
in Da,
ua
- fa
on EX,
(3.25)
a - g« on E., ava where
a - 1,2,
is a smooth portion of
Eu
small perturbation of
8D.,
and
E1
is assumed to be a
No data are given on the remainder of
E2.
8D2.
As is usual in problems of this type one defines D - Ds n D2
(3.26)
and then defines subdomains chosen surfaces
p(x) - 8.
uidx + 1 JD1
DS
of
D
by a family of appropriately
Using the constraint
u2dx 6 M2.
(3.27)
D2
Persens obtained the continuous dependence inequality
1p13 d k1M2v(B)[a1Nf1-f2NE1+a21grads(f1-f2)OEs lu1-u22
-v(B)
+ a91g1 -g,NE +a4M26 ]1
(3.28)
1
where the various terms are defined as before and function of
B
which satisfies for
v(8)
is an explicit
0 f B < 1,
0 4 v(B) < 1.
Persens
[12]
(3.29)
also
considered
the
problem
of
continuous
dependence on geometry for the Dirichlet initial boundary value problem of
linear
elastodynamics
elasticities.
In
without
a
definiteness
assumption
on
this problem the constraint restriction was
the more
severe, but results were quite similar to those obtained by Crooke and Payne [6] for the backward heat equation.
IV
CONTINUOUS DEPENDENCE ON MODELING: SOME EXAMPLES As mentioned earlier it is impossible to characterize the
error made in setting up a mathematical model of a physical problem. For
ill-posed problems we know that any error
in modeling may
lead to
Payne: Geometric and modelling perturbations
124
instabilities unless "solutions" are adequately constrained.
Since the
modeling errors are unknown we can only give examples and determine constraints sufficient to ensure stability against the modeling errors indicated in the examples. The
first
study
this
of
type
of
modeling error
in
an
ill-posed problem was perhaps that of Payne and Sather [11] who compared
the solution of the backward heat equation with that of a singularly
A few years later Adelson
perturbed well-posed hyperbolic problem.
[1,2] considered a number of quasilinear Cauchy problems for elliptic
One wishes to compare an
systems of which the following is typical. appropriately constrained solution
beAv+v=ul nu=o
of
v
in DCB2n (4.1)
u = f, grad u = g
v=f, grade =q where the Cauchy surface number
and
b
E
on E is as in (3.25),
constant
a
is
1!
(positive
appropriately constrained solution
a
or
is a small positive negative),
with
an
of
w
nw=0 in D
w=f on E
-
(4.2)
grad w = g assuming of course that the data of L2
are close to that of
w
v.
Using
constraints, Adelson obtained comparison results similar to (3.28)
in the case
b < 0.
constraints on
v
(Here of course
s = 0).
For
b > 0
more severe
were required and the resulting continuous dependence
inequality was less sharp.
The results of Ames
[3,4]
might also be
continuous dependence on modeling results. inequalities
relating,
for
instance,
initial-boundary value problem for
interpreted as
Ames developed comparison
solutions
of
the
Dirichlet
the backward heat equation with
solutions of various related well posed singular perturbation problems introduced when employing the quasireversibility method for stabilizing ill-posed problems.
Payne: Geometric and modelling perturbations
125
Finally, we mention some recent results of Bennett [5].
Cauchv problem for the minimal surface equation
I)
Here one is concerned with the solution
[11 +
IouI2]-%u j, j - 0
1n
u
of the problem
D C An,
u - ef(x) gradu - eg(x) where
E
(4.3)
on E,
is defined as before to be a smooth portion of
Bennett
8D.
compared the solution of this problem with that of
nh-0 inD h -f(x) on
(4.4)
E,
gradh - g(x) assuming
JE{If-fI2 + Ig-gl21ds 4 Key for some positive
y.
(4.5)
Setting
w - u - eh
(4.6)
and imposing constraints of the form
Igradulfi1l + e2lgradul2]dx 4 Mie4 J
(4.7)
D
and JD [u2
+ e2h2]dx 4 M21
(4.8)
he derived the following continuous dependence result for
j
w2dx 4 Ke4V(8)
,
y = 2
0 < S < R1 < 1.
D8
where
He actually obtained results for a range
is as in (3.28).
v(S)
A
of values of
y.
We
The constant note
that
in
of course depends on the M.
K
order
to
stabilize
the
Dirichlet
initial-boundary value problem for the backward heat equation against most
other
sources
of
error
it
was
sufficient to
impose
an
L2
Payne: Geometric and modelling perturbations
Regarding
constraint.
the
126
surface
minimal
equation
problem
as
a
modeling perturbation of the backward heat equation we note that here a
much stronger constraint was sufficient constraint,
imposed.
Admittedly
(4.6)
is
only
a
it seems highly unlikely that continuous
but
dependence could be established under a significantly weaker constraint on
u.
A second problem considered by Bennett is of some interest,
The end problem for the one dimensional nonlinear heat
ii)
equation
In this example one wishes to compare the solution of au
a
au
a
0 < x < a,
at = ax [p([8x]2)ax],
(4.9)
u=Ef(t) au
on x - 0, t1 < t < t2,
- Eg(t)
ax
t > t,
with that of the ill posed problem
ah
at-nh=0, 0<xt1
h = f(t)
ah
A
(4.10)
onx=0, t1 -T 0
since Iu(t)I < 1. If equality holds then y(t) = 0 and Iu(t)I most) every t
1 for (al-
(O,T). This contradicts (S).
To prove (ii) we choose functions un with progressively faster oscillations. Let
un (t) _ +1 for t
(2k T,22n1 T), (9)
un(t) _ -1 for t where k
(
2k+l
2k+2
{0,1,...,n-1}. The state equation (S) yields lyn(t)I < 2n and
hence J(un) < -T + T(2n)2 so that (ii) follows.
The reason that (M) has no solution is that the "optimal" solu-
tion would have to satisfy two incompatible conditions, namely u = 0 (to ensure y = 0) and u2 a 1. The approximate solutions develop fast oscilla-
tions to avoid this incompatibility. In our case we have u2 a 1 and we claim that un is close to zero. Obviously un is not close to zero in the
sense that
Iun-OI is small. Nevertheless the primitive yn of un is small
and it is yn which enters the optimization problem. Hence we are led to measure the distance between two functions by means of their primitives rather than by the difference of their values. We define a new distance between two (bounded measurable) functions f and g by
d(f,g) =
sup
O>E, then the elastic energy of the network will be of order
EL
ti
Ce Ld 602
(15)
.
We now consider the covering network as a network of conductors gij=mij, with boundary conditions Vi 0 on the external boundary and V.=64 internal boundary.
0
on the
The production rate of Joule heat will then be of
order
WL
ti
oe Ld-2 602
(16)
The different power of L appearing in Eqs. (15) and (16) is a result of the different physical dimensions of the elastic modulus C and the conductivity o.
From the earlier discussion, we deduce that WL < EL.
We now allow L to decrease until it becomes less than When L2.
There also remains a challenge to prove that
Eq. (17) is actually an equality for all 2 0 a.e. in n , then the energy density (per unit reference volume) is given by W(F) - W(F,e)
:
- WFA)T(FA),e).
It follows from (1.1) that W(RFM) - W(F) for all
R e 0+(3),
orthogonal GI
(1.2)
F e M+ " a
group,
(AMA-11M a GL(Z)s
and M9X9
and
M e GI
where
0+(3)
is the proper
(F E MI x a I det F > 0)
and
det M - 11.
As is customary in nonlinear elasticity, we suppose that Is
bounded
below
and
Borel
measurable
and,
in
order
to
W
prevent
Fonseca: Stability of elastic crystals
189
interpenetration of the matter, we further assume that +m, when
W(F)
det F -a 0+
with
Moreover,
W(F) - +m
if and only if
if
is
W
det
differentiable
F < 0. in
the
M+ " s
first
Plola-Kirchhoff stress tensor is given by S(F) = aF(F)
for all
F e M+X S. It is the behaviour of
the
invariance
help
that
(1.2)
with respect to
W
us
to
together with
8
understand
the
onset
of
cubic-tetragonal transitions and the appearance of twinned states in the
martensite phase at
low temperature
(cf.
Basinski
Christian
&
[3],
Ericksen [8], Fonseca (10], Kinderlehrer (15], Wayman & Shimizu [19]).
As discussed by James
is composed of
a twinned configuration
[14],
several morphologically different phases that are symmetry related and energetically
equivalent,
(interfaces)
where
meeting
at
deformation
the
surfaces
of
gradient
composition
suffers
jump
discontinuities.
However, due to the invariance of discrete
group
Gi,
are
we
with
faced
under the infinite
W
serious
analyzing equilibria and stability problems.
In
difficulties fact,
when
as shown by
Ericksen [9) and Kinderlehrer [15], the presence of twinned co- existent metastable states leads to the breakdown of ellipticity of the nonlinear system of equilibrium equations -div S(Vu) - f Also,
the
minimization
a.e. in Cl
of
the
.
total
energy
functionals
cannot be
achieved by direct methods of the calculus of variations (cf. Ball [1]) since,
as
W
is bounded on several directions, there is failure of
growth at infinity and the total stored energy functional
u
L W(vu(x))dx
Is not sequentially weakly Kinderlehrer
[4) and Fonseca
*
lower semicontinuous [10]).
Moreover,
(cf.
the usual
Chi pot
&
relaxation
Fonseca: Stability of elastic crystals
190
techniques (cf. Dacorogna [5]) cannot be applied because of condition (1.3).
As we will show next,
it turns out that these variational
problems are very unstable (cf. Chipot & Kinderlehrer [4], Fonseca [11], [12] and Fonseca & Tartar [13]).
2.
VARIATIONAL PROBLEMS AND RELAXATION
we assume that
Throughout this section,
(1.2) and
(1.3)
hold.
we discuss
Firstly,
briefly
stability properties
the
of
solutions of the dead loading traction boundary value problem. If
f
:
629
fl
is the body force per unit volume
reference configuration and if
t
is the surface traction per
629
aft
:
in the
unit area of the undeformed configuration, we seek minimizers of the total energy functional
E(.) which is given by
E(u): = JnW(vu)dx - Jnf.u dx -
t.u dS. Jan
As
proved
Fonseca
in
[11],
only
stresses
residual
can
provide minima, namely Theorem 2.1 ([11]) Let
be a bounded
fl c 629
1
(aft;629)
f f dx + f W
f c Lq(f1;629) and
satisfy the compatibility condition
t dS - 0. an
fl If
domain, and for
P
1- q-,q t E W
let
+ q - 1,
1 < p < +°^,
Ct
is continuous then
E(u) > -
inf u e W1,P ((2;629) if and only if
a.e. in
f -0
fl
and
t = 0
a.e. on
an.
Remark 2.2 (1)
p - 1
or
The
p - +=
conclusion
of
Theorem
whenever the problem
2.1
holds
for
the
cases
Fonseca: Stability of elastic crystals
-div T = f Tu = t
in n,
on 8n,
T e L1(n;M3"3), where
admits a solution to
191
u
is the outward unit normal
an. (2)
When there is no loading,
inf
jnW(vu)dx
u E W1.P(n;IR3) is attained at some
W(vuo) =
uo e W1.p(n;IR3)
if and only if
min W(F) a.e. X E n. Fe M3 X 3
The nature and number of such natural In fact,
complicated.
if
states can be very
is differentiable in
W
M+ x3
it may be
possible to construct infinite collections of twinned stable solutions of the homogeneous problem div S(Vu) = 0 S(vu).u = 0
in n,
on an.
It can be shown also that metastable states are subjected to severe restrictions (cf. Fonseca [11]).
E(.) with respect to the norm
In fact, relative minimizers of
11.11°, + 11.11,
,P,
for
1
p < +-,
must
satisfy a balance law similar to the balance of angular momenta, where the deformation is replaced by its gradient.
drawn from this necessary condition;
Several conclusions can be
in particular,
it implies that no
piecewise homogeneous deformation is a relative minimizer unless there is no loading.
As shown by Ball
& Murat
[2],
W'°D - quasiconvexity
is a
necessary condition for the sequential weak * lower semicontinuity of a multiple integral. u
Hence, since
JW(vu)dx
is not sequentially weak * lower semicontinuous, it might be helpful to identify the lower proved that
W1,°° - quasiconvex envelope of
W, QW.
In [12], we
Fonseca: Stability of elastic crystals
192
QW(F) = h(det F)
for all
F E Ms X',
sub-energy function
where
(2.3) h
is
lower convex envelope of the
the
given by
0
¢(s): - inf (W(F)I det F - s), Moreover,
showed
we
that
s e R.
the
characterization
given
by
Dacorogna [6] of the quasiconvex envelope of a continuous bounded below
and everywhere finite function is still valid for
W
satisfying (1.2)
and (1.3), namely QW(F) - Z(F) for all
F e M+ " ',
(2.4)
where
Z(F): - inf {
1
meas n in J
W(F + v{(x))dxl(E W1,'(n;R')).
A similar result was obtained by Chipot & Kinderlehrer [4], who proved that
Z(F) - h(det F)
for every F e M+ " ' In Fonseca & Tartar [13], the pure displacement problem was studied.
On
taking
into
account
(2.3),
necessary
and
sufficient
conditions were sought for the existence of minimizers of I(u): - Jnh(det vu)dx - Jnf.u dx
where
f e L'(n;R')
and
u E u0 + W' o
We must notice, however, that on setting 1(u): - InW(vu)dx - Inf.u dx,
«: = inf {I(u)I u e uo + Wl,°(n;R')) and
R: - inf {I(u)I u E uo + it was not possible to show that a
o
a - B.
In fact on letting
inf {I(u)I u e uo + W',°((I;R')
and u is piecewise affine)
193
Fonseca: Stability of elastic crystals
and
0': = inf {I(u)I u e uo + Wo'
((l;A3)
and u is piecewise affine)
it follows from (2.3) and (2.4) that < a < S
= « .
However, due to (1.3) we have that when
h(s) -. +-
s + 0+,
and so a density argument could not be devised allowing us to conclude «
that
«.
We define the set of admissible deformations H = {u e Wt'
(a:R')I det Vu > 0 a.e. in (2
and ulan = uolan).
Theorem 2.5 ([13))
and let s
0+.
in
n.
be an open bounded strongly Lipschitz domain
n c R3
Let
h:(0,+°)
(a) If
be convex and bounded below with
A2
Assume that
f e Lt(n;R9)
and that
is such that
u e H
h(s)
when
uo a C((I;R3) is one-to-one
I(u) < +W
and I(u) S I(v)
for
all v e H, then vuTf
(i)
is a gradient in
(ii) there exists every finite collection
D C n
D'(n;R3),
with meas
such that for
aD = 0
(x1,...,xn,xn +i = xt) C n \ D
one has
n
f(xi).(u(xi) - u(xi+ 1)) >0, =
i=1 (iii)
uo(n)
A
there
exists
a
convex
function
G:
convex
such that JOG(u(x))dx a
A2
and
f(x) a &G(u(x))
a.e.
x e n
and (iv)
if,
in addition, there exists
1nh(a det Vu) det vu dx > then
a e
(0,1) such that
hull
Fonseca: Stability of elastic crystals
194
G(u(x)) - h(det Vu(x)) - h'(det vu(x)) det Vu(x) + C a.e. x e n
for some constant C e (b)
JR.
u e H
Let
be such that
I(u) < +-
with
inh'(det Vu) det Vu dx > If uo(n)
IR
there
exists
convex
a
function
G: convex
hull
such that
G(u(x)) - h(det Vu(x)) - h (det Vu(x)) det vu(x) a.e.
x e n
with
f(x) a 8G(u(x))
x e a
a.e.
then I(u) < I(v)
for all
v E H.
Remarks 2.6
(1) The necessary conditions are obtained regardless of the boundary conditions.
In fact,
it is possible to generalise part (a) of
Theorem 2.2 as follows: Let
3 < p < +m ,
and n 8f12 - 0
u E Hp
t e L1 (8n2;R )
and let
an - and u ant
with Assume that
Hp: - (u E W1.p(n;IR3)1u1an1 - uo).
and that there exists
ulaa - ullan.
where
ul a C(n;J3)
one-to-one in
n
such that
If
J(u): - Jnh(det Vu)dx - Jnf.u dx - Jan t.u dS < 2
and if there exists
e > 0
J(u) < J(v) then (1), (ivy)
such that
for every
(ii) and (iii) hold.
if there exists
v c Hp
with IN - ullLp < E
Moreover,
0 < a < 1 < S
- < Jnh'(a det Vu) det Vu dx
for which and
Jnh'(R det vu)det Vu dx < +m
then
G(u(x)) - h(det Vu(x)) - h'(det Vu(x)) det Vu(x) + C for some constant C e A. (2) The conditions (1) and (ii) of Theorem 2.5 may be used to
195
Fonseca: Stability of elastic crystals
detect body forces
for which
f
is not attained in
inf I(u)
As
H.
uE H an example, let VfTVu Also, if
Then from
f e Wloc (n%IR3)
it follows that
is a symmetric matrix.
(2.7)
f E C'(f;R'))
is smooth enough (e.g.
f
(i)
then (ii)
and (2.7)
yield
VfTVu is a positive symmetric matrix and so,
det of > 0 a.e. in f2 In particular,
(2.8)
.
excludes compressive body
(2.8)
forces of the type
f(x) --kx, k> 0. From the theorem above, we may conclude that always achieved in the case where
In
I(u)
uE
uo = identity, f = -kes
with
is
k > 0
(gravity field) and the material is "strong", i.e. h(5)
+W when
(2.9)
s
5
If
(2.9)
fails,
it
is
possible to construct an example where the
material breaks down for
k
bigger than a certain critical
kc
(cf.
Fonseca & Tartar [15]).
REFERENCES [1]
"Convexity conditions nonlinear elastostatics", Arch.
BALL, J.M.
[2]
337-403. BALL, J.M. & MURAT, F.
[3]
225-253. BASINSKI, Z.S. & CHRISTIAN, J.W.,
[5]
existence Mech.
Anal.
theorems 63
in (1977),
"W1' -quasiconvexity variational and problems for multiple integrals", J. Funct. Anal. 58 (1984),
by twin boundary movements [4]
and Rat.
in
"Crystallography of deformation Indium-Thallium alloys", Acta
Metallurgica 2 (1954), 101-116. CHIPOT, M. & KINDERLEHRER, D. "Equilibrium configurations of crystals", to appear DACOROGNA, B. "Quasiconvexity and relaxation of non convex problems in
the calculus of variations",
102-118. DACOROGNA, B.
J.
Funct.
Anal
46
(1982),
"Remarques sur les notions de polyconvexite, quasi-convexite et convexite de rang 1", J. Math. Pures et Appl. 64 (1985), 403-438. (7] ERICKSEN, J.L. "Special topics in elastostatics", Advances in Applied Mech. 17 (1977), 188-244. [8] ERICKSEN, J.L. "Some phase transitions in crystals", Arch. Rat. [6]
Fonseca: Stability of elastic crystals
196
Mech. Anal. 73 (1980), 99-124. [9] ERICKSEN, J.L. "Twinning of crystals", IMA Preprint #95, Univ of Minn., Minneapolis, 1984. Thesis, (10] FONSECA, I. Variational Methods for Elastic Crystals. Univ. of Minn., Minneapolis, 1985.
[11] FONSECA, I. Variational methods for elastic crystals", Arch. Rat. Mech. Anal., to appear. (12] FONSECA, I. "The lower quasiconvex envelope of the stored energy function for an elastic crystal", to apear. [13] FONSECA, I. & TARTAR, L. "The displacement problem for elastic crystals", in preparation. [14] JAMES, R.D. "Finite deformation by mechanical twinning", Arch. Rat. Mech. Anal. 77 (1981), 143-176. [15] KINDERLEHRER, D. "Twinning of crystals II". IMA Preprint #106, Univ. of Minn., Minneapolis, 1984. (16] MORREY, C.B. "Quasi-convexity and the lower semi-continuity of multiple integrals", Pacific J. Math. 2 (1952), 25-53. (17] PARRY, G. "Coexistent austenitic and mrtensitic phases in elastic Proceedings of the 5t International Symp. on crystals", Continuum Models of Discrete Systems, (SPENCER, A.J.M. ed. to appear). [18] PITTERI, M. "On the kinematics of mechanical twinning in crystals", Arch. Rat. Mech. Anal. 88 (1985), 25-57. [19] WAYMAN, C.M. & SHIMIZU, K. "The shape memory ('Marmen') effect in alloys", Metal Science Journal 8 (1972), 175-183.
OPTIMAL BOUNDS FOR CONDUCTION IN TWO-DIMENSIONAL, TWO-PHASE, ANISOTROPIC MEDIA
G.A. Francfort Laboratoire Central des Ponts & Chaussees, 58 bld Lefebvre - 75732 PARIS Cedex 15 - France. F. Murat Laboratoire d'Analyse Numerique, Universite Pierre & Marie Curie 5, Place Jussieu - 75230 PARIS Cedex 05 - France.
INTRODUCTION
This paper is concerned with the determination of the set of all possible effective conductivities of a two-phase anisotropic material
with arbitrary phase geometry. Since Hashin & Shtrikman's original bounds on the set of possible isotropic effective tensors of a two-phase material
with isotropic phases due attention has been paid to the case of isotropic phases
(cf. Hashin (1983), Tartar (1985), Kohn & Milton (1985), Francfort
& Murat (1986), Ericksen, Kinderlehrer, Kohn & Lions (1986) and references therein).
The case of polycrystalline media has been considerably less investigated (cf. Schulgasser (1977)). In a two-dimensional setting, Lurie & Cherkaev (1984) addressed the problem of characterizing the set of all anisotropic effective conductivity tensors of a two-phase material with anisotropically conducting phases in arbitrary volume fraction. In the present paper (which describes the results of Francfort & Murat (1987)), we revisit Lurie & Cherkaev's bounds and derive a complete characterization in the two-dimensional case. We consider two homogeneous and anisotropic conducting materials. If they are positioned in a common reference
configuration, there exists
an orthonormal basis el, e2 of R2 such that the conductivity tensors Al and A2 of the two phases read as
Al = ale, ®e1 +a2e2 ®e2,
A2 = BielOei+82e2®e2, and we assume with no loss of generality that
(1)
Francfort & Murat: Optimal bounds for conduction
198
0 0 and/or *n' -0' (13)
,p(d) = if {cpnd+cp2yn}, y0 and 'V'n >0, the inequality X1,X2 n +Vn2cp' n implies that A1,A2 0 and/or b >0,
a2 aX1X2 +b},
D< W ={(al,a2) E[ a,s] 2 C SalX2}, I
(0
={(Xl,A2)E[a,6]2 I c>ala2}
In view of the above computations, the set K(y,S,cp,4) decomposes as
=D( (y)0 D> (6)
n
cp',V'S0
nn
L< (O) is H-stable, which is the object of the next subsection
Francfort & Murat: Optimal bounds for conduction
4.2.
208
A few results of the theory of H-convergence. Our purpose here is not to give a detailed exposition of the
theory of H-convergence, but rather to state and/or derive the few results needed in the proof of Theorem 3. Further details may be found in Francfort & Murat (1987).
The following theorem, specific to the two-dimensional case, is central to our analysis.
THEOREM 4. Let A belong to .AG([ a, R] 2) and H-converge to an element A0 of ,A(,([ a, a] 3). Then A
A
detEA H-converges to
det A
This result traces back to Keller (1964). A proof was presented by Kohler & Papanicolaou (1982) in a probabilistic setting. The proof presented below is due to L. Tartar.
Proof of Theorem 4. Let we, qE be as in Definition 2 (cf. (4)(6)). If R denotes the rotation of angle - 7/2, we set
Then, as c tends to zero,
we:
wo = Rqo
qE
qo = Rwo,
(15)
weakly in [L2(0)]2. Furthermore, recalling (6),
curl wE _ - div qE, (16)
div qE
= curl wE,
lie in a compact set of Hloc(D).
Francfort & Murat: Optimal bounds for conduction
209
Finally,
w
B
qe
e
(17) e
Theorem 1 applies to Be and yields the existence of a subse-
quence BE,
of Be and of an element Bo of ,A(,([ a, R] 2) such that Be, H-conas e' tends to zero. In view of (15)-(17) and the very defi-
verges to B 0
nition of H-convergence (cf. Definition 2), we conclude that
qo = Bowo'
or still that (I - tRB0RA0)wo = 0,
where I stands for the identity mapping on R2.
A classical argument in the theory of H-convergence permits to choose w0 as an arbitrary vector of R2 (at least locally), from which it is easily concluded that
Bo(x) =RAo1(x) tR almost everywhere.
The identity
RC
1
tc tR = det C
which holds true for any invertible two by two matrix yields the desired result
As announced in Remark 9, the H-stability of .A(,(K(y,d,(p,,p)) will result from the H-stability of four kinds of sets. Specifically, the following lemmae hold in the context of Definition 7:
LEMMA 1. The sets .A(,(D (c)) and .A(,(D., (r)) are H-stable
LEMMA 2. If a>0 and b>0, the set .A(,(L>- (a,b)) is H-stable. If ab'm0, the set .A(,(L< (a,b)) is H-stable
210
Francfort & Murat: Optimal bounds for conduction
Lemma 1 easily results from Theorem 4 together with the following comparison lemma
(Tartar (1979a))
:
LEMMA 3. Let AE and Be belong to .X([a,R]2) and H-converge
A
and B 0
respectively. If
0
AC(x) 1, the solution is constant:
In the lower triangle it can be reconstructed from its level lines, and it is * = x+y. * a 1.
That solution has
1v*1 = (1+1) 1/2
and violates
In fact the constraint jaJ < 1 and the boundary a.n = 1 require that a = n along the bottom; the level lines (which are the streamlines in a flow problem) Iv*
< 1.
condition
must exit vertically. * = r, with
remainder of
The minimizing function becomes
in the quarter-disk r < 1. In the the solution is again * a 1. The level
Iv*I a 1 0
curves become circular arcs instead of straight lines. In this example, where * = g = 0 at the origin, the level curve yt had to be at least a distance t away. It could not come closer to the origin, because
(v*I < 1
forbids it.
at another,
If
= 0
at one point and
$ = t
then the distance between those points is not less than t. This constraint on yt, reflecting the constraint 17+1 < 1 on itself, is the key to its construction.
234
Kohn & Strang: The constrained least gradient problem
THE CONSTRUCTION OF THE LEVEL CURVES Our problem is to minimize f lv$Idx = f lytldt. We approach it by minimizing each individual length lytI. Then we show that the curves
yt
which solve this family
of one-dimensional problems, with t as parameter, fit together as the level curves of a function >a(x,y) which is minimizing. The boundary points at which on
yt.
Without a constraint on
u = g = t
must lie
would connect
Iv*I, yt
those points by straight lines of minimum total length.
The
condition 1v$ < 1 imposes an extra requirement: yt cannot enter the disk of radius It-g(Q)j around any
boundary point
Each level curve avoids a set which is
Q.
a union of open disks. have
and
* = t
is less than
If it entered that set we would at points whose distance apart
$ = g(Q)
It-g(Q)I, contradicting
jv.1 < 1.
The key is that it suffices to consider only pairs of points, one of which lies on the boundary. struction of
yt
The con-
is independent of the construction of
(we give a further example below). that the distance between yt and
ys
The problem is to show is at least It-sI. ys
This construction can be restated, more precisely but more awkwardly, in terms of
Et = {x: +(x) > t).
That
set must contain the union Dt
g(Q)>t
B(Qs8(Q)-t)
and it must not intersect the union Gt
Here
B(Q,r)
of the boundary point Et n Gt = 0 the disks
B(Q,t-g(Q))
g(Q) 0,
t.
Informal proof.
as close as possible:
Suppose
A e y0
and
B e yt
are
d(A,B) = d(yo,yt).
B
is a boundary point (or if the shortest path from A to B touches the boundary, when 0 is not convex) then from the construction d(A,B) > t. or
A
(1)
If
(2)
If not (1), the tangent to
the tangent to
yt
at
B.
straight pieces of
y0
and
yO
at
A
In case both A
is parallel to and
yt, we can translate
B A
are on and
B
for some until (1) occurs or until (say) d(Q,A) = Ig(Q)I Q. Then A is on one of the circles boundary point The which y0 cannot enter, and yo is tangent to it. points
A,B,Q
parallel (and (3)
If
are collinear because the tangents are Q is not between A and B or 11) occurs).
g(Q) < 0
we have the expected situation
d(A,B) > d(B,Q) - d(A,Q) > t + Ig(Q)I - Ig(Q)I = t.
Similarly if B is on the circle of radius g(R) - t > 0 around a boundary point R, we conclude that d(A,B) > d(A,R) - d(B,r) > t. recognize that
yo
and
yt
For the remaining cases we may consist of unions of
disjoint curves (as in the example below) and we pick out the components yo and yt which contain A and B. They connect boundary points at which g(A1) = g(A2) = 0 and g(B1) = g(B2) = t-
g(Q) > 0 and the points occur in the order Between B1 and Q, g has an even number of zeros. Therefore in addition to A there is a 1 point A3 from which a different component y" crosses (4)
Suppose
QAB
(Fig. 3a).
to the other side QB2. If y" is below yo this contradicts d(A,B) = dist(y0,yt). If y" is above this contradicts d(Q,y0) > g(Q) = d(Q,A).
y0
Similarly there is no point, collinear in the order ABR, with
d(B,R) = t - g(R) > 0.
236
Kohn & Strang: The constrained least gradient problem
We assume the generic case in which
Note:
g - t and
g
and
have an even number of (simple) zeros on the boundary yo
and
yt
are disjoint unions of smooth eurves;
otherwise perturb to ygyt+e. Suppose finally that the points occur in the order A still lies on the circle of radius g(Q) > 0 around Q. Because B is the point on yt closest to A, it must lie on a circle of radius Jg(R)-tJ around some (5)
ABQ.
boundary point R, collinear with AB and below yt (R = Q is likely but in principle could yield another all candidate.) These properties of R lead to the same conclusion, for g(R) > t in case (3) and g(R) < t in case (4),
that d(A,B) > t.
This completes the proof (not minimal!) that d(yO,yt) > t. Similarly d(ys,yt) > Is-ti. Therefore there is a function
.(x)
with these as its level sets and
with iv*1 < 1. That function solves the constrained least gradient problem. Example 2:
Suppose
a.n = f = 2xy = sin 20. l
'ff
01
is the unit circle and 1 g = cos 20 =
Its integral is
(y2-x2), the boundary value for
i
.
We look first at the minimization of fhai2dx = fivsi2dx. The solution is the harmonic function (y2-x2), or a=(*y,-*x) = (y,x).
whole of
Its support is the
0, and its level curves are hyperbolas. Next we minimize
straight lines.
fiaidx ; the level curves are
For any value in the range
- .. < t
0.
As
U a
is
square
result,
integrable
in
U 6 H1(st)
the vicinity
the existence of singularities
amounts to the existence of solutions of the form (1.2) with
0 < Re a
1
it,
for a non-convex point.
i.e.
It is not difficult to obtain a physical interpretation of this result.
We consider the flux lines tangent to U
is
an
analytic
subdomain of
Q. In
function
and
other words,
At the interior of
grad u.
consequently
it
cannot
which push to each other. Moreover
(1.6)
vanish in
the flux lines must go all
then in the non-convex corner, the boundary "disturbs"
0 = - o U = - div (grad U)
st,
over
S2
a ;
the flux lines
263
Leguillon & Sanchez-Palencia: Singularities in elliptic non-smooth problems
i.e.
behaves as an incompressible fluid, and
grad U
becomes
Igrad UI
infinite at points where the flux lines touch to each other.
In the case of an elliptic equation for a composite medium (i.e.
containing an
interface
r )
the criterion for the existence of
a singularity becomes the non-convexity of the boundary
to the refracted fluxes.
with respect
E
Indeed, let us consider the elliptic equation
in divergence form
ax
(ai'(x) aaUU) = 0
i
with piecewise constant coefficients, taking the
constant
domains
values
in
22
(Fig.
Q1,
define the gradient a
au
axi
ai - aij axj
equation
(1.7)
bution sense for
in
sense
= 0
(g t)
= 0
in
the
distri-
belonging locally
and
sz 1
(an)
where the bracket denote the jump across n
U
s22
and
the
transmission conditions
Fig. 4
(U)
flux
amounts to (1.7) in the classi-
H1
cal
(1.8)
We
the
au then,
to
and
of
4).
by
gi
E1
g
each
r
= 0
and the indices
t
and
denote the tangential and normal components.
If we consider solutions with region
stl,
to (1.7)
02,
(1.8)
implies a
a
and
g
refraction law across
constant in each r.
Now, we add
the Neumann boundary condition
an = 0
(1.9)
on
ast
and we see that the above mentioned piecewise constant solution is consis-
tent with a boundary with sides Then,
the
refracted flux
E1, E2
parallel to
lines are analogous
in
a1, a2 (Fig.
4).
this problem to the
264
Leguillon & Sanchez-Palencia: Singularities in elliptic non-smooth problems
straight lines of the Laplacian. The singularity criterion is the following
:
Send a flux refracted flux
a2
with respect to
a1
across
v
we
parallel
l1
in
r. Then, if have
E2
to
E1
and compute the
is convex (resp. non-convex)
no singularity
(resp.
singularity)
Fig.
5.
Non-singular
Singular Fig. 5
We see in particular that the anisotropy plays an important role in the existence of singularities.
For Dirichlet
boundary conditions the
(instead of Neumann)
criterion is the same with
a
replaced by the normal to
g
(as
u = 0
at the boundary implies that the gradient is normal to it).
the
2.
NUMERICAL COMPUTATION OF SINGULARITIES
In
elasticity
refracted
known.
It
is
fluxes
one)
simple
a
criterion
for the existence of
singularities
a,
and
in
particular
is
to
not
the existence of such
satisfying
0 < Re a
0
A
C=
2[ f' (a) - a y"(a)]
B=
,
8 [11(a) -a'Q"(a) +
3
2
(a)
The discrete equationsof motion are the Euler-Lagrange equations associated with the Lagrangian density d2v
m
L - Z 2 m(avn/at) 2
- 4)
.
We obtain thus
- [A (en-en+l) - B(en-en+l) + C(en-en+l)].
n
(8)
d t2
On taking the continuum limit such that vn+l=v((n+l)a)=v(na)+a(av/axl+.., we evaluate en =(av/ax)-..
,
en-en+l
en-en+l ,etc
'
.
Being careful
to retain all relevant terms, from eqns.(8) we obtain the following partial differential equation 2
at
v
2
2
v =0 a--2] ax ax
- cT[l + Eg(vx) + I E
(9)
for the continuous transverse displacement v , with c,2r
- A/ a a2 > 0
E ^+ - 3B/A or 5C/A , s2= a2A/ 36 B,
,
(10)
g(vx) =(av/ax)2 + I (av/ax)4
,
'S = 5C/ 3B
,
e =
m/a3
Equation (9) is derivable from the following total Lagrangian L
f
=(2e a 1
2
(atave
2
-[
A ! 2
1
av 2
(ax) +
a2 a2v 2
4 (a2) +
B (av)4 + rx 4
which, following avenue 2
,
a3
av
a33 aX
C (av)6])dx 6
ax"
can be deduced directly from the continuum
limit of the discrete Lagrangian . Equations (9)-(10) deserve the following comments . First, eqn.(9) is a nonlinear hyperbolic dispersive equation
that is very close to the Boussinesq equation for which g(vx) = av/ax; see Maugin,1985,p.91. Thus we call it a modified Boussinesq equation (MBE) Second
,
.
taking the space derivative of eqn.(9), introducing the appropriate
nondimensionalization and setting e = vx , we can write the result as
280
Maugin: Solitons in elastic solids
a2e
__
ai.
a2
a2 ax-2
ax2(
at2
CcSe) '
S_ Se
__
a
a (a
ae
ax a e
),
(12)
x
where
a\1'=2 Ke-4 e3+6 e5 ae
2 ae
,l'_'i'(e,ex,T).
(13)
Equation (12) is a one-dimensional version for pure shear motions of the second-gradient theory of elastic solids of Mindlin and Tiersten (1962), but with an energy density of the sixth order in e and quadratic in the strain gradient. Being sixth order, this may admit minimizers such as twins with two oppositely sheared configurations (martensites) and a zeroshear austenite phase of an obviously differing energy level and stability
behaviour. The quantities T and
r are intrinsic stresses of the first
and second order as introduced in the
full continuum approach using
method 3 in which the density of virtual power of internal forces would a
priori
be written as a linear continuous form on the velocities of
successive strain gradients [e.g., p(i,= -(V e + r.v & + .. ); cf (1973)]
.
Germain,
The first-order phase-transition picture of our elastic material
is complete with A or K positive and other signs specified by (13) and K written as K = K+(T-Tc)
,
K+ > 0 , where Tc is the transition tempera-
ture . When this temperature dependence is discarded, then
egns.(12)-
(13) are exactly those obtained by Falk(1982,1983a,b) in statics or in dynamics (Falk,1984,1986). However, we have here a means of evaluating all phenomenological coefficients from microscopic parameters through egns.(7'.
Accompanying boundary conditions (here limit conditions at the ends of a finite x-interval) follow from method 3 and are consistent with (12).Falk (1984) has studied the static solutions of (12) on a finite interval as well as the solitary-wave solutions in the dynamical case on the complete real line.Non-propagating "solitary-wave"solutions of the front type may be interpreted as nuclei of one phase, one martensite (or austenite), in a
matrix of the other martensite (or one martensite), while propagating kink (hump) solutions may be interpreted as domain walls. The kinks occur only at rest in the absence of external force . When driven by an external force, the domain-wall motion obeys a Rankine-Hugoniot equation (in close analogy
with shock waves) while a generalized Maxwell area construction holds in the stress(d)-strain(e) diagram (Falk,1986). Finally, the type of eqn.(9)
allows one to answer the question whether the obtained solitary waves are true solitons or not
.
Indeed,
depending on the expression of g(vx) and
Maugin: Solitons in elastic solids
281
the type of fourth-order derivative, equations of the type of (9) are known in the literature as"improved" and "modified improved" Boussinesq equations (for short, IBE and MIBE). Such equations occur, for instance, in nonlinear electric transmission lines and in the study of solitary
waves propagating along a nonlinear elastic rod of circular cross section (Soerensen et al,1984). They are only nearly integrable in the sense that: (i) they exhibit solitary-wave solutions, (ii) they possess a finite number of conservation laws,(iii) they reduce to exactly integrable equations in the small-amplitude limit soliton-like behaviour
,
,
and (iv))they numerically exhibit an almost
only little linear radiation being created during
the interaction of their solitary-wave solutions. This was shown numerically by Bogolubsky (1977) and Iskandar and Jain (1980). Equation (9) is of the same type and, therefore, its solitary-wave solutions are not true solitons, but they are not so far from such a behaviour. Obviously, any further coupling with another parameter such as an electric or a magnetic one
,
as indicated in the third column of Table l,will certainly not
improve the situation, but rather aggravate
it
.
Nonetheless, it remains
to perform the complete analytical and numerical study.
CONCLUSION The qualitative result enunciated in the foregoing paragraph and that mentioned before are tantamount to saying that the solitary waves that can be shown to exist in the three ferroic states of Table 1 for deformable crystals and that are supposed to represent domain walls in motion are never true solitons although the various cases are described by differing systems of equations. But these systems have the common feature
of
only nearly integrable. Consequently, the interaction (e.g.,
head-on collision
,
annihilation and reappearance, with a change of
phase) of the domain-wall solutions is always accompanied by linear radiations and this results from the very form of the relevant equations and the inherent physics and is not an artifact of computations.
Acknowledgments. This work was supported by Theme "Nonlinear Waves in Electromagnetic Elastic Materials of the et Mdthodes Num6riques Performantes" (M.P.B., C.N.R.S., Paris, France).
Maugin: Solitons in elastic solids
282
References
Aizu K.(1970).Possible species of ferromagnetic, ferroelectric and ferroelastic crystals. Phys.Rev., B2 754-772. Berdichevskii V. and Truskinovskii L.(1985). Energy structure of localization. In Local effects in the analysis of structures, ed. P. Ladeveze ,pp.127-158, Elsevier:Amsterdam. Paris :Hermann. Boccara N.(1976). Symetries Bogolubsky I.L.(1977). Some examples of inelastic soliton interaction. Comp.Phys.Comm.,13, 149-155. phase transitions. Falk F.(1982).Landau theory and martensitic J.Phys.Coll.C4., 43 ,C4-3 - C4.15. Falk F.(1983a).One-dimensional model of shape-memory alloys. Arch.Mech., ,
35
,
63-84.
Falk F.(1983b).Ginzburg-Landau theory and static domain walls in shapememory alloys. Zeit.Phys.B.Condensed matter,51,177-185. Falk F.(1984). Ginzburg-Landau theory and solitary waves in shape-memory alloys. Zeit.Phys.B.Condensed matter, 54, 159-167. Falk F.(1986).Driven kinks in shape-memory alloys. In Trends in applications of Pure mathematics to mechanics, ed.E.Krtner and K. Kirchgassner,pp.164-167, Berlin:Springer-Verlag. Germain P.(1973).La methode des puissances virtuelles en mecanique des milieux continus.I.Thdorie du second gradient. 12, 235-274. Iskandar I. and Jain P.C.(1980).Numerical solutions of the improved Boussinesq equation. Proc.Indian Acad.Sci.(Math.Sci.), 89, 171-181.
Klassen-Naklyudova M.V.(1964).Mechanical twinning of crystals. New York: Consultants Bureau. Magyari E.(1984).Mechanically generated solitons in nematic liquid crystals. Zeit.Phys.B.Condensed matter, 56, 1-3. Maugin G.A.(1980).The method of virtual power in continuum mechanics:Application to coupled fields. Acts Mechanics, 35, 1-70. Maugin G.A.(1985).Nonlinear electromechanical effects and applications. Singapore: World Scientific Publ., Maugin G.A.(1986).Solitons and domain structure in elastic crystals with a microstructure:mathematical aspects. In Trends in applications of pure mathematics to mechanics, ed.E.Krtsner and K.Kirchgassner pp.195-211, Berlin:Springer-Verlag. Maugin G.A.and Miled A.(1986a). Solitary waves in elastic ferromagnets. Phys.Rev., B33, 4830-4842. Maugin G.A. and Miled A.(1986b).Solitary waves in micropolar elastic crystals . Int.J.Engng.Sci., 24 (in the press). Maugin G.A. and Pouget J.(1986).Solitons in microstructured elastic materials-Physical and mechanical aspects. In Continuum models of discrete systems (6),ed.A.J.M.Spencer, Rotterdam:A.A.Balkema. Maugin G.A.and Pouget J.(1980).Electroacoustic equations for one-domain ferroelectric bodies. J.Acoust.Soc.Amer., 68, 575-587. Mindlin R.D. and Tiersten H.F.(1962). Effects of couple stresses in linear elasticity. Arch.Rat.Mech.Anal., 11, 415-448. Pouget J.(1986).Ondes solitaires dans les milieux 6lastiques In Proc.Interdisciplinary workshop on Nonlinear excitations, ed. M.Remoissenet, pp.360-375, Dijon,France:Publ.L.0,R.C.,University of Dijon. Pouget J., Askar A. and Maugin G.A.(1986".Lattice model for elastic ferroelectric crystals:Microscopic approach. Phys.Rev., B33, 6304,
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Pouget J. and Maugin G.A.(1985b).Solitons and electroacoustic interactions in ferroelectric crystals.II.Interactions between solitons and radiations. Phys.Rev., B31, 4633-4651. Pouget J.P.(1981).Les transformations ordre-desordre. In Les transformations ed.V.Gabis and M.Lagarde de phase dans lea solides francaise de et cristallopp.125-243,Paris: ,
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Soerensen M.P., Christiansen P.L. and Lomdahl P.S.(1984'l. Solitary waves on nonlinear elastic rods.I. J.Acoust.Soc.Amer., 76, 871-879. J.C. and Tol6dano P.(1986'. The Landau theory of phase transitions. Singapore: World Scientific Publ.. Wadhawan V.K.(1982'). Ferroelasticity and related properties of crystals. Phase transitions, 3 , 3-103. Yomosa S.(1985'). Solitary excitations in DNA double helices. In Dynamical problems in soliton systems, ed.S.Takeno, pp.242-247, Berlin: Springer-Verlag.
ON THE DYNAMICS OF STRUCTURAL PHASE TRANSITIONS IN SHAPE MEMORY ALLOYS M. Niezgodka Systems Research Institute, Polish Academy of Sciences, Newelska 6,
01-447 Warsaw, Poland; partially supported by the A. v. Humboldt Foundation while visiting Universitat Augsburg J. Sprekels Institut fur Mathematik, Universitat Augsburg, Memminger Str. 6, 8900 Augsburg, West Germany; partially supported by the Deutsche Forschungsgemeinschaft
1. Introduction. In this paper we consider the nonlinear system of differential equations
(1.1a)
Utt - IT, (0, e)[x - µuxxt = f(x, t),
(1.1b)
- @[TO(0, e)[t - µu2t - kOxx - ak9xxt = A(x, t),
(1.1c)
e = ux,
for (x, t) E fl x (0, T[ (where fl = (0, 1) and T > 0), together with the initial and boundary conditions (1.1d)
u(x, 0) = uo (x), ut (z, 0) = u1(z), O(x, 0) = Oo(z), z E S2,
(1.1e)
u = 0, on r x [0, T],
(1.1f)
kO = kj (Or - O), on r x [0, T].
Here, I' = {0, 1} and O = -Ox at x = 0 and O = Ox at z = 1, respectively. The positive constants k, a, k1i µ and the functions f, A, Or, uo, u1, Oo are given.
Problem (1.1) constitutes a onedimensional model for the structural phase transitions in shape memory alloys (Niezgodka & Sprekels 1985). The involved quantities have their usual physical meaning, i.e.: u - displacement, O - temperature, e - (linearized) strain, f - loads, A - heat sources; the Helmholtz free energy @(@, e) has to be specified. (1.1a) reflects the momentum balance and (1.1b) the energy balance. The mass density p is assumed constant and normalized to unity; accordingly, the equation of continuity is ignored. The terms -µuxxt in (1.1a) (and -/curt in (1.1b)) and -ak9xxt in (1.1b) indicate the presence of viscous stresses and a short thermal memory, respectively.
Niezgodka & Sprekels: Dynamics of structural phase transition
285
2. Phenomenology and Free Energy. The load-deformation (o - e) curves of shape memory alloys exhibit a strong dependence on temperature: There exists a critical temperature O, such that the following shapes are observed:
a)OO,
Or
e
Fig. 1: temperature-dependent a-s-curves Due to the form of the a-s-curves, a shape memory effect results: Loading above the yield load at 0 < O, results in a residual deformation after unloading; if the deformed material is then heated to a temperature above O,, it creeps back to its virginal state (= original shape) since this is for 0 > Oe the only possible unloaded configuration. In a series of papers (Achenbach & Muller 1984, Muller 1979, Muller & Wilmanski 1980; see also I. Muller's contribution to this volume), I. Muller and his co-workers constructed a macroscopic model, based on statistical mechanics. In particular, the hysteresis effects were ascribed to structural phase transitions between different stable configurations of the crystal lattice, highly symmetric austenite (A) and its two sheared versions, the martensitic twins (denoted M+ and M_). The basic variable in this approach is the Helmholtz free energy IQ = 'B(O, e), assumed to be a function of temperature and macroscopic strain alone. The o-E-curves resulting from the theory have the form
a) O < O,
c)O>> O,
b) O > O,
e
Fig. 2
e
Niezgodka & Sprekels: Dynamics of structural phase transition
286
The branches corresponding to W (e, e) < 0 are unstable. Thus the system follows the dotted lines and a hysteresis is observed. In our initial-boundary value problem (1.1), the free energy function 'Q(49, e) has still to be specified. The simplest polynomial expression such that IFE matches the behaviour depicted in Fig. 2, is the LANDAU-DEVONSHIRE-form, valid
for small values of lei and 1e - e,1: (2.1)
W(e, c) = to(e) + !cl (e - ec)62 - ,C2E4 + /C366.
Here ic, > 0, i = 1, 2, 3, are appropriately chosen. In the sequel we work with the general expression W(e, c), tacitly assuming that, for lei and 1e-e,1 small, W is given
by (2.1). For the mathematical treatment it suffices to postulate some asymptotic conditions:
(H1)
T E C3(R x R);
(H2)
W®(e, e) = 0 , on (-oo, 0] x R;
(H3)
- OWee (e, e) > 0, on R x R;
(H4)
(1 + O2)IWee(e, E)I < c1 + c2161 , on R x R;
(H5)
IW (8,6)1 :5 C3, on R x R;
(H6)
R6 (91 6)I C C4 + C5161,
(H7)
on R x R; :5 c6 + c7 lei,
for all e c R.
Here c;, i = 1,
, 7, are fixed positive constants. In Niezgodka & Sprekels (1985) and NiezgSdka et. al. (1986) the hypotheses (H1) - (H7) have been formulated individually for the components W; of IQ in the two cases 'k (0, 6) = To (e) + W 1(e)62 + W2 (6), and
IQ (e, e) = 'I`o(e) + IF1 (e)e2 + 12 (e, 6),
respectively. For the case (2.2), typical admissible forms for To, W1i tiIr2 are depicted
in Fig. 3. In particular, we may choose Wo such that -'Yo(e) ti e4 for a --e 0+
and -To(8) - e, for a -' +oo, respectively; this reflects the DEBYE-law for the temperature dependence of the specific heat.
287
Niezgodka & Sprekels: Dynamics of structural phase transition
0
0
Fig. 3.: Admissible forms of TO, 'yt, '@2
3. Existence and Uniqueness. For t > 0 we use the notations: Sgt =Sax (0,t), II' II =norm in L2(f2), 1119(t)11 2 = 02(0, t) + 02(1, t), Yt = L2 (0, t; H' (0)), Zt = L2(0, t; L2 (fl)). We define: Definition: A pair (u, O) is called a "weak solution" of (1.1) on fix [0, T] for T > 0, if the following is satisfied: (3.1)
u E H2(0,T;L2(0)) n H'(0, T; H' (0) n H2(12));
(3.2)
0EH'(O,T;L2(0))nH'(0,T;L2(I'))nL2(O,T;H'(0)); Oo(x),u(x,0) = uo(x),ut(x,0) = ul(x), a. e. on 12;
(3.3) (3.4)
(i)
(ii)
with e = uy there hold
JnT
f[utt - ( E(O, E))2 - µuyxt - f)pdxdt = 0,
fT f[_O(We(8ie))t S +Put(uxxtc+uxtSy)+
for all n E ZT,
Niezgodka & Sprekels: Dynamics of structural phase transition
+ k0 +
288
+ akOxtcx + ) ]dxdt
f f[ki(0 - Or) + akl (Or - Or,t )]Sdxdt = 0,
for all SEYT. There holds:
Theorem Let in addition to (H1) - (H7) the following assumptions on the data of (1.1) be satisfied:
(H8)
uo EHI (f') n H2(f) , U1 Ell' (f2);
(H9)
Oo E HI(fl), Oo(x) > 0 on fl;
(H10)
A E L2(0, T; L2(f))) , for any T > 0, A(x, t) > 0, a. e.;
(H11)
f E L2(0, T; L2(f2)), for any T > 0;
(H12)
OrEHI(0,T), for any T>0 Or(t) > 0 and Or,t(t) > 0, a. e.
Then problem (1.1) has a global unique weak solution (u, O) such that (3.5)
0, e E C(fl x [0, oo)); O(x, t) > 0, on fl x [0, oo).
Proof: See Niezgbdka & Sprekels (1985) and Niezgbdka et. al. (1986) for the existence; the uniqueness proof is analogous to that in Hoffmann & Songmu (1986)11
Remarks: 1. According to (3.5), e does not jump across the phase transition front (which should be the case for the order parameter of a first order phase transition). This stems from the smoothing effects of viscosity and thermal memory and from the
one-dimensionality.
2. Clearly O(x, t) > 0 on fl x [0, T], for some T > 0. 3. If the data are smooth, (u, O) is a classical solution.
4. Numerical Simulations. We report about simulations carried out in Alt et. al. (1985) for the case that µ = 0,
a = 0, f - 0, A - 0, fZ = [-1,+1] and (1.1e, f) replaced by a = IIe = or(t), 0 = Or(t), on the boundary. As potential we chose (4.1)
IF(0,e) =e2+ 2(5-02)(1+
e2)-I/2,
Niezgodka & Sprekels: Dynamics of structural phase transition
289
which exhibits the behaviour depicted in Fig. 1 with O, = v. Moreover, uo(x) = sox (eo > 0), ul - 0, Oo =_ 1; i.e., at t = 0 the material is completely in the M+-phase.
We only display a case where M+ -. M_ and M± --> A transitions occured. The experiment extended over 800 time steps during which ar, Or had the form:
800
t
Or 800
t
Fig. 4
Figs. 5 - 28 show the evolution of e, O, u, where each picture shows 500 time steps. Since if exceeds the yield limit, in Figs. 5 - 8 we see stress-induced M+ --> M_ transitions, accompanied by temperature outbursts due to released latent heat (Figs. 9 - 12) and the formation of sharp interfaces between M+ and M_ (Figs. 13 - 16) along which s is discontinuous. Upon unloading at t = 150, a superposition effect induces another transition nucleating from the center at x = 0. After the stabilization at t = 400, a load below the yield limit is applied, together with a sharp increase of Or. This results (see Figs. 17 - 28) in Mf -, A transitions, until at t = 800 a configuration is reached which corresponds to a state close to the origin in the o-E-plane; i.e., the material is a stressed austenite, and we observe shape memory.
Niezgodka & Sprekels: Dynamics of structural phase transition
290
7
DEFORMATION
Fi
DEFORMATION
BOUNDARY DATA:
8= 1.00
,
o= 0.00
;
tE
6
291
Niezgodka & Sprekels: Dynamics of structural phase transition
F
DEFORMATION
1.0
.0.0
-1.0
4 0
Kn
8=
BOUNDARY DATA :
,
o= FIT-;
tE
200,
FFi"
DEFORMATION
1.0
.0
III
I II III I III161111 .
III i III I I I I I'' i I I II I I IIII! I!
.0
lT
iE
son eon
e= i
nn
.
o= a. ni
tE
]0. 400
292
Niezgodka & Sprekels: Dynamics of structural phase transition
Fig. 9
TEMPERATURE
BOUNDARY DATA:
TEMPERATURE
8= 1.00
,
a =
0.00
;
tE
0,
1001
Fig. 10
293
Niezgodka & Sprekels: Dynamics of structural phase transition
Fig. 11
TEMPERATURE
8= 1.00
a=
,
tE
;
3001
,
Fig. 12
TEMPERATURE
A= 1.00
,
a= 0. 00
;
tE
,
4001
294
Niezgodka & Sprekels: Dynamics of structural phase transition
F_13
DISPLACEMENT
UNDARY -DATA
e
{
,
o=
tE
Fig 14
DISPLACEMENT
DUUNDARY DATA:
e= 1.00
,
o= U-UU
;
tE
,
Niezgodka & Sprekels: Dynamics of structural phase transition
Fig. 15
DISPLACEMENT
BUUNDARY DATA:
8=
295
,
as
U. UO
;
tE
,
Fig. 16
DISPLACEMENT
296
Niezgodka & Sprekels: Dynamics of structural phase transition
Fig. 17
DEFORMATION
600
UOUNDWRY-DM
tE
e
Fig. 18
DEFORMATION
BOUNDARY DATA :
[ 400, 500]
6=
1.80
,
o= 0.10
;
tE
,
6091
Niezgodka & Sprekels: Dynamics of structural phase transition
297
Fig. 19
DEFORMATION
Fig. 20
DEFORMATION
8= 1.80
,
o= 0.10
;
tE
,
Niezgodka & Sprekels: Dynamics of structural phase transition
TEMPERATURE
TES p ATURE
298
ELI- 21
Fig_. 22
Niezgodka & Sprekels: Dynamics of structural phase transition
Fig. 23
TEMPERATURE
Fig. 24
TEMPERATURE
BOUNDARY DATA :
299
8=
1. 80
,
a= 0. 10
;
tE
700,
300
Niezgodka & Sprekels: Dynamics of structural phase transition
Fig. 25
DISPLACEMENT
BOUNDARY DATA: e=
DISPLACEMENT
,
Q= 0.10
;
tE
400,
5001
Fig 26
301
Niezgodka & Sprekels: Dynamics of structural phase transition
Fig. 27
DISPLACEMENT
9= 1.80
DISPLACEMENT
,
o= 0.10
;
tE
,
Fig. 28
Niezgodka & Sprekels: Dynamics of structural phase transition
302
References: Achenbach, M. & Muller, I. (1984). Shape memory as a thermally activated process. Berlin: Hermann-Fottinger-Institut, TU Berlin, Preprint.
Alt, H. W., Hoffmann, K.-H., Niezg6dka, M. & Sprekels, J. (1985). A numerical study of structural phase transitions in shape memory alloys. Augsburg: Institut fur Mathematik, Universitat Augsburg, Preprint No. 90. Hoffmann, K.-H. & Songmu, Z. (1986). Uniqueness for nonlinear coupled equations arising from alloy mechanism. West Lafayette, Indiana: Center for Applied Mathematics, Purdue University, Techn. Report No. 14. Muller, I. (1979). A model for a body with shape-memory. Arch. Rat. Mech. Anal. 70, 61 - 77.
Muller, I. & Wilrnanski, K. (1980). A model for phase transition in pseudoelastic bodies. Il Nuovo Cimento 57B, 283 - 318. Niezgodka, M. & Sprekels, J. (1985). Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys. To appear in: Math. Meth. in the Appl. Sci. Niezgodka, M., Songmu, Z. & Sprekels, J. (1986). Global solutions to a model of structural phase transitions in shape memory alloys. To appear in: J. Math. Anal. Appl.
ON THE HOMOGENIZATION OF SOME FREE BOUNDARY PROBLEMS Jose-Francisco Rodrigues University of Lisbon and C.M.A.F. Av. Prof.Gama Pinto, 2. 1699 LISBOA CODEX (PORTUGAL)
1. Introduction
In this article we present some results and open problems related to the stability, with respect to the homogenization, of the coincidence sets and free boundaries associated with elliptic and parabolic variational inequalities. They can be considered as partial contributions to an interes ting and deep problem posed by J.L.Lions several years ago (see [L1,2]).
For each parameter e >0, we consider the Dirichlet unilateral problem (1.1)
u '> 0 , LEuE >f e and u£ (LEuE-fE) =0 uE = g E
on
a. e. in Q
E,
where, in the elliptic case, we denote (if v=v(x), x eS2, open and bounded)
(1.3)
LEv = -
and
x
Q =S2cRn, E = 8S2 Lipschitz;
i
and, in the parabolic case, (if v=v(x,t)),
(1.4)
)x J
and Q=S2 x]0,T[cRn+1, E=2Qx]O,T[US2x{O}. 1
In both cases, we assume the coefficients aid=ai(x),
x 6 S2, satisfy (1.5) a1J
VE ERn
and Jac
I
<M,
a.e.
x e02,
where M >a >0 are constants independent of e>0. Under appropriate conditions on the data fE and gE (gC >0), see [KS] or [BL] for instance, it is well known that
Rodrigues: Homogenization of some free boundary problems
304
the variational inequality (1.1)(1.2) admits a unique continuous solution ue(u6=ue(x) or uE=ue(x,t), respectively in the el liptic or parabolic cases). In general, the domain Q is "a pos teriori"divided into two parts: an open (in Q) subset, the con tinuation set, where {uE
(1.6)
a.e. in Ae =
LEuE = f£
>0},
and its complement in Q, the coincidence set, separated in Q by the so-called free boundary denoted, respectively, by
(1. 7)
IE
-
{ue=0}
(closed) and
(E
- ale Il Q, (c > 0) .
The problem (1.1)(1.2) is well-behaved for the homoge nization of the coefficients, that is, if the matrices {aid}
H-converge to la?.) as E--0 (see [L1] , [BLP] or [T]) and fe->fo, J
eo
g ;g
under appropriate conditions, one has uE--u°
.
It is then
natural to discuss in what sense and under which conditions one can expect IE->I°*and
Pe,(o
In the next section we give general conditions in order to obtain the convergence of the coincidence sets in Hausdorff distance (see [CR1] and [R1]) and we discuss its applica-
bility to the homogenization. In the Section 3 a different inde pendent convergence is given in terms of their characteristic
functions. It improves previous results of [M2] [(R2] and [Rl]. In the following section we apply the preceding results to a ty pical free boundary model, the one-phase Stefan problem. Finally one concludes with some related open questions, namely in the dam
problem and in the two-phases Stefan problem. 2. Convergence in Hausdorff Distance
Recall the Hausdorff distance defined by
h(I,J) =inf {6 >0 : IcB6(J)
(2.1)
and
J cBd(I)},
where B6(I)= U {y:lx-yl O) to (1.1)(1.2) satisfies fE 0 and ri= 0. Now (4.12) follows immediately by remarking that from (4.10) one has un+l
W
JT
Io)
r
IX e Xiojdtdx = J
S2 0
I
I
ISe-S°ldx
U
.
1S
REMARK 6. This result was essentially established in
[R1] for the special case of periodic homogenization and it gi ves an answer to a problem posed in [L1]. More references to the
one-phase Stefan problem can be found in the survey article
[R3]. n REMARK 7. In the case of one-dimensional space varia ble, if Q=]a,i$[
,
under appropriate assumptions (see [KS] and
[R1]) one can show that the free boundary (PE is a Lipschitz cur
ve in the coordinates obtained from (x,t) by rotating through 'R/4. Then the sets IEsatisfy an exterior cone property uniformly
in E, which together with 1c-10 (Hausdorff or in measure) implies the uniform convergence of the free boundary. 0 REMARK 8. In higher dimensions, under certain conditions
and assuming aid independent of one direction, for instance x1, it can be shown than the free boundary ((t)=8I£(t) fl 2 admits a
local representation (P E(t)
: xl =E(x'
,t), t >0, x' = (x2,...,xn),
(E >0).
In this case, as in Theorem S one also obtains the convergence of $E(t) to 0° (t) in L. 1 This holds easily for the elliptic case
Rodrigues: Homogenization of some free boundary problems
312
n=O and it can be extended to the parabolic one n>O by refining the conclusion (3.12) in the form pn(IE(t) :I°(t))+0, for fixed
t >0 (see [Rl]). A similar application was done in [CR2] for the variational inequality approach to the dam problem with horizon tal layers. 11 5. Related Open Problems
Without being complete let us mention the following PROBLEM 1.
In [L2], J.L.Lions reported some numerical
experiments of Bourgat (Laboria/INRIA, France) on the approxima tion of the solution and the free boundary of an elliptic obsta cle problem in composite media by using homogenization techniques, which suggested a good convergence, and he has raised the problem of not only the convergence of the free boundaries D but also if, in some appropriate sense, it could be estimated in terms of the parameter e. This problem seems to be still open. 0 PROBLEM 2. Analogously, the same question has been for
mulated for the boundary obstacle problem, or Signorini problem, which consists of (5.1)
(5.2)
in
LeuC= fE
uE >0,
3uE/3VE >g6,
S2
u6(auE/avE-gE) =0 on 3c2
where 313vE =aijni3/axj denotes the conormal derivative. In the elliptic case, as well as in the corresponding parabolic one,
there remains open the question of the convergence of the boundary coincidence sets
Je } Jo
,
where
J. = {x E 3c2 : ue (x) = 0) .
I1
PROBLEM 3. The similar problem to (1.1)(1.2) for higher
order elliptic (or parabolic) operators can also be posed. For instance, if (5.3)
Lev = (aijkivx x x x k Q i j
,
(i,j,k,Q =1,2)
the unilateral problem (1.1) with appropriate boundary conditions,
corresponds to a nonhomogeneous plate constrained against a pla ne. The convergence of the contact region
{u6=O}
with respect
to homogenization is an open problem.LI PROBLEM 4. In the already mentioned steady dam problem
Rodrigues: Homogenization of some free boundary problems
313
with a rectangular geometry and a two dimensional permeability in the special isotropic case aid=aE6ij with aC(xl,x2) _ E
E
al(xl)a2(x2), the problem for the pressure pE=pc(xl,x2) is transformed into a variational inequality of the form (1.1)(1.2)
for the Baiocchi's transformation uE (xl,x2)=Jb a2(p)pE(x1,ri)drl. The y
corresponding homogenization can be found in [CR2] {pE
ponds to the physical situation whenever
and it corres
> 0} = {uE > O}. However {pE>0}
in the case of horizontal layers it may happen that
{u,>0}
and we must use a direct and more general formulation due to
Alt (see [A] or [R2]). In general, if 30=P1 U r2 U r3 , the hydros tatic pressure is given by h >0 on r3 and h=0 on r2, e=(0,1) de notes the vertical direction and aE the matrix {aid}, the problem consists of:
"Find a pair (pE,YE) cEl(S2) xL_(2), such that, (5.4) (5.5)
pE >0
0<X {PE:
-
in
S2
,
pE = h E
> 0}
-< Y - 0,
(5.6)
on
a.e. V
r2UT3'
in
0
,
c H1(Q)
:
fQ
CIr V2)". Those d'Etat, Univ.Paris VI, 1976.
[M2]
F.MURAT - Private communication, Paris (1980).
[R1]
J.F.RODRIGUES - "Free boundary convergence in the homogenization o the one phase Stefan problem", Trans.Amer. Math.Soc.274(1982), 297-305.
[R2]
J.F.RODRIGUES - "Some remarks on the homogenization of the dam problem", manuscripta math. 46(1984), 65-82.
[R3]
J.F.RODRIGUES - "The variational inequality approach to the one phase Stefan problem" Acta Appl.Math. (1986).
[T]
L.TARTAR - "Cours Peccot,
College de France, Paris, 1977 (see also these Proceedings).
THE POINT INTERACTION APPROXIMATION, VISCOUS FLOW THROUGH POROUS MEDIA, AND RELATED TOPICS J. Rubinstein Department of Mathematics Stanford University Stanford, CA 94305
I.
INTRODUCTION TO THE P.I.A.
The Point Interaction Approximation is a technique to analyse boundary value problems in domains with many tiny obstacles.
It was
introduced by Foldy [4] for the problem of multiple scattering of waves. The general framework is as follows: and cut out of it
N
We take a smooth domain
B1, B2, BN.
holes
D
R3
in
In the domain of interest
N
DN=D - UBi
(1.1)
1
we consider the prob l em
If G8
L[uN] = f
in
M[uN] = 0
on
9Bi
MO[uN] = 0
on
aD
DN
(1.2)
is the Green's function of (1.2), we can write uN =
GN[f].
The essence of the P.I.A. is to approximate N
GN(x.y)
qY G(x.wi)
= G(x,y) + S
=
HN(x,y)
(1.3)
1
where
G
is Green's function of
D
appropriately chosen "charges", and
(with respect to wi
L),
qi
are
are the centers of the holes.
In
fact Foldy assumed (1.3) as a good representation of the medium, and proceeded to look for smooth (macroscopic, effective) approximation for it. There are 3 questions one has to answer while applying (1.3): (i)
(ii)
(iii)
Evaluate Prove
from the microstructure.
qi nGN
-
(1.4)
HNn < < 1
Show that there is a smooth operator nJ-HNn > 1.
Note that we have as yet left the norms
Rigorous results in the sprit of (1.4) were recently
obtained by Figari, Orlandi and Teta [2] for the Laplace equation, using estimates derived by Ozawa [6].
Although we formulated the P.I.A.
for
linear operators, it can be used successfully for nonlinear problems as well.
Caflisch et. al [1] applied it (formally) to study wave propagation
in bubbly liquids, and the problem of diffusion in regions with many
melting holes was recently studied by Figari, Papanicolaou and Rubinstein [3].
In the next section we apply the P.I.A. to Stokes equations and
derive rigorously the equations associated with Brinkman and Darcy. Detailed proofs are given in [7] and [8]. discussed at section III.
Some related problems are
Rubinstein: The point interaction approximation
318
APPLICATION TO FLOW IN POROUS MEDIA
II.
Using the notation of the preceding section, we let stokes operator in
DN, UN
the local velocity field and
be the
L
M[f] = f, i.e.
= VP + f vuN-O
ub1N
auN
= 0
uN
Here
is the viscosity and
u
R =
Each hole has radius
this corresponds to
R
and on
D.
(In the context of homogenization theory,
.
where
c
9B
is a nonnegative parameter.
x
0
on
(2.1)
is the size of the cell).
a
G(x,y)
is as before the Green's function (in fact it is called here Oseen's tensor)
0, and we let
in
wise.
t
be the 3Nx3N matrix which is composed of
G, s.t. the (i,j) block is
blocks of
We also denote by
G(wi, wj)
for
NxN
ij and zero other-
the matrix G(x,W1)
We write (1.3) in matrix form:
1 N(x,Y) = G(X,Y) where
a
T( ,Y)(. * +
I)-1
(x,
),
a = 6truaO .
Geometrical assumptions: 1)
2)
are bounded away from
{wN}
min
CN-1+v
IwN -
,
2D (uniformly in
N).
v < 1/3
i *j 3)
N-2
- wJ I-3+g < c
0.
i *j (Note that assumption (2) guarantees that there is no overlapping). Theorem 1 [7] Let
f c CO(D)
e > 0, there is
,
{WN}
NO
satisfy
s.t.
(1) - (3), and
for every
N > NO
a > 0.
Then for every
(2.2)
319
Rubinstein: The point interaction approximation 1
n(HN -
GN)(f)o2 z c N
6.
nNn < 1
Now, under the conditions of Theorem 1 we can show that (matrix norm), and hence we can expand {wN}
(R t + I)-1
in (2.2).
We let
be i.i.d. random variables with probability density function
p(x).
Then we can analyse the statistics of (2.2) and prove
Theorem 2 [7] Let
p(x) e C(D), f
and
x
P N {N°nv - uNa2 < e} = 1
lim
V e > 0,
° c
.
Then
as in Theorem 1.
P is the probability measure induced by
space of configurations and
p(x)
uev - av - 62rpas p(x)v, = vp + f
(2.3)
v v=0 in v=0 where (2.3) is Brinkman's equation.
on the
~
solves
v
D
on
aD,
It holds in the capacity scaling, i.e.,
for very dilute systems - the porosity is
- 1 - 0(N-2).
It is well
known that flow in dense materials is macroscopically described by Darcy's equation.
While the P.I.A. surely breaks down at high concentrations, it
still holds slightly "above" the capacity scaling: Let by
R = ao N-1+y, y e (0,1/60), and set
a = Nyr.
We denote
U(x,r) the solution of Darcy's equation
r U + 6,rua0p U = -VP - f .
(2.4)
V U=0 U
.
n=0
in on
aD
,
and expand uN = N-y uN,0 + N-2y uN,1 + ...
.
0
320
Rubinstein: The point interaction approximation
Then [8] lim
PN {nuN,O - UNQ < e} = 1
N+. for every
e > 0, where Q = {v e
L2(D);
v
v = 0,
v
(See also the related work of Sanchez-Palencia [10])
n1aD = 01
(2.6)
Rubinstein: The point interaction approximation
321
LOW ORDER CONNECTIONS
III.
There is a large body of work on this difficult problem, and all the results here so far are formal.
Hoping to get better
understanding of the origin for such corrections and rigorous estimates,
we start at the level of the
we suggest to consider a simpler problem:
P.I.A. and continue to the averaged equation.
know only the distribution density of
To first order we needed to
{wj}.
It is natural to conjecture
that the 2 point correlation function will be needed to evaluate further We proceed formally:
terms.
Let
he distributed uniformly in
(wj}
Pr (w. e dx) =
V = vol
dx V
Pr (wj a rj + dxI wi = ri) =
We assume scale
respect to the configurations sphere centre at function
x.
and
C
F(x) decays over a length
will denote averaging with x
(i.e. configurations
D (w.r. to
of
(3.1)
(D)
V [1 - F(rj - ri)ldx
F(x) = F(Ixl), and that
t (the correlation length).
0
means averaging given a Cx).
Let
ON
o), and consider in
G
be the Green
the microscopic
problem:
uN
=0
,
Au = - f in
DN
for Ix - wjl =
d
)f (x,' )dx' - Pj
Ix,-Wj I-d
x
,
e
aD
d x,
f
(3.5)
and condition (3.3) to get w(y) = y G(s) + n f G(y-x')F(y-x') x,dx'
Since we assumed that
decays fast we get
F
f G(y-x')F(y-x')dx']
y ~ [G(6)]-1
and thus we close (3.4) (3.6)
1-bet2 where we scale as usual
e = G 6), and
b = R 2
r G(x)F(x)dx
(b = 0(1), e = 0(1)).
This is the "random analog" of the Hasimoto result for periodic structures. (Hasimoto [5], Saffman [9]).
If we set
R = 0( dq), then the correction is obstacles.
0(eq) where
s
is the volume fraction of the
Rubinstein: The point interaction approximation
We stated precisely the approximations we used. (recall that we start at the P.I.A. level) is (3.5).
323
The most crucial one Since this is exact
for periodic structures it seems natural to look first for a rigorous justification of (3.6) in that case.
Acknowledgement
This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the NSF and the ARO.
Rubinstein: The point interaction approximation
IV.
324
REFERENCES
[1]
R.E. Caflisch, M.J. Mikisis, G.C. Papanicolaou and L. Ting, J. Fluid Mech 153 (1985) 259-273.
[2]
R. Figari, E. Orlandi and S. Teta, J. Stat. Phys. 41 (1985), 465.
[3]
R. Figari, G. Papanicolaou and J. Rubinstein (In preparation)
[4]
L.L. Foldy, Phys. Rev. 67 (1945), 107.
[5]
H. Hasimoto, J. Fluid Mech., 5 (1959), 317.
[6]
S. Ozawa, Comm. Math. Phys., 91 (1983), 473.
[7]
J. Rubinstein, J. Stat. Phys., 1986, (to appear).
[8]
J. Rubinstein, Proceedings of the IMA Workshop on Infinite particle systems and hydrodynamics limits, Springer-Verlag (to appear).
[9]
P.G. Saffman, Stud. Appl. Math., 52 (1973), 115.
[10] E. Sanchez-Palencia, Int. J. Eng. Sci. 20 (1982), 1291.
THE VANISHING VISCOSITY-CAPILLARITY APPROACH TO THE RIEMANN PROBLEM FOR A VAN DER WAALS FLUID
M. Slemrod* Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, New York 12180-3590
0.
INTRODUCTION
In this note we sketch a proof for the solvability of the Riemann problem for the isothermal motion of a van der Waals fluid.
The problem is of physical interest in that it simulates the behavior in a shock tube of a compressible fluid which may exhibit two different phases, say liquid and vapor.
The main tools of the analysis are
(1) Dafermos's [1] formulation of the vanishing viscosity method for the Riemann problem and (ii) resolution of the viscous system via bifurcation ideas suggested in the paper of Hale [2].
Rigorous proofs will
appear elsewhere. 1.
PROBLEM FORMATION We wish to solve the Riemann initial value problem for the
isothermal motion of a van der Waals fluid.
We denote by u the velocity,
w the specific volume, p the pressure, x the Lagrangian mass variable, and t the time.
The equations of one dimensional inviscid motion are
then
ut + p(w)x = 0
wt - ux=0 In addition we prescribe Riemann initial data uR
w(x,O) =
u(x,0) = U
+
R
wR + wR
x0
+
Here UR, uR, wR, wR are constants.
This research was sponsored in part by the Air Force Office of Scientific Research, USAF, Contract/Grant No. AFOSR-85-6239. The U. S. Government's right to retain a nonexclusive royalty free license in and to copyright this paper for government purposes is acknowledged.
326
Slemrod: The vanishing viscosity-capillarity approach
The graph of p is given in Figure 1*. It resembles a typical van der Waals isotherm when the temperature is below critical. case p'
In this
> 0 on (a,a) when (1.1) is elliptic and p' < 0 on (b,a), (a,o.)
when (1.1) is hyperbolic.
We denote the two values A,B as the two
values of specific volume that yield the Maxwell equal area rule B
f (p(s) - p(A))ds = 0 A
(1.3)
We then choose our Riemann data as
uR=uu
wR=A+uw ,
,
(1.4)
uR= uu where u
,
wR=B+uw
,
u+, w , w+ are constants and u > 0 is a small parameter.
We now sketch a proof which shows that this two phase Riemann problem has a solution for small u > 0 which is obtained via viscosity-capillarity limits. In order to resolve (1.1), (1.4) we imbed (1.1) in the "viscous" system (e > 0) ut + p(w)x = CO tuxx
,
(1.5)
wt - Ux = ell twxx The terms on the right hand side simultaneously attempt to capture the effects of viscosity and capillarity (see [31) and preserve the structure of solutions of the Riemann problem in terms of the independent variable
= x. t
The program is to show that for every fixed e > 0 (1.5) admits a solution uE (c1, wE (r;) satisfying the ordinary differential equations -CU, + p(w)' = eu U", (1.6)
- w' - u' = eu w", and boundary conditions
u(-m) = uR, w(-°) = wR,
u(+c.)
=
uR, w(+m) = wR
.
(1.7)
If, in addition, we can show uE (c), wE (E) have total variation bounded independently of E > 0 it will follow from Helly's theorem that WE (c),
*See page 335
Slemrod: The vanishing viscosity-capillarity approach
w
327
possess a subsequence which converges boundedly a.e. as E + 0+ to
functions
which are solutions to the Riemann problem, (1.1),
(1.4). 2.
PRELIMINARY ANALYSIS It is convenient to introduce the new independent variable
where a is constant yet to be determined.
In this case (1.6) becomes
-u(e+a)u' + p(w)' = cu",
(2.1)
-u(e+a)w' - U' = cW", and
'
denotes differentiation with respect to s.
identical boundary conditions in e
conditions in E go into u(--) = uR,
As p > 0 the boundary
u(-oo) = wR,
u(+-) = uR,
w(+o") = wR .
(1.2)
When u = 0 (2.1) becomes p(w)'
= cu" (2.3)
-u
ew
which may exhibit both homoclinic and heteroclinic orbits.
In particular
(2.3) exhibits the classical van der Waals solution wm0)(e)
u(0)(6),
which is the heteroclinic connection between the equilibrium points
u=0, w=A
and
u=0, w=B
of (2.3) i.e., um0)(0), wm0)(e) satisfies (2.3) and boundary conditions
(0) um
(0)
0, wm
A,
(0) um
(0)
(+o) = 0, wm
(+001 = B.
(2.4)
On the other hand (2.3) exhibits other non-oscillating solutions, namely homoclinic orbits.
We denote the homoclinic orbit which
connects the point
u=0, w=A+lv to itself by um(E;u),
Again, this means that um(g;u), wm(E;P)
satisfies (2.3) and boundary conditions
Slemrod: The vanishing viscosity-capillarity approach
328
0
(2.5) wm(-°^,ii) = A + uv Here v is as yet undetermined scalar.
(Since v = 0 is a possibility, we
don't specify conditions at e = +-. Of course if v = 0, um(+co;u) = 0, wm(+m;u) = B while if v * 0, um(+°,u) = 0, wm(+oo;u) = A + uv.) 3.
THE FINITE DOMAIN PROBLEM We consider (2.1) on the finite domain (-L,L) with boundary
conditions
um0)(M
u(L) =
w(L) =
) + uu+
w(0)(L)
u(-L) =
w(-L) =
+ uw+ um0)(-L)
u(0)(-L)
+ uu_
+ pw
We shall show that (2.1), (3.1) can be solved for u(6;L), w(6,L) which are bounded independently of L.
In this case the functions
u(9;L), w(9;L) can be extended to the infinite domain
by setting
u(e;L) = u(0)(L) + uu+
w(e;L) =
for E > L
wm0)(L)
;
+ uw+ (3.2)
u(e;L) =
u(0)(-L)
+ uu
for :;