Bemelmans J. et. al. (editor) Elliptic and Parabolic Problems: Rolduc and Gaeta 2001
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60
Theorem 3.1 For any £
0
> 0, r > 0 there exists a constant C independent of £ such that
Q
\
u
i - •"?|i,(o,T)xn
(o
< j
T
- (3.14)
In particular u\
—
u\
—>
0 in L
1
((0,T) x Q^,).
Before to give the proof of Theorem 3.1 let us introduce some lemmas.
Lemma 3.1 Let i e {1,2} and u\ be a solution to (3.12). Then there exists C = C(X,p)
a constant independent of
£
such that
l
u
<lt~(o,r;i
2
(n,))"
/o
+ 1 ||V<4||^
(
dt<C(^ + l). (3.15)
Proof of Lemma 3.1. We just consider v = u\ in (3.12). We derive
2^Kl2,n, + J A{t,x,u\)Vu\Vu\dx =
{ft,u\).
Using (3.8) it follows that
\j
t
\4\lsi
t
+
A\Vu\\\
2
2nt
<
|//|.||V«j||
2in/
.
(3.16)
The rest of the proof is then absolutely identical to the proof of Lemma 2.2. We leave the
details to the reader. This completes the proof of the Lemma.
•
We turn now to the
Proof of Theorem 3.1. Since w is a positive function, by (3.9) for any e > 0 there exists a
S =
<5
e
< e such that
r-£r
=
l-
(3-17)
We define then F
€
by
'0 if
z<S
e
f
z
ds
/ -^j-^ H6
€
<z<e, (3.18)
Js,
w
2
(s
s
c
w
J
(s)
1 if
e
< z
It is clear that F
e
defined this way is a piecewise (^-function such that F'
t
the derivative
of F
e
, belongs to L°°(R). It follows that if we denote by p the function introduced in
(2.24)-(2.26), for l
x
£ (0,£) it holds that
F
eP
2
= F
t
(u\ - ul)p
2
(£) G Hl$l
t
). (3.19)
61
Taking v =
F
e
p
2
as test-function in (3.12) and performing the difference of the equations,
we get
(
d
-{u\
- uj), {u\ - uj)p
2
} + / A(t,x,u
l
e
)X>ulV{F
e
p
2
}dx-
/ A{t,x,u
2
)Vu}V{F
e
p
2
}dx = 0. (3.20)
Jilt
liii
Let us set
H
e
(z)= ['F
c
(t)d£.
Jo
For smooth functions it holds that
d
/ H
e
(u\ - u\)p
2
dx = (^(«J - uj)p, F
c
(u\ - u
2
t
)pj.
dt ./n
t
By density this holds also for
Up 5 11/) <1S
above. For simplicity we will denote Aij(t,x,u\)
resp.
Aij(t,x,uj) by Aij(uj) resp.
A
it
j(uj).
Then, from (3.20) we derive
H
€
(u\
- uj)p
2
dx+
i fit
-I
/ A(u))VuJVp
2
•
F
c
dx - / A{uf)Vu
2
t
Vp
2
•
F
e
dx =
Jilt Jfic
- J A(u\)VulVF
e
-p
2
dx+ [ A(u])Vu
2
VF
f
_-
p
2
dx.
(3.21)
Jn
t
Jn
t
/ n, J
n
t
Since p
2
is independent of Xi we obtain, setting Vjr, = (dxi
E--=j
t
f H
t
(u\ - u
2
)p
2
dx+
/ Ai{u})VulV
Xl
P
2
-F
e
dx- / AxiuDVufoxtp
2
•
F
e
dx =
Jiii Jiii
- I A{u\)V{u\-u\)WF
(
-p
2
dx- I (A{u\) - A(u|))V^VF
e
•
p
2
dx.
(3.22)
in, Jilt
Using (3.8), (3.10) we derive easily
E<-\
[ \V{u} - u\)\
2
F'
t
p
2
dx
Jiii
+ C f uj{u\-u
2
t
)\Vu
2
\F[\^(u\-u
2
)\p
2
dx. (3.23)
J [6<u\-u
2
t
<e]
{[8 < u\ - u
2
< e] denotes the set {x G Q
t
\ S < (u\ - u
2
)(x) < e} where F[ = ^ •)
Therefore we have
E < -A f \V{u\ - u
2
t
)\
2
F^p
2
dx + C [ |V«?|v^|V(uJ - u
2
)\p
2
dx. (3.24)
Jn
t
Jiii
62
Using in the last integral the Young inequality
a
^
ffl2 +
£
62
'
we get, since in the above integral we in fact integrate on
[5
< u\
—
u
2
< e].
E<-\
[ |V(i4 - u
2
)\
2
F'
e
p
2
dx + ^ [ \V(ul - u
2
)\
2
F'
e
p
2
dx
Jn,
2 J
n
Jn,
J
n
c
C
2
f \V4\
2
P
2
dx
ZX
J[S<u\~u\<t}
<
C
-1
2-^
•l[S<u\-u}<t]
- f \Vu]\
2
p
2
dx. (3.25)
J[6<u\-v?,<t\
Going back to (3.22) and integrating in t we obtain
/ H
e
(u\-u
2
t
)p
2
dx+ I I A(i4)VuJV
Xl
p
2
-F
e
dxdt
Jn,
Jo Jn,
/ / AxiuDVufoxiP
2
-F
(
dxdt <
Jo Jn,
lo Jn,
C
2
f*
If
2A J
0
J\s
<u
\-v?
t
<i}
\Vu
2
t
\
2
p
2
dxdt. (3.26)
Letting e
—>
0 in the above inequality and noting hat
F
e
^XR+,
H
t
{z)^z
+
,
where ( )
+
denotes the positive part of a function and x«+ the characteristic function of
K
+
we obtain
/ {u\-uf)
+
p
2
dx+ I I A
1
(u
l
e
)Vu
1
e
V
Xl
p
2
dxdt
Jn,
Jo
J{u)-u?>o]
/ / A
1
{u
2
l
)Vu
2
V
Xl
p
2
dxdt< 0 (3.27)
Jo J\u\-u
2
>0]
u\-u
2
>0]
for a.e. t E (0,T). We set then
a,ij(t,x,z) = / aij(t,x,s)ds. (3.28)
Jo
For u e H
1
^), it holds that for a.e. t e (0,T),
d
i:j
{t,x,u) € H\tt) (3.29)
and for every k
—
1,..., n
ru
d
Xk
a,ij(t,x,u) = a,ij(t,x,u)d
Xk
u+ d
Xk
aij(t,x,s)ds. (3.30)
Jo
63
With this, if we denote by E' the difference of the two last integrals in (3.27) we have
E' = / Y2
aij(ul)d
x
u\d
Xi
p
2
dxdt-
Jo
i=i,...,
P
•A«,
1
-«?>°]
/ zJ / a
ij
(u
2
e
)d
x
u
2
e
d
Xi
p
2
dxdt. (3.31)
Jo
i=i,...,
P
-
/
W-"?>o]
;P
.7
=
1,...,71
J=l n
(For simplicity we drop the dependence in t, x in the coefficients.) Thus, by (3.30) we
obtain
J'= / J2 { d
Xj
a^{u\) - I d
Xj
a
iij
(s)ds}d
Xi
p
2
dxdt-
Jo
i^..,
P
J
H-n]>o)
{ Jo J
j=l,...,n
/ V] / \d
Xj
aij(uj) - d
x
a,ij(s)ds>d
Xi
p
2
dxdt.
j =
l,...,n
Hence,
J =
l,...,n
— / d
Xj
a,ij(s)dsd
Xi
p
2
>
dxdt
Jul
' J
= / Yl I \
d
^AhA
u
\
+
w
) -
Kii^dxiP
2
Jo
i=i,...,
P
Jn
' >•
j =
l,...,Tl
—
/ d
Xj
aij(s)dsd
Xi
p
2
} dxdt (3.33)
where we have set to = (uj
—
u
2
)
+
. Integrating by parts we obtain
E>
= / Y, { ~(«ij(
u
£ +
w
) - Hj(
u2
e))dl
iXi
p
2
- / d
Xj
a,ij(s)dsd
Xi
p
2
\ dxdt.
Jo
i=i,.../
fl
<
I
7
"? J
j =
l,...,n
(3.34)
Indeed, since u\
—
uj = 0 on (—£,£)
p
x eta (see (3.4)) and 9
Xi
p
2
= 0 on
d(—l,l)
p
x w, it
holds that
(a
it
j(u
2
e
+ w) - a
itj
(u
2
))d
Xi
p
2
= 0 on d£V
Using the definition of
a
it
j,
we rewrite E' under the form
E'=
/ - Y" M / aij(s)ds92..,..p
2
+ / cVa^^dsc^p
2
Uzdi.
j=l,...,n
64
Since the a^- and d
Xk
a,ij are bounded and
K/(f)|,Kp
2
(f)|<f
(3.35)
where
C is a
constant independent of £i, we derive for some constant
C =
C(p,A,A')
since p = 0 outside (—H.i,l\f
:
\E'\
<Y f f
wdxdt
= - f f {u\-
uj)
+
dxdt. (3.36)
*-i
Jo
Jn
tl
"i
JO
in,,
Recalling the definition of E' and going back to (3.27) we obtain for some C independent
of£i
/ {u\
-
u})
+
p
2
dx
<j- / (u\-
u
2
e
)+dxdt (3.37)
thus by (2.24)
/ {u\
—
uj)
+
dx < —
/ /
(wj
—
uj)
+
dxdt.
Jn
tl
/
2
*i Jo Jfi
(l
Integrating in
i it
follows that
/
/
(uj
-
ufj
+
dxdt <
— \ j [u\-
u
2
)
+
dxdt.
Since clearly — u\, —uj are solution of similar equations we arrive to
/
/
\u\
-
u\\dxdt < —
/
\u\
-
u
2
e
\dxdt. (3.38)
Jo
Jn
tl/2
*-i Jo
Jn
tl
Setting £i
=
£/2
k
we get
/
/
\u\ -u\\dxdt < —
/ /
\u\
-
u\\dxdt. (3.39)
Jo Jn
l/2
k+i
*•
Jo
Jn
t/2k
Iterating this formula we obtain
rT
r
n i-T
/
/
Wi
-
u
2
t
\dxdt <
— / /
\u\
-
u\\dxdt
Jo Jn
t/2k+
i
l
Jo Ja
t
-
¥*
{[ L
Wt
~
uj]2dxdt
)'
2
Ti/2|
^
|i/2
(by the Cauchy-Schwarz inequality). Using now Lemma 3.1 we derive easily that
f
j
\u\
-
u\\dxdt <
^T
l
'M -
u\|
1
=.(o,T;W
(
n
/
))r
1
/
2
|n
/
|
1
/
2
Jo
Jn
t/2k+1
*•
4(
<2
°
+
1
)
1,V/2
65
for some constant C independent of L Choosing then k in such a way that k+l-a-p/2
>
r and
I
such that £/2
k+1
>
l
0
we obtain
/
f \u\-u\\dxdt<y
(3.40)
Jo
Jn
lo
£
with C independent of L This completes the proof of the theorem.
•
Acknowledgments. The work of the authors has been supported by the Swiss Na-
tional Science Foundation under the contract #20-58856.99. We are very grateful to this
institution for this help.
References
[1] M. Chipot: Elements of nonlinear analysis, Birkhauser, (2000).
[2] M. Chipot: £ goes to plus infinity, Birkhauser, (2002).
[3] M. Chipot, A. Rougirel: On the asymptotic behaviour of the solution of elliptic
problems in cylindrical domains becoming unbounded, to appear.
[4] M. Chipot, A. Rougirel: On the asymptotic behaviour of the solution of parabolic
problems
in
cylindrical domains of large size in some directions, Discrete Contin.
Dyn. Syst. Ser. B 1 (2001), 3, p. 319-338.
[5] M. Chipot, A. Rougirel: Remarks on the asymptotic behaviour of the solution
to parabolic problems in domains becoming unbounded, Nonlinear Analysis, Vol. 47,
Issue 1 (2001) p. 3-12.
[6] M. Chipot, A. Rougirel: Sur le comportement asymptotique de la solution de
problmes elliptiques dans des domaines cylindriques tendant vers I'infini, C. R. Acad
Sci.
Paris, t. 331, Srie I, p. 435-400, 2000.
[7]
R.
Dautray, J.L. Lions: Mathematical analysis and numerical methods for sci-
ence and technology, Springer-Verlag, (1988).
On some variational problems with an infinite number
of wells
A. Elfanni
Universitat des Saarlandes
Fachbereich 6.1- Mathematik
Postfach 15 11 50
D-66041 Saarbriicken, Germany
Email: elfanni@math.uni-sb.de
Abstract
We give a necessary and sufficient condition for nonexistence of minimizers for
some variational problems allowed to have an infinite number of potential wells and
subject to some special linear boundary conditions. Namely the gradient of the
boundary condition is related by a convex relation to some wells spanning a hy-
perplane. We also prove that the minimizing sequences generate a unique Young
measure which is supported by the wells related to the boundary data. The varia-
tional problems we consider in this note are certainly special cases of the complete
work of Priesecke [F.] but the condition we present here does not involve the com-
putation of the convex envelopes.
1 Introduction
Let fi denote a bounded domain of R" (N > 2) with Lipschitz boundary T. Let
ip : K
N
—> R be a continuous function and W{tp) the set of wells of tp i.e.
W{<p) = { w G R
N
I
<p{w)
= 0 }. (1.1)
We assume that
<p
satisfies
ip(w) >0 Vw<?W(ip). (1.2)
If H denotes the following hyperplane of R
w
H =
{w£K
N
/wfj,
= 0} (1.3)
(w
•
n denotes the scalar product of w and fi)
we assume that there exist
w\,...,
w
N
N elements of W(^>) such that
w
{
eHVi =
l,...,N
(1.4)
66
67
0 G ri^CoK),^) (1.5)
where Co(tu
i
)^
=1
is the convex hull of the uVs, ri
H
(Co(«;
i
)-I
1
) denotes its interior rela-
tively to the topology of H.We denote by W„
1,0
°(fi) the set
W^°°(Q)
= {ve
W
1
-
00
^)
I v(x) = 0 on T}.
Then we consider the following minimization problem
inf / <p(Vv(x)) dx. (1.6)
It is well known that
inf / v(Vv(x)) dx = |0| tp*'{0).
(|Q| is the Lebesgue measure of Q and
tp**
is the convex envelope of
<p
(see [D.])). Since
0
G
CO(UJ;),
one has using the convexity of
ip**
<p"(0)=0.
Remark that due to (1.4), (1.5) one has
H = Span(wi)
i=
i,...
>A
r = Span(uij -
WI)J
=2
,...,JV.
(1-7)
where Span(a;)i denotes the vectorial space spanned by the vectors <ij e R", By (1.5)
and (1.7), there exist «j € (0,1),
%
= 1,...,
TV
unique such that
N N
t=l i=l
We adopt the following notations
H
+
= {weIl
N
/ w-n>0}, TJ
+
= {w<EK
N
J
W/J,>0}
H. = {weR
N
/w-
l
i<0}, Tl- = {w£~R
N
I w-n<0}
W(<p>)*
= %)\K i =
l,...,N}
The case where W(ip) spans a proper subspace of R
N
was studied by M. Chipot and C.
Collins (see for example [C], [C.C.]). In this paper, we are concerned by the case where
W(<p)
spans the whole R
w
. One can then select N wells and a hyperplane H such that
(1.4) holds. By making possibly a translation we can assume that (1.5) also holds.
68
2 A necessary and sufficient condition for nonexis-
tence of minimizers.
In this section we give a necessary and sufficient condition for nonexistence of minimizers.
The problem we study in this note is certainly a special case of the interesting work of
G. Friesecke (see [F.]) but our condition does not involve the computation of the convex
envelope of
ip
(see the four-well antiplane shear problem studied below.)
Theorem 1. We assume that
0^%), (2.1)
then the infimum in (1.6) is not attained if and only ifW{<p) C H
+
or W(<p) C H_.
Proof.
Let us prove that the above condition is sufficient. We assume for example that
W(ip) C H
+
. Let u be a minimizer of (1.6) then one has
Jn
tp(Vu(x))dx = 0. (2.2)
in
Hence by (1.1), (1.2) one has
VM(I) e W{ip) C
~H+
a.e. in
Q,.
Therefore
Vu(i)
•
n > 0 a.e. in Q. (2.3)
Using the following variant of Poincare's inequality
/ \u(x)\dx <C \Vu{x)
•
n\dx (2.4)
Jn
Jn
btains
[ \u{x)\dx<C ( Vu{x)dx-ti. (2.5)
in Jn
Integrating by part one gets
Vu(x)dx = 0. (2.6)
in
where C is a constant one obtains
/
Jn
in
Combining (2.5)-(2.6) one deduces that
u = 0 in fi.
Since 0 £ W(</>) one has by (1.2)
ip(0) > 0
so that (2.2) is impossible.
Now assume that there exist u>_i e W($) n H
+
and w
0
G W($) n H_. Since
0 G ri
H
{Co(wi))
i=h N
there exist k wells (k > N
—
1) among the Wi's that we can assume to
that
Span(wj)i
=
_i,
0
,...,j:
= R
N
0 e Int(Co(u)i)
i=
_
li0>
...,
fc
)
where Int(A) denotes the interior of A. Therefore there exist B-i,B
0
,
that
k k
t=-l i=-l
Then we consider the following function
it
(a;) = f\
Wi •
x + 1, x e R
N
where f\ denotes the infimum of functions. It is clear that
V«(i) =
Wi
a.e. in R" (i =
—l,0,...,k).
Due to (2.8) one has
k
f\ Wi-x<0, VieR".
We denote by 5 the following open subset of K
N
k
S = {x e R
N
I u(x) = /\wi-x + l>0}.
We claim that 5 is bounded. Indeed one has
k
f\wi-x
+ l>0 V x€ S.
i=-l
Then
k
Y
Wi •
{-x) < 1
V
x e S
where V denotes the supremum of numbers. Thus
\x\ \/
Wi
•
t£ <
1 V
x e S.
Hence
k
\x\ inf \/
Wi •
y < 1
V
x e S
S6S»-'
.
v