Electronic Journal of Differential Equations, Vol. 1997(1997), No. 12, pp. 1–18.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp (login: ftp) 147.26.103.110 or 129.120.3.113
Behaviour near the boundary for solutions of
elasticity systems ∗
V. N. Domingos Cavalcanti
Abstract
In this article we study the behaviour near the boundary for weak
solutions of the system
u00 − µ∆u − (λ + µ)∇(α(x) div u) = h ,
with u(x, t) = 0 on the boundary of a domain Ω ∈ Rn , and u(x, 0) = u0 ,
u0 (x, 0) = u1 in Ω. We show that the Sobolev norm of the solution in an
ε-neighbourhood of the boundary can be estimated independently of ε.
1
Introduction
Let Ω be a bounded domain in Rn with a C 3 -boundary Γ, and let ν(x) be the
unit exterior normal of Γ at a point x. For T > 0, we denote by Q the finite
cylinder Ω×]0, T [, and by Σ its lateral boundary Γ×]0, T [. For an open subset
Γ0 of Γ, Σ0 denotes Γ0 ×]0, T [. For each ε > 0, ωε denotes the ε-neighbourhood
of Γ0 in Ω defined by
[
ωε =
B(x, ε) ∩ Ω ,
x∈Γ0
where B(x, ε) is the open ball with center x and radius ε. Functional spaces
and their inner products are denoted as follows
H = [L2 (Ω)]n ,
V = [H01 (Ω)]n ,
(u, v)H =
n
X
(ui , vi )L2 (Ω)
i=1
n
X
∀u, v ∈ H
(∇ui , ∇vi )L2 (Ω)
((u, v))V =
∀u, v ∈ V .
i=1
∗ 1991 Mathematics Subject Classifications: 93B05, 93C20, 35B37.
Key words and phrases: Behaviour near the boundary, controllability, elasticity system
c
1997
Southwest Texas State University and University of North Texas.
Submitted April 1, 1997. Published July 31, 1997.
1
2
Behaviour near the boundary
EJDE–1997/12
This article is concerned with the behaviour near the boundary for weak
solutions of the system
u00 − µ∆u − (λ + µ)∇(α(x) div u) = h in Q
u = 0 on Σ
u(0) = u0 , u0 (0) = u1 in Ω ,
(1.1)
with
{u0 , u1 , h} ∈ V × H × L1 (0, T ; H) ,
(1.2)
where λ, µ > 0 are the Lamé constants, and α ∈ C (Ω) is a real function such
that for all x in Ω and some α0 , α(x) ≥ α0 > 0.
Recall that if u is the unique solution of the above system, for the energy
function
1
E(t) =
1 0 2
µ
(λ + µ) 1
|u (t)|H + ||u(t)||2V +
|α 2 (x) div u(t)|2L2 (Ω)
2
2
2
there exists a positive constant c such that
sup E(t) ≤ cE0 ,
t∈[0,T ]
where
E0 = ||u0 ||2V + |u1 |2H + ||h||2L1 (0,T ;H) .
(1.3)
We shall prove that the H norm of the solution u of (1.1) in an ε-neighbourhood of the boundary can be estimated independently of ε. It will be done
by studying first the behaviour of ∇u in the same neighbourhood. We use
the method developed by J.P. Puel and C. Fabre [7] for solutions of the wave
equation, which is an extension of the multipliers method introduced by F.
Rellich for elliptic equations and used by C. Morawetz in hyperbolic equations.
Their method was also applied to Schrodinger equations, and to equations for
vibrating beams (cf. C. Fabre [4] and C. Fabre - J. P. Puel [6]).
The interest in the above result lies in the fact that this estimate, combined
with other results, allow us to obtain the boundary control as the limit of internal
controls. Then, provided that the internal controls exist we obtain the boundary
control passing to the limit as ε → 0. This is a way of getting boundary control
when we only have the guarantee that the system is internal controllable.
To use this kind of argument is a powerful tool since we act where it is
convenient and then pass to the limit. In this direction we can cite the work by
M. E. Bradley and M. A. Horn [1] where they control the Von Kármán system
w00 − γ 2 ∆w0 + ∆2 w + b(x)w0 = [w, χ(w)], by showing that the boundary control
which stabilizes its solution when γ 6= 0 also stabilizes the solution to the system
obtained in the limit as γ → 0. Here γ is a parameter which is proportional to
the thickness of the Von Kármán plate.
EJDE–1997/12
2
V. N. Domingos Cavalcanti
3
Geometrical Properties of Ω
Using the fact that Ω is a bounded domain of Rn with a C 3 −boundary Γ, we
obtain the following lemma.
Lemma 2.1 There exist open sets U1 , . . . Um and a positive constant ε0 which
satisfy
S
• For each ε in ]0, ε0 [, ωε ⊂ m
i=1 Ui , where ωε denotes the closure of ωε in
Rn .
• For each x ∈ ωε ∩ Ui , there exists a unique (y, z) ∈ (Γ ∩ Ui )×]0, ε[ such
that x = y − zν(y); i = 1, ..., m.
• There are functions Fi−1 : x → (w, z), C 2 -diffeomorphisms defined from
ωε ∩ Ui onto their image; i = 1, . . . , m.
Proof. Due to the regularity of Γ, which is the C 3 -boundary of Ω, by M. Do
Carmo [2] (section 2.7, proposition 1) for each point p ∈ Γ and Xp : U → Γ,
a C 3 -parametrization of a neighbourhood of p, there exist a neighbourhood
Wp ⊂ Xp (U ) of p in Γ and εp > 0 such that the segments of the normal lines
through points q ∈ Wp , with center at q and length 2εp , are disjoint; that is,
Wp has a tubular neighbourhood.
Considering 0 < 2rp < εp where B2rp (p)⊂ Bp , Wp = Bp ∩ Γ and Bp is an
open set of Rn we obtain:


[
Ω=Ω∪Γ⊂Ω∪
Brp (p) .
p∈Γ
Using the compactness of Ω, there exist p1 , . . . , pm such that
Γ⊂
m
[
Brpi (pi ).
i=1
Defining
Ui = B2rpi (pi );
i = 1, . . . , m and ε0 =
1
min {rp1 , . . . , rpm } ,
2
Sm
we conclude that for every ε ∈]0, ε0 ], ωε ⊂ i=1 Ui .
On the other hand, we observe that for each pi ∈ Γ the related C 3 - parametrization Xi implies that the function
Fi : Xi−1 (Γ ∩ Ui )×]0, ε[→ ωε ∩ Ui
(w, z) 7→ Fi (w, z) = Xi (w) − zNi (w) ,
4
Behaviour near the boundary
EJDE–1997/12
where ε ∈]0, ε0 [ and Ni (w) is the normal vector at point Xi (w), is a C 2 diffeomorphism.
Moreover, according to the choice of the open sets Ui we conclude that for
every x ∈ ωε ∩ Ui where ε ∈]0, ε0 ] there exist a unique normal projection y of x
on Γ and unique z ∈]0, ε[ such that x = y − zν(y).
2
Remark. It follows from the above lemma that if x ∈ ωε ∩ Ui for sufficiently
small ε, the normal projection of x onto Γ is uniquely defined. Furthermore,
p(x) = Xi (ω) = Fi (ω, 0) ,
where Fi (w, z) = Xi (w) − zNi (w).
Lemma 2.2 Let det JFi (w, z) denote the determinant of the Jacobian matrix
of Fi at (w, z). Then
(i) there exist positive constants ε0 , m, M , such that ∀(w, z) ∈ Xi−1 (Γ ∩ Ui ) ×
[0, ε0 ],
m ≤ | det JFi (w, z)| ≤ M
(ii) | det JFi (w, 0)| = 1, ∀w ∈ Xi−1 (Γ ∩ Ui )
(iii) the function (w, z) → | det JFi (w, z)| is a C 1 -function.
(iv) for function v on ωε ∩Ui , define v̂(w, z) = (v ◦Fi )(w, z). If v ∈ H 1 (ωε ∩Ui )
then
∂
v̂(w, z) = −∇v(Fi (w, z)) · ν(Fi (w, 0) .
(2.1)
∂z
Proof. Since Fi (w, z) = Xi (w) − zNi (w) it follows that | det JFi (ω, 0)| =
||Ni (ω)|| and consequently | det JFi (ω, 0)| = 1, ∀ω ∈ Xi−1 (Γ ∩ Ui ). Besides,
taking into account the regularity of the boundary of Ω, we get that det JFi (ω, z)
is a C 1 −function. For a εi > 0 small enough we obtain that det JFi (ω, z) does
not change its sign on ]0, εi[, which allow us to conclude (iii). Combining the
two results obtained in (ii) and (iii) we get (i). Finally, from the regularity
∂
of the functions Fi (w, z), observing that ∂z
Fik (w, z) = −Nik (w) and from the
identity
n
X
∂
∂v
∂
(v ◦ Fi )(w, z) =
(Fi (w, z)) Fik (w, z) ,
∂z
∂xk
∂z
k=1
which holds for regular functions v, we obtain (iv).
2
EJDE–1997/12
3
V. N. Domingos Cavalcanti
5
Fundamental Identity
In this section we prove an identity that is essential in the proof of our main
result.
Lemma 3.1 If u = (u1 , ..., un ) is the solution of (1.1)-(1.2), for every vector
valued function g ∈ W 2,∞ (Ω, Rn ) collinear to the normal vector on Γ, we have
Z
n
n Z
X
X
∂ui ∂ui ∂gk
2µ
dx dt + µ
(∇ui · ∇(div g))ui dx dt
∂xj ∂xk ∂xj
i=1 Q
i,j,k=1 Q
n Z
X
∂ui ∂gj
+2(λ + µ)
α(x) div u
dx dt
∂xj ∂xi
i,j=1 Q
Z
+(λ + µ)
α(x) div u(u · ∇(div g)) dx dt
(3.1)
= 2
Q
n Z
X
hi (∇ui · g) dx dt +
i=1 Q
n Z
X
i=1
|
+µ
Σ
i=1
Z
∇α · g(div u)2 dx dt − 2
+
n Z
X
i=1
Proof.
Q
n Z
X
i=1
hi ui divg dx dt
Q
∂ui 2
| g · νdΓdt + (λ + µ)
∂ν
+(λ + µ)
+2
n Z
X
Z
α(x)(divu)2 g · νdΓ dt
Σ
n Z
X
i=1
u0i (0)(∇ui (0) · g) dx −
Ω
n Z
X
i=1
u0i (T )(∇ui (T ) · g) dx
Ω
u0i (T )ui (T ) div g dx
Ω
u0i (0)ui (0) div g dx .
Ω
n
For initial data u0 , u1 , h ∈ H01 (Ω) × H 2 (Ω) × V × L1 (0, T ; V ) let
2∇u · g = (2∇u1 · g, . . . , 2∇un · g) ,
where
2∇ui · g = 2
n
X
∂ui
gk .
∂xk
k=1
Multiplying (1.1) by 2∇u · g we obtain
n Z
n Z
X
X
u00i (2∇ui · g) dx dt − µ
∆ui (2∇ui · g) dx dt
i=1
Q
−(λ + µ)
n Z
X
i=1
Q
i=1
Q
∂
(α(x) div u) (2∇ui · g) dx dt
∂xi
(3.2)
6
Behaviour near the boundary
n Z
X
=
EJDE–1997/12
hi (2∇ui · g) dx dt .
Q
i=1
Integrating by parts with respect to t, by Gauss Theorem,
2
n Z
X
i=1
u00i (∇ui · g) dx dt
Q
= 2
n Z
X
u0i (T ) (∇ui (T ) · g) dx − 2
i=1 Ω
n Z
X
i=1
u0i (0) (∇ui (0) · g) dx (3.3)
Ω
(u0i ) div g dx dt .
2
+
i=1
n Z
X
Q
Using the fact that
∂ui
∂ui
= νk
∂xk
∂ν
by Green and Gauss Theorems we have
−µ
n Z
X
i=1
on Γ ,
(3.4)
∆ui (2∇ui · g) dx dt
Q
= −µ
n Z
X
i=1
−µ
Q
n Z
X
i=1
|∇ui |2 div g dx dt + 2µ
Σ
∂ui
∂ν
(3.5)
Z
n
X
i,j,k=1
2
Q
∂ui ∂ui ∂gk
dx dt
∂xj ∂xk ∂xj
g · ν dΓdt .
Finally, by (3.4) and by Gauss theorem we have
n Z
X
∂
(α(x) div u) (2∇ui · g) dx dt
(3.6)
∂xi
i=1 Q
n Z
X
∂
= 2(λ + µ)
α(x) div u
(div u)gk dx dt
∂xk
Q
k=1
n Z
X
∂ui ∂gk
+2(λ + µ)
α(x) div u
dx dt .
∂x
k ∂xi
i,k=1 Q
Z
2
−2(λ + µ) α(x) (div u) g · ν dΓ dt .
Z Σ
Z
2
2
= −(λ + µ)
(div u) (∇α(x) · g) dx dt − (λ + µ)
(div u) α(x) div g dx dt
Q
Q
Z
2
+(λ + µ)
α(x) (div u) g · ν dΓdt .
−(λ + µ)
Σ
EJDE–1997/12
V. N. Domingos Cavalcanti
n Z
X
+2(λ + µ)
α(x) div u
Q
i,k=1
Z
7
∂ui ∂gk
dx dt .
∂xk ∂xi
2
−2(λ + µ)
α(x) (div u) g · ν dΓdt .
Σ
Replacing (3.3), (3.5) and (3.6) in (3.2) it follows that
n Z
n Z
X
X
0
2
ui (T ) (∇ui (T ) · g) dx − 2
u0i (0) (∇ui (0) · g) dx
i=1 Ω
n Z
X
+
i=1
(u0i ) div g dx dt − µ
2
Q
n Z
X
i=1
i=1
Q
(3.7)
Ω
2
|∇ui | div g dx dt
2
n Z X
∂ui
∂ui ∂ui ∂gk
dx dt − µ
g · ν dΓdt
∂xj ∂xk ∂xj
∂ν
i=1 Σ
i,j,k=1 Q
Z
Z
2
2
−(λ + µ)
(div u) (∇α(x) · g) dx dt − (λ + µ)
(div u) α(x) div g dx dt
+2µ
Z
n
X
Q
+2(λ + µ)
=
i=1
∂ui ∂gk
α(x) div u
dxdt − (λ + µ)
∂xk ∂xi
Q
i,k=1
n Z
X
Q
n Z
X
Z
2
α(x) (div u) g · ν dΓdt
Σ
hi (2∇ui · g) dx dt .
Q
Pn
∂g
Let u div g = (u1 div g, . . . , un div g), where ui div g = j=1 ui ∂xjj . Now, multiplying (1.1) by u div g we get
n Z
n Z
X
X
u00i ui div g dx dt − µ
∆ui ui div g dx dt
Q
i=1
−(λ + µ)
i=1
n Z
X
=
n Z
X
Q
Q
i=1
∂
(α(x) div u) ui div g dx dt
∂xi
hi ui div g dx dt .
Q
i=1
Using integration by parts we obtain
n Z
X
u00i ui div g dx dt
i=1
=
Q
n Z
X
i=1
−
u0i (T )ui (T ) div g dx −
Ω
n Z
X
i=1
(3.8)
Q
(3.9)
n Z
X
i=1
|u0i | div g dx dt .
2
Ω
u0i (0)ui (0) div g dx .
8
Behaviour near the boundary
EJDE–1997/12
and by Green’s Theorem
−µ
n Z
X
∆ui ui div g dx dt
(3.10)
Q
i=1
n Z
X
= µ
2
|∇ui | div g dx dt + µ
Q
i=1
n Z
X
i=1
∇ui · ∇ (div g) ui dx dt.
Q
Since u = 0 on Γ, by Gauss Theorem we have
−(λ + µ)
n Z
X
i=1
Q
Z
∂
(α(x) div u) ui div g dx dt
∂xi
(3.11)
2
α(x) (div u) div g dxdt
= (λ + µ)
Q
Z
α(x) (div u)u · ∇ (div g) dx dt .
+(λ + µ)
Q
Then, from (3.8)-(3.11), it follows that
n Z
X
i=1
−
u0i (T )ui (T ) div g dx −
Ω
i=1
n Z
X
|u0i | div g dx dt + µ
2
i=1 Q
n Z
X
u0i (0)ui (0) div g dx
Ω
2
|∇ui | div g dx dt
Q
α(x) (div u)2 div g dx dt
Q
Z
α(x) (div u)u · ∇ (div g) dx dt
+(λ + µ)
n Z
X
(3.12)
Z
∇ui · ∇ (div g) ui dx dt + (λ + µ)
Q
i=1
i=1
n Z
X
i=1
+µ
=
n Z
X
Q
hi ui div g dx dt
Q
n
Adding (3.7) and (3.12), and using that H01 (Ω) ∩ H 2 (Ω) × V × L1 (0, T ; V )
is dense in V × H × L1 (0, T ; V ), we obtain (3.1).
2
Lemma 3.2 Let u be the solution of (1.1) with initial data satisfying (1.2),
and let vr = θr u, where {θr }1≤r≤m is a C ∞ partition of the unity relative to
the open sets U1 , . . . , Um . Then vr = (vr1 , . . . , vrn ) is the solution to
vr00 − µ∆vr − (λ + µ)∇(α(x) div vr ) = hr
vr = 0 on Σ
vr (0) = θr u0 , vr0 (0) = θr u1 on Ω ,
in Q
(3.13)
EJDE–1997/12
V. N. Domingos Cavalcanti
9
where
hr = θr h − 2µ∇θr · ∇u − µu∆θr − (λ + µ)α(x)divu∇θr − (λ + µ)∇(α(x)u · ∇θr )
and supp vr ⊂ Ur × [0, T ]. Furthermore, if ε0 is the minimum between the two
epsilons found in Lemmas 2.1 and 2.2, the function
 R R
Pn
2
 1 T
ε ∈]0, ε0 ]
i=1 |∇vri (x, t) · ν(p(x))| dx dt,
ε 0
ωε ∩Ur
2
G(ε) =
RT R
Pn ∂vri 
ε = 0,
i=1 ∂ν dΓ dt
Γ∩Ur
0
where Fr (Wr ×]0, ε[) = ωε ∩ Ur , is continuous on [0, ε0 ].
Proof.
The continuity of G on [0, ε0 ] follows from Lemma 2.2.
2
Lemma 3.3 Let δ be a positive number. Then there exists a real number γ∈
]0, δ[, and there exist positive decreasing functions ρε ∈ W 2,∞ (0, ε), where ε =
δ + γ, such that
1
ρε (ε) = 0 ρ0ε (ε) = 0 and ρ0ε = − , in [0, δ] .
(3.14)
δ
Furthermore,
C2
C3
||ρε ||L∞ (0,ε) ≤ C1 , ||ρ0ε ||L∞ (0,ε) ≤
, ||ρ00ε ||L∞ (0,ε) ≤ 2
(3.15)
ε
ε
for positive constants C1 , C2 , C3 , and
Z
9
γ ε 00
|ρ (z)|2 z 2 dz =
.
(3.16)
2 0 ε
16
Proof.
Let γ be the positive value when solving for x in
x 1 x
9
(3.17)
1 + + ( )2 = .
δ
3 δ
8
q
7
3
Then γ =
6 − 1 2 δ which belongs to the interval ]0, δ[. Put ε = γ + δ, and
define ρε : [0, ε] → R, by
γ
1 − 2δ
− zδ ,
z ∈ [0, δ]
ρε (z) =
1
2
(δ
+
γ
−
z)
,
z ∈ [δ, δ + γ] .
2γδ
Then ρε ∈ W 2,∞ (0, ε) and satisfies (3.14),(3.15). To show that ρε satisfies
(3.16), we use (3.17) as follows
Z
Z
γ ε 00
γ δ+γ 1 2
2 2
|ρ (z)| z dz =
z dz
2 0 ε
2 γ
γ 2δ2
1
γ
γ2
1+ + 2
=
2
δ
3δ
9
=
.
16
10
Behaviour near the boundary
EJDE–1997/12
4
Behaviour of the Solution u in ωε ×]0, T [
Our goal in this section, which contains the main result of this work, is to
study the behaviour of the solution u of the elasticity system given in (1.1) in
ωε ×]0, T [.
Theorem 4.1 There exist positive constants C and ε0 such that every solution
u of (1.1) where (1.2) holds with α1 < µ/(3n(λ + µ)) satisfies
Z
n Z
1X T
|∇ui (x, t) · ν(p(x))|2 dx dt ≤ cE0 ∀ε ∈]0, ε0 ] .
ε i=1 0 ωε
Furthermore, C and ε0 depend only on the positive number T, the function α(x),
the geometry of Ω, and the Lamé constants.
Proof. Without loss of generality assume that Γ0 = Γ. Let u be the solution
of (1.1) and define vr = θr u where {θr }1≤r≤m is a C ∞ partition of the unity
relative to the open sets U1 , . . . , Um given in lemma 2.1. According to Lemma
3.2, vr is the solution of (3.13) and the function G is continuous on [0, ε0 ]. Let
δ0 ∈ [0, ε0 ] be a value such that
G(δ0 ) = max G(ε) .
ε∈[0,ε0 ]
If δ0 = 0, we obtain
max G(ε) =
ε∈[0,ε0 ]
n Z
X
T
Z
Γ∩Ur
0
i=1
∂vri 2
∂ν dΓ dt .
But, considering the trace theory developed by M. Milla Miranda [9], we get
∂vri
∂ui
= θ|Γ
∂ν
∂ν
and consequently,
∗
max G(ε) ≤ c
n Z
X
ε∈[0,ε0 ]
i=1
0
T
Z ∂ui 2
dΓ dt .
Γ ∂ν
On the other hand, proceeding as in J. L. Lions [8] (chapter IV - pages 224, 225),
2
i
that is, using the multiplier method, we obtain that ∂u
∂ν ∈ L (Σ), i = 1, 2, ..., n
and
n Z T Z X
∂ui 2
∂ν dΓdt ≤ cE0 .
0
Σ
i=1
Then, we conclude that
max G(ε) ≤ cE0 .
ε∈[0,ε0 ]
EJDE–1997/12
V. N. Domingos Cavalcanti
11
If δ0 ∈] ε20 , ε0 [, we have
Z
n Z
2 X T
max G(ε) ≤
|∇vri (x, t)|2 dx dt ≤ cE0
ε0 i=1 0 ωδ0 ∩Ur
ε∈[0,ε0 ]
where c, in both cases, is the desired constant. We can then restrict ourselves
to the case 0 < δ0 < ε20 .
As δ0 > 0, according to lemma 3.3, there exists a real number γ ∈]0, δ0 [. Put
ε = γ + δ0 , then there exists a decreasing function ρε ∈ W 2,∞ (0, ε) such that
ρε (ε) = 0,
ρ0ε (ε) = 0,
and ρ0ε = −
1
in [0, δ0 ]
δ0
(4.1)
and (3.15) and (3.16) are satisfied. Let us consider now the following vector
valued function
gε (x) = ρε (z)ν(y),
Noting that
∂z
∂xk
x ∈ ω ε ∩ Ur ,
where x = y − zν(y).
(4.2)
= −νk we have
∂gεj
∂νj
(x) = −ρ0ε (z)νk (p(x))νj (p(x)) + ρε (z)
(p(x)),
∂xk
∂xk
x ∈ ωε .
So, we obtain
div gε (x) = −ρ0ε (z) + ρε (z) div ν(p(x)),
x ∈ ωε
and
∇(div gε )(x) = ρ00ε (z)ν(p(x)) − ρ0ε (z) div(ν(p(x)) + ρε (z)∇(div ν(p(x))) .
Next, we use identity (3.1) taking as function g the family of functions gε
defined in (4.2), replacing u by vr the solution of (3.13) and observing that
supp vr ⊂ Ur × [0, T ].
From the above equalities, Lemma 2.2, and taking into account that Fr (Wr ×]0, ε[) =
ωε × Ur , we get by lemma 3.1 the following expression
2
Z ε
n Z T Z
X
∂v̂ri
0
−2µ
ρε (z) (w, z, t) | det JFr (w, z)|dz dw dt
(4.3)
∂z
Wr 0
i=1 0
Z ε
n Z T Z
X
∂v̂ri
−µ
ρ00ε (z)
(w, z, t)v̂ri (w, z, t)| det JFr (w, z)|dz dw dt
∂z
Wr 0
i=1 0
n Z T Z
X
α(x)ρ00ε (z) div vr (vr · ν(p(x))) dx dt
+(λ + µ)
i=1
= R1 + R2 .
0
ωε ∩Ur
12
Behaviour near the boundary
where R1 is given by
Z TZ
n
X
−2µ
ωε ∩Ur
i,j,k=1 0
n
XZ T Z
−µ
ωε ∩Ur
0
i=1
n Z
X
−2(λ + µ)
T
−2
+
Z
ωε ∩Ur
α(x)ρε (z) div vr
Wr
Z
Z
∂vri ∂νj
(p(x)) dx dt
∂xj ∂xi
α(x)ρε (z) div vr (vr · ∇(div ν(p(x)))) dx dt
ρε (z)ĥri (w, z, t)
i=1 0
n
XZ T
i=1
Z
T
Z
ωε ∩Ur
Z ε
0
n Z
X
T
Z
−(λ + µ)
∂vri ∂vri ∂νj
(p(x)) dx dt
∂xj ∂xk ∂xk
ρε (z)(∇vri · ∇(div ν(p(x))))vri dx dt
0
i,j=1
Z
ρε (z)
EJDE–1997/12
0
∂v̂ri
(w, z, t)v̂ri (w, z, t)| det JFr (w, z)|dz dw dt
∂z
ε
ρε (z) div ν(Fr (w, 0))ĥri (w, z, t)| det JFr (w, z)|dz dw dt
0
0
Wr
n
X
∂vri 2
|
| + (λ + µ)α(x)| div vr |2 }ρε (0)dΓ dt
∂ν
Σ
i=1
Z TZ
+(λ + µ)
ρε (z)(∇α(x) · ν(p(x)))| div vr |2 dx dt
{µ
+
+2
i=1
−
+
ε
ρε (z)v̂ri (w, z, T )
n Z
X
n Z
X
ωε∩Ur
Z
Wr
i=1
−2
0
n Z
X
0
Z
ε
ρε (z)v̂ri (w, z, 0)
Wr
0
i=1
ωε∩Ur
i=1
ωε∩Ur
n Z
X
0
−2(λ + µ)
∂v̂ri
(w, z, 0)| det JFr (w, z)|dz dw
∂z
ρε (z) div ν(p(x))vr0 i (T )vri (T ) dx
ρε (z) div ν(p(x))vr0 i (0)vri (0) dx
and R2 is given by
Z
n Z T Z
X
−µ
i=1
∂v̂ri
(w, z, T )| det JFr (w, z)|dz dw
∂z
ε
Wr
0
n Z
X
T
i=1
0
ρ0ε (z) div ν(Fr (w, 0))
∂v̂ri
(w, z, t)v̂ri (w, z, t, ) ×
∂z
| det JFr (w, z)|dz dw dt
Z Z ε
∂v̂ri
α(Fr (w, z))ρ0ε (z) div vr (Fr (w, z), t)
(w, z, t) ×
∂z
Wr 0
νi (Fr (w, 0))| det JFr (w, z)|dz dw dt
EJDE–1997/12
V. N. Domingos Cavalcanti
Z
T
Z
α(x)ρ0ε (z) div ν(p(x)) div vr (vr · ν(p(x))) dx dt
+(λ + µ)
−
+
n Z
X
i=1 0
n Z
X
i=1
0
T
Z
13
ωε ∩Ur
ε
ρ0ε (z)ĥri (w, z, t)v̂ri (w, z, t)| det JFr (w, z)|dz
0
Z
Wr
ωε ∩Ur
ρ0ε (z)vr0 i (T )vri (T ) dx −
n Z
X
i=1
ωε∩Ur
dw dt
ρ0ε (z)vr0 i (0)vri (0) dx .
Note that the expression for R1 collects the terms which involves ρε (z) and the
expression R2 the terms related to ρ0ε (z).
Taking into account the regularity of the functions α, ρε and of the normal
vector ν, and considering that
∂vri
∂ui
= θr |Γ
∂ν
∂ν
div vr |Γ = θr |Γ div u|Γ
and
(4.4)
we have a positive constant C1 such that |R1 | ≤ C1 E0 .
On the other hand, observing that:
∂vri
∂v̂ri
(x) = −
(w, z)νj (Fr (w, 0)) + Dw v̂ri Dxj w
∂xj
∂z
where
Dw v̂ri Dxj w =
n−1
X
k=1
∂v̂ri
∂wk
(w, z)
,
∂wk
∂xj
we can write
div vr (x) =
n
X
∂v̂rj
{−
(w, z)νj (Fr (w, 0)) + Dw v̂rj Dxj w}.
∂z
j=1
(4.5)
Using the Hölder inequality, we have
Z
|v̂ri (w, z, t)| ≤ z
2
0
z
2
∂v̂ri
∂z (w, s, t) ds.
From this inequality, and using in (4.5) an analogous argument to the one used
by C. Fabre and J. P. Puel in [6] (page 196) we conclude that there exists C2 > 0
such that
|R2 | ≤ C2 E0 .
(4.6)
From (4.3), (4.4) and (4.6) we find a positive constant C3 independent of ε,
δ0 and u for which,
2
Z ε
n Z T Z
X
∂v̂ri
−2µ
ρ0ε (z) (w, z, t) | det JFr (w, z)|dz dw dt
(4.7)
∂z
Wr 0
i=1 0
14
Behaviour near the boundary
≤ µ
n Z
X
Z
T
0
i=1
Z
Z
Wr
T
ρ00ε (z)
0
Z
−(λ + µ)
ε
∂v̂ri
(w, z, t)v̂ri (w, z, t)| det JFr (w, z)| dz dw dt
∂z
α(x)ρ00ε (z) div vr (vr · ν(p(x))) dx dt + C3 E0 .
ωε ∩Ur
0
EJDE–1997/12
But using (4.5) we have
Z
T
Z
−(λ + µ)
0
= (λ + µ)
α(x)ρ00ε (z) div vr (vr · ν(p(x))) dx dt
ωε ∩Ur
n Z T
X
Z
Z
i,j=1
0
Wr
Z
Z
Z
T
−(λ + µ)
Wr
0
ε
α(Fr (w, z))ρ00ε (z)
0
ε
(4.8)
∂v̂ri
(w, z, t)νi (Fr (w, 0))v̂rj (w, z, t) ×
∂z
νj (Fr (w, 0))| det JFr (w, z)| dz dw dt
n
X
α(Fr (w, z))ρ00ε (z)
Dw v̂ri Dxi w ×
0
i=1
(v̂r (w, z, t) · ν(Fr (w, 0)))| det JFr (w, z)| dz dw dt .
Using the Cauchy-Schwarz inequality and (4.1) we prove that
(λ + µ)
n Z
X
i,j=1
Z
T
0
Wr
Z
1
γ
Z TZ
ε
α(Fr (w, z))ρ00ε (z)
0
∂v̂ri
(w, z, t)νi (Fr (w, 0)) ×
∂z
v̂rj (w, z, t)νj (Fr (w, 0))| det JFr (w, z)| dz dw dt
(4.9)
2
Z Z εX
n ∂v̂ri
1/2
∂z (w, z, t) | det JFr (w, z)| dz dw dt} ×
Wr δ0 i=1
Z εX
n
|ρ00ε (z)|2 |v̂ri (w, z, t)|2 | det JFr (w, z)| dz dw dt}1/2
T
≤ (λ + µ)α1 n{
0
{γ
0
Z
Wr
δ0 i=1
9
≤ (λ + µ)α1 nG(δ0 )1/2 ( G(δ0 ) + C4 E0 )1/2 ,
4
where the last inequality comes from (3.16) and the identity
v̂rj (w, z, t)| det JFr (w, z)|1/2
Z z
Z z
∂v̂rj
∂
1/2
=
(w, s, t)| det JFr (w, z)| ds +
v̂rj (w, s, t) (| det JFr (w, s)|1/2 )ds .
∂z
∂z
0
0
Now, using an analogous argument to the one used in C. Fabre and J.P. Puel
[6](page 197) we get
(λ + µ)|
RT
i=1 0
Pn
R
Wr
Rε
0
α(Fr (w, z))ρ00ε (z)[Dw v̂ri Dxi w]×
(v̂r (w, z, t) · ν(Fr (w, 0)))| det JFr (w, z)| dz dw dt | ≤ C5 E0 .
(4.10)
EJDE–1997/12
V. N. Domingos Cavalcanti
15
So, by (4.8)-(4.10) we have
Z
T
Z
−(λ + µ)
ωε ∩Ur
0
α(x)ρ00ε (z) div vr (vr · ν(p(x))) dx dt
(4.11)
1
3
1
(λ + µ)α1 nG(δ0 ) + C6 (λ + µ)α1 nG(δ0 ) 2 E02 + C7 E0 .
2
≤
Proceeding in the same way we did to obtain (4.9) we get
µ
n Z
X
Z
Z
Wr
0
i=1
≤
T
ε
ρ00ε (z)
0
∂v̂ri
(w, z, t)v̂ri (w, z, t)| det JFr (w, z)| dz dw dt
∂z
3
1/2
µG(δ0 ) + C8 µG(δ0 )1/2 E0 .
2
(4.12)
Finally, using the fact that ρ0ε (z) ≤ 0 we have
−2µ
n Z
X
i=1
T
0
Z
Wr
Z
0
ε
∂v̂ri
ρ0ε (z) ∂z
2
(w, z, t) | det JFr (w, z)| dz dw dt
≥ 2µG(δ0 ) .
(4.13)
Replacing the inequalities (4.11)-(4.13) in (4.7) we obtain
1 3 (λ + µ)
1/2
2µG(δ0 )
− n
α1 ≤ C9 G(δ0 )1/2 E0 + C10 E0 .
4 4
µ
Since α1 <
µ
3n(λ+µ) ,
there exists ζ0 > 0 such that
0 < ζ0 <
1 3 (λ + µ)
− n
α1
4 4
µ
and consequently there exists C > 0, independent of ε, u and δ0 such that
max G(ε) ≤ CE0 .
ε∈[0,ε0 ]
So, in any case, there exists C > 0 independent of ε, u and δ0 such that
1X
ε i=1
n
Z
0
T
Z
ωε ∩Ur
|∇vri (x, t) · ν(p(x))|2 dx dt
≤ C{||u0 ||2V + |u1 |2H + ||h||2L1 (0,T,H) },
∀ε ∈]0, ε0 ] .
P
But since u = m
r=1 vr , Theorem 4.1 is proved.
From the above theorem we obtain the following result.
2
16
Behaviour near the boundary
EJDE–1997/12
Theorem 4.2 There exist positive constants C and ε0 such that every solution
µ
u of (1.1) where (1.2) holds with α1 ≤ 3n(λ+µ)
satisfies
Z
n Z
1 X T
|ui (x, t)|2 dx dt ≤ CE0 , ∀ε ∈]0, ε0 ] .
ε3 i=1 0 ωε
Furthermore, C and ε0 depend only on the real positive number T , the function
α(x), the geometry of Ω, and the Lamé constants.
Proof. Let u = (u1 , . . . , un ) be the solution of (1.1) with (1.2) and let ε0 be the
minimum between the two epsilons found in Lemmas 2.1 and 2.2. Considering
{θr }1≤r≤m a C ∞ partition of the unity relative to the open sets U1 , . . . , Um
Pm
given in Lemma 2.1 we obtain u = r=1 uθr . Then, for every ε ∈]0, ε0 ] we
obtain
Z TZ
Z TZ
m
X
|ui (x, t)|2 dx dt =
|
ui θr |2 dx dt
0
0
ωε
≤ C11
ωε r=1
m
XZ T Z
≤ C11
m Z
X
Z
Z
|ui |2 dx dt
ε
|ûi (w, z, t)|2 | det JFr (w, z)| dz dw dt
m Z
X
Wr
T
Z
0
Z
ε
|ûi (w, z, t)|2 dz dw dt ,
Wr
0
r=1
Z
ωε ∩Ur
T
0
r=1
≤ C12
T
0
r=1
= C11
ωε ∩Ur
0
r=1
m Z
X
|ui |2 |θr |2 dx dt
0
where ûi (w, z, t) = ui (Fr (w, z), t). Since
Z z
∂ ûi
ûi (w, z, t) =
(w, s, t) ds ,
0 ∂s
it follows that
Z
|ûi (w, z, t)| ≤ z
2
0
z
2
∂ ûi
∂s (w, s, t) ds .
Consequently from (4.14) we have
Z
T
Z
|ui (x, t)|2 dx dt
0
ωε
≤ C12
m Z
X
r=1
0
T
Z
Z
Z
ε
(z
Wr
0
0
z
2
∂ ûi
ds)dz dw dt
(w,
s,
t)
∂s
(4.14)
EJDE–1997/12
V. N. Domingos Cavalcanti
17
2
∂ ûi
≤ C12
z
∂s (w, s, t) ds dz dw dt
Wr 0
0
r=1 0
2
Z ε
Z ε
m Z T Z
X
∂ ûi
= C12 (
z dz)
∂s (w, s, t) ds dw dt)(
Wr 0
0
r=1 0
Z ε
m Z T Z
X
= C13 ε2
|∇ui (Fr (w, s), t) · ν(Fr (w, 0))|2 ds dw dt
m Z
X
r=1
T
Z
0
Z
Wr
Z
ε
ε
0
where the last equality comes from (2.1). Therefore,
Z
T
Z
|ui (x, t)| dx dt
2
0
≤ C14 ε
2
ωε
m Z
X
r=1
T
Z
≤ C14 ε
T
Z
ωε ∩Ur
0
Z
|∇ui (x, t) · ν(p(x))|2 dx dt
|∇ui (x, t) · ν(p(x))|2 dx dt .
2
0
ωε
Then, we obtain the desired result from Theorem 4.1, in view of
1
ε3
Z
T
Z
|ui (x, t)|2 dx dt ≤ C14
0
ωε
1
ε
Z
T
Z
|∇ui (x, t) · ν(p(x))|2 dx dt.
0
ωε
Acknowledgments. I want to express my sincere gratitude to my adviser
Professor Manuel Milla Miranda, to Luiz Adauto Medeiros for his stimulating
discussions, and to Jean Pierre Puel and Caroline Fabre for their helpful comments. I should also thank God, my family and my friends for the strength
received.
References
[1] Horn, M. A. and Bradley, M. E., Global stabilization of Von Karman plate
with boundary feedback acting via bending moments only, Nonlinear Analysis, V.25, N.4, (1995), 417-435.
[2] Do Carmo, M., Differential Geometry of Curves and Surfaces, Prentice
Hall, New Jersey, 1976.
[3] Domingos Cavalcanti, V. N., Asymptotic Behaviour of the Elasticity System
Solutions, Doctorate Thesis, Federal University of Rio de Janeiro, Rio de
Janeiro, 1995.
[4] Fabre, C., Comportement au voisinage du bord des solutions de quelques
équations d’évolution linéaires, Thèse de Doctorat de l’Université Pierre et
Marie Curie - Paris VI, Paris, 1990.
18
Behaviour near the boundary
EJDE–1997/12
[5] Fabre, C., Équation des ondes avec second membre singulier et application
à la controlabilité exacte, C. R. Acad. Sci. Paris, 310-I (1990), 813-818.
[6] Fabre, C. and Puel, J. P., Behaviour near the boundary for solutions of the
wave equation, Journal of Differential Equations 106 (1993), 186-213.
[7] Fabre, C. and Puel, J. P., Comportement au voisinage du bord des solutions
de l’équation des ondes, C. R. Acad. Sci. Paris, 310-I (1990), 621-625.
[8] Lions, J. L., Contrôlabilité Exacte - Perturbations et Stabilisation de
Systémes Distribués, Tome 1, Masson, Paris, 1988.
[9] Milla Miranda, M., Traço para o dual dos Espaços de Sobolev, Bol. Soc.
Paran. Mat. (2a série), vol. 11, No. 2, 131-157, 1990.
[10] Puel, J. P., Côntrolabilité exacte et comportement au voisinage du bord des
solutions de l’équation des ondes, Lectures at IM - UFRJ, Rio de Janeiro,
1991.
[11] Zuazua E., Controlabilidad Exacta y Estabilización de la Ecuación de Ondas, Lectures Notes, IM-UFRJ, Rio de Janeiro, 1990.
[12] Zuazua E., Contrôlabilité approchée du systéme de l, élasticitée avec contraintes sur les contrôles, C. R. Acad. Sci. Paris, 319,(1994), 61-66.
Valéria Neves Domingos Cavalcanti
Universidade Estadual de Maringá
87020-900 Maringá-PR, Brazil
Email address: valeria@gauss.dma.uem.br