Journal of Inequalities in Pure and
Applied Mathematics
COERCIVENESS INEQUALITY FOR NONLOCAL
BOUNDARY VALUE PROBLEMS FOR SECOND ORDER
ABSTRACT ELLIPTIC DIFFERENTIAL EQUATIONS
volume 4, issue 2, article 43,
2003.
AISSA AIBECHE
Department of Mathematics,
Faculty of Sciences
University Ferhat Abbas,
Route de Scipion, 19000,
Setif, Algeria.
EMail: aibeche@wissal.dz
Received 09 December, 2002;
accepted 17 March, 2003.
Communicated by: Z. Nashed
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
036-03
Quit
Abstract
In this paper we give conditions which assure the coercive solvability of an
abstract differential equation of elliptic type with an operator in the boundary
conditions, and the completeness of generalized eigenfunctions. We apply the
abstract result to show that a non regular boundary value problem for a second
order partial differential equation of an elliptic type in a cylindrical domain is
coercive solvable.
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
2000 Mathematics Subject Classification: 35J05, 35P20, 34L10, 47E05.
Key words: Ellipticity, Abstract Differential Equation, Coerciveness, Non Regular Elliptic Problems.
Aissa Aibeche
Contents
1
2
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solvability of the Principal Problem . . . . . . . . . . . . . . . . . . . . .
3.1
Homogeneous Problem . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Non Homogeneous Problem . . . . . . . . . . . . . . . . . . . . .
4
Solvability of the General Problem . . . . . . . . . . . . . . . . . . . . . .
5
Completeness of Root Functions . . . . . . . . . . . . . . . . . . . . . . . .
6
Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
4
7
7
13
18
21
24
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
1.
Introduction
Many works are devoted to the study of hyperbolic or parabolic abstract equations [16, 18, 9]. In [16, 20] regular boundary value problems for elliptic abstract equations are considered. A few works are concerned with non regular
problems.
In this paper we establish conditions guaranteeing that non local boundary
value problem for elliptic abstract differential equation of the second order in an
interval is coercive solvable in the Hilbert space L2 (0, 1; H). A coercive estimates, when the problem is regular, was proved in [1, 3]. The considered problem is not regular, since the boundary conditions are non local, similar problems
have been considered in [4, 5, 7, 22]. Moreover, we prove the completeness of
root functions. The completeness of root functions for regular boundary value
problems are proved in [1, 6, 10, 14] and in the book [22].
The obtained results are then applied to study a non local boundary value
problem for the Laplace equation in a cylinder.
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
2.
Notations and Definitions
Let H be a Hilbert space, A a linear closed operator in H and D(A) its domain.
We denote by L(H) the space of bounded linear operators acting on H, with
the usual operator norm, and by Lp (0, 1; H) the space of strongly measurable
functions x → u(x) : [0, 1] → H, whose pth −power are summable, with the
norm
Z 1
p
p
kuk0,p = kukLp (0,1;H) =
ku(x)kpH dx < ∞, p ∈ [1, ∞] .
0
Now, introduce the Lp (0, 1; H) vector-valued Sobolev spaces Wp2 (0, 1; H1 , H),
where H1 , H are Hilbert spaces such that H1 ⊂ H with continuous embedding
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Wp2 (0, 1; H1 , H) = {u : u00 ∈ Lp (0, 1; H) and u ∈ Lp (0, 1; H1 )}
Aissa Aibeche
kukWp2 (0,1;H1 ,H) = kukLp (0,1;H1 ) + ku00 kLp (0,1;H) < ∞.
Title Page
and
Let E0 , E1 be two Banach spaces, which are continuously injected in the Banach space E, the pair {E0 , E1 } is said to be an interpolation couple. Consider
the Banach space
E0 + E1 = {u : u ∈ E, ∃uj ∈ Ej , j = 0, 1, with u = u0 + u1 } ,
kukE0 +E1 =
inf
ku0 kE0 + ku1 kE1 ,
u=u0 +u1 ;uj ∈Ej
JJ
J
II
I
Go Back
Close
Quit
and the functional
K(t, u) =
Contents
inf
u=u0 +u1 ;uj ∈Ej
ku0 kE0 + t ku1 kE1 .
Page 4 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
The interpolation space for the couple {E0 , E1 } is defined, by the K-method,
as follows
(
)
Z ∞
p1
(E0 , E1 )θ,p = u : u ∈ E0 + E1 , kukθ,p =
t−1−θp K p (t, u)dt
<∞ ,
0
0 < θ < 1, 1 ≤ p ≤ ∞.
(
(E0 , E1 )θ,∞ =
)
u : u ∈ E0 + E1 , kukθ,∞ = sup t−θ K(t, u)dt < ∞ ,
t∈(0,∞)
0 < θ < 1. Let A be a closed operator in H. H(A) is the domain of A provided
with the Hilbertian graph norm
kuk2H(A) = kAuk2 + kuk2 , u ∈ D(A).
kukpθ,p
Z
=
Aissa Aibeche
Title Page
If −A is the infinitesimal generator of the semigroup exp(−xA) which is analytic for x > 0, decreasing at infinity and strongly continuous for x ≥ 0, then
the following holds [19, p. 96]:
n
o
(H, H(An ))θ,p = u : u ∈ H, kukpθ,p < ∞ ,
where
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Contents
JJ
J
II
I
Go Back
Close
∞
−n(1−θ)p−1
t
n
p
p
kA exp(−tA)uk dt + kuk ,
Quit
0
0 < θ < 1, n ∈ N, 1 ≤ p < ∞ and kukθ,p its norm.
Page 5 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
Let H and H1 be Hilbert spaces such that the imbedding H1 ⊂ H is continuous and H1 = H. Then (H, H1 )θ,2 , is a Hilbert space, we denote it by (H, H1 )θ .
It is known that (H, H1 )θ = H(S θ ), where S is a self-adjoint positive-definite
operator in H [17].
+∞
1 R
Let F f (σ) = (2π)− 2
exp(iσx)f (x)dx be the Fourier transform of the
−∞
function f.
Lemma 2.1. [22, p. 300] Let A be a self adjoint and positive definite operator
in H. Then
i
h
i
h
1
1 1. ∃ω > 0, Aα exp −x (A + λI) 2 ≤ C exp −ωx |λ| 2 for all α ∈ R,
x ≥ x0 > 0, |arg λ| ≤ ϕ < π, where C does not depend on x and λ.
i h
R1
1
α− 14 2
α
2α− 12
2
2
2. 0 (A + λI) exp −x (A + λI) u ≤ C A
u + |λ|
kuk
1
for all α ≥ 14 , |arg λ| ≤ ϕ < π, and u ∈ D Aα− 4 , where C does not
depend on u and λ.
α
−β 3. A (A + λI) ≤ C (1 + |λ|)α−β , for all 0 ≤ α ≤ β, |arg λ| ≤ ϕ <
π, where C does not depend on λ.
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
3.
3.1.
Solvability of the Principal Problem
Homogeneous Problem
Consider in the Hilbert space H the boundary value problem for the second
order abstract differential equation
(3.1)
L(D)u = −u00 (x) + Au(x) + A(x)u(x) = f (x)
L1 u = δu(0)
(3.2)
x ∈ (0, 1),
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
= f1 ,
L2 u = u0 (1) + Bu(0) = f2 ,
Aissa Aibeche
A, A(x), B are linear operators and δ is a complex number.
Looking to the principal part of the problem (3.1), (3.2) with a parameter
(3.3)
00
L(D)u = −u (x) + (A + λI)u(x) = 0
L1 u = δu(0)
(3.4)
x ∈ (0, 1),
= f1 ,
L2 u = u0 (1) + Bu(0) = f2 .
Theorem 3.1. Assume that the following conditions are satisfied
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
1. A is a self-adjoint and positive-definite operator in H.
Page 7 of 29
2. δ 6= 0.
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
3. B is continuous from H(A1/2 ) in H(A) and from H in H(A1/2 ).
Then the problem (3.3), (3.4) for f1 ∈ (H, H(A)) 1 , f2 ∈ (H, H(A)) 3 and
4
4
for λ such that |arg λ| ≤ φ < π, |λ| −→ ∞, has a unique solution in the
space W22 (0, 1; H(A), H), and for the solution of the problem (3.3), (3.4) the
following coercive estimate holds
(3.5)
ku00 kL2 (0,1;H) + kAukL2 (0,1;H) + |λ| kukL2 (0,1;H)
3 3
≤ C A 4 f1 + |λ| 4 kf1 kH
H
1 1
+ A 4 f2 + |λ| 4 kf2 kH ,
H
Aissa Aibeche
where C does not depend on λ.
Proof. The solution u, belonging to W22 (0, 1; H(A), H), of the equation (3.3)
is in the form
1
(3.6)
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
−xAλ2
u(x) = e
Contents
1
−(1−x)Aλ2
g1 + e
Title Page
g2
with Aλ = A + λI and g1 , g2 ∈ (H, H(A)) 3 .
4
Indeed, let u ∈ W22 (0, 1; H(A), H) be a solution of (3.3). Then we have
1
1
D − Aλ2
D + Aλ2 u(x) = 0.
JJ
J
II
I
Go Back
Close
Quit
Note by
1
v(x) = D + Aλ2 u(x).
Page 8 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
1
From [22, p. 168] v ∈ W21 (0, 1; H(A 2 ), H) and
1
2
(3.7)
D − Aλ v(x) = 0.
So
1
2
v(x) = e−(1−x)Aλ v(1),
(3.8)
where, according to [19, p. 44],
1
1
v(1) ∈ H(A 2 ), H 1 = H, H(A 2 ) 1 .
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
2
2
From (3.7) , (3.8) we have
1
−xAλ2
Aissa Aibeche
Z
x
1
2
1
2
e−(x−y)Aλ e−(1−y)Aλ v(1)dy
0
1
1
1
1
1
2
− 12
−xAλ2 −Aλ2
−xAλ
−(1−x)Aλ2
−e
e
v(1),
u(0) + Aλ
e
=e
2
u(x) = e
u(0) +
where u(0) ∈ (H(A), H) 1 [19, p. 44]. Now,
2
1
1
A 2 : (H, H(A)) 3 → (H, H(A)) 1 = H, H(A 2 ) 1
4
4
2
is an isomorphism. Consequently the last inequality is in the form (3.6).
Let us show the reverse, i.e. the function u in the form (3.6) with g1 and
g2 in (H, H(A)) 3 , belongs to W22 (0, 1; H(A), H). From interpolation spaces
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 29
4
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
properties see [15], [19, p. 96] and the expression (3.6) of the function u we
have
kukW 2 (0,1;H(A),H)
2
+1
≤ AA−1
λ

2 ! 21
 Z 1
1
Aλ e−xAλ2 g1 dx
×
+

Z
0
1
0

2 ! 12 
1
Aλ e−(1−x)Aλ2 g2 dx

Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
≤ C kg1 k(H,H(Aλ )) 3 + kg2 k(H,H(Aλ )) 3
4
4
(3.9) ≤ C(λ) kg1 k(H,H(A)) 3 + kg2 k(H,H(A)) 3 .
4
Aissa Aibeche
4
The function u satisfies the boundary conditions (3.4) if

1

 δg1 + δe−Aλ2 g2
= f1 ,


1
2
1
−Aλ2
−Aλ e
Contents
JJ
J
1
1
2
g1 + Bg1 + Aλ g2 + Be
which we can write in matrix form as:

 
δI 0
0

+
(3.10) 

1
1
1
2
B Aλ2
−Aλ2 e−Aλ
Title Page
−Aλ2
g2 = f2 ,
II
I
Go Back
1
−Aλ2
δe
Be
1
−Aλ2



g1
!
f1
=
g2
Close
!
.
f2
Quit
Page 10 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
The first matrix of operators is invertible, its inverse is
 1

I
0
δ

.
(3.11)
1
− 12
1 −2
− δ Aλ B Aλ
Multiplying the two members of (3.10) by the matrix inverse (3.11), we get the
following system:

1

 g1 + e−Aλ2 g2 = 1δ f1


1
−Aλ2
e
g1 + g2 =
−1
− 1δ Aλ 2 Bf1
+
−1
Aλ 2 f2
we can solve it by Cramer’s method, because the coefficients of the linear sys1
2
tem are bounded linear operators. The determinant is givenby I + e−2Aλ which
−2A 21 is invertible as a little perturbation of unity, in fact e λ ≤ q < 1.
Hence the solution is written as

 g1 = 1δ f1 + R11 (λ)f1 + R12 (λ)f2 ,
(3.12)
 g = − 1 (I + T (λ)) A− 12 Bf + (I + T (λ)) A− 12 f + R (λ)f ,
2
1
2
21
1
λ
λ
δ
where Rij (λ) are given by

1
1
− 12 −A 2

1
1
−2Aλ2

λ B,
R
(λ)
=
−
(I
+
T
(λ))
e
+
(I
+
T
(λ))
A
e

11
λ
δ
δ



1
1
−
2
R12 (λ) = − (I + T (λ)) Aλ 2 e−Aλ ,




1

 R (λ) = 1 (I + T (λ)) e−Aλ2 ,
21
δ
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
and satisfy kRij (λ)k → 0 when |λ| → ∞. (I + T (λ)) is the inverse of I +
1
2
e−2Aλ obtained from the corresponding Neumann series.
Finally the solution u is given by
1
−xAλ2
u(x) = e
1
f1 + R11 (λ)f1 + R12 (λ)f2
δ
1
1
−1
−(1−x)Aλ2
+e
− (I + T (λ)) Aλ 2 Bf1
δ
+ (I +
−1
T (λ)) Aλ 2 f2
+ R21 (λ)f1 .
From the assumptions of Theorem 3.1 and the properties of interpolation spaces,
the following applications are continuous,
−1
(I + T (λ)) Aλ 2 B : (H, H(A)) 3 7−→ (H, H(A)) 3 ,
4
4
Aissa Aibeche
Title Page
Contents
−1
(I + T (λ)) Aλ 2 : (H, H(A)) 1 7−→ (H, H(A)) 3 .
4
4
Then we have the estimates
− 21
(I + T (λ)) Aλ Bf1 (H,H(A)) 3
≤ C kf1 k(H,H(A)) 3
−1
(I + T (λ)) Aλ 2 f2 (H,H(A)) 3
4
JJ
J
II
I
Go Back
4
4
and
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Close
≤ C kf2 k(H,H(A)) 1 .
4
Quit
Page 12 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
Set u(x) = u1 (x) + u2 (x) + u3 (x), where
1
1
2
u1 (x) = e−xAλ f1 ,
δ
1 −(1−x)A 12 −1
λ
(I + T (λ)) Aλ 2 Bf1 ,
u2 (x) = − e
δ
1 − 12
−(1−x)Aλ2
u2 (x) = e
(I + T (λ)) Aλ f2 .
Then
ku1 kW 2 (0,1;H(A),H)
2
1
−xA 12 1
1 2
−xA
Ae λ f1 Aλ e λ f1 .
+
=
|δ| |δ| L2 (0,1;H)
L2 (0,1;H)
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
However, from Lemma 2.1, we have
3 1
3
4 Aλ e−xAλ2 f1 4 kf k
≤
C
A
f
+
|λ|
1
1 H .
L2 (0,1;H)
H
Non Homogeneous Problem
L0 (λ, D)u = −u00 (x) + Aλ u(x) = f (x)
II
I
Go Back
Consider, now, the principal problem for the non homogeneous equation with a
parameter
(3.13)
Contents
JJ
J
Similarly we obtain bounds for u2 and u3 .
3.2.
Title Page
x ∈ (0, 1),
Close
Quit
Page 13 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
L10 u = δu(0)
(3.14)
= f1 ,
L20 u = u0 (1) + Bu(0) = f2 .
We have the result.
Theorem 3.2. Suppose the following conditions satisfied
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
1. A is a self-adjoint and positive-definite operator in H.
2. B is continuous from H(A1/2 ) in H(A) and from H in H(A1/2 ).
3. δ 6= 0.
Aissa Aibeche
Then the problem (3.13), (3.14), for f, f1 and f2 in L2 (0, 1; H), (H, H(A)) 3
4
and (H, H(A)) 1 respectively, and for λ such that |arg λ| ≤ φ < π, |λ| −→
4
∞, has a unique solution belonging to the space
(1, ∞), and the following coercive estimate holds
(3.15)
Wp2 (0, 1; H(A), H),
for p ∈
ku00 kL2 (0,1;H) + kAukL2 (0,1;H) + |λ| kukL2 (0,1;H)
3 ≤ C kf kL2 (0,1;H) + A 4 f1 H1 3
1
4 4
4
+ |λ| kf1 kH + A f2 + |λ| kf2 kH ,
H
where C does not depend on λ.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
Proof. In Theorem 3.1, we proved the uniqueness. Let us now show that the
solution of the problem (3.13), (3.14) belonging to Wp2 (0, 1; H(A), H) can be
written in the form u(x) = u1 (x) + u2 (x), u1 (x) is the restriction to [0, 1] of
u
e1 (x), where u
e1 (x) is the solution of the equation
(3.16)
L0 (λ, D)e
u1 (x) = fe(x),
x ∈ R,
with fe(x) = f (x) if x ∈ [0, 1] and fe(x) = 0 otherwise. u2 (x) is the solution of
the problem
(3.17)
L0 (λ, D)u2 = 0, L10 u2 = f1 − L10 u1 , L20 u2 = f2 − L20 u1 .
The solution of the equation (3.16) is given by the formula
Z
1
b
(3.18)
u
b1 (x) = √
eiµx L0 (λ, iµ)−1 fe(µ)dµ
2π R
b
where fe is the Fourier transform of the function fe(x), L0 (λ, s) is the characteristic pencil of the equation (3.16) i.e. L0 (λ, s) = −s2 I + A + λI.
From (3.18) and Plancherel equality it follows that:
|λ| ku1 kL2 (0,1;H) + ku001 kL2 (0,1;H) + kAu1 kL2 (0,1;H)
(3.19)
≤ |λ| kb
u1 kL2 (R;H(A)) + kb
u001 kL2 (R;H) + kAb
u1 kL2 (R;H)
≤ |λ| F −1 L0 (λ, iµ)−1 F fe(µ)
L2 (R;H)
−1
2
−1 e
+ F (iµ) L0 (λ, iµ) F f (µ)
L2 (R;H)
−1
−1 e
,
+ F AL0 (λ, iµ) F f (µ)
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 15 of 29
L2 (R;H)
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
where F is the Fourier transform.
From condition (1) of Theorem 3.2, for |arg λ| ≤ ϕ < π and |λ| sufficiently
large, we have
L0 (λ, iµ)−1 = (A + λI + µ2 I)−1 (3.20)
≤ C(1 + λ + µ2 )−1 ≤ C |µ|−2
(3.21)
AL0 (λ, iµ)−1 = A(A + λI + µ2 I)−1 ≤ C
(3.22)
|λ| L0 (λ, iµ)−1 = |λ| (A + λI + µ2 I)−1 ≤ C |λ| (1 + λ + µ2 )−1 ≤ C.
Aissa Aibeche
Title Page
Then it follows that
|λ| ku1 kL2 (0,1;H) + ku001 kL2 (0,1;H) + kAu1 kL2 (0,1;H) ≤ C kf kL2 (0,1;H) .
Since u1 ∈
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
W22 (0, 1; H(A), H)
and from [19, p. 44] we have
u01 (0) ∈ (H(A), H) 3 = (H, H(A)) 1 ,
4
JJ
J
II
I
Go Back
4
Close
u1 (0) ∈ (H, H(A)) 3 .
4
Quit
Therefore L10 u1 ∈ (H, H(A)) 3 and L20 u1 ∈ (H, H(A)) 1 .
4
Contents
4
Page 16 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
From Theorem 3.1, the problem (3.17) , when |arg λ| ≤ φ < π, |λ| →
∞, has a solution u2 (x) which is in W22 (0, 1; H(A), H). Now, we have to find
bounds for the following terms
3
3
4
4
A L10 u1 = A u1 (0) ,
H
H
3
3
|λ| 4 kL10 u1 kH = |λ| 4 ku1 (0)kH ,
1
1
1
4
A L20 u1 ≤ A 4 u01 (1) + A 4 Bu1 (0)
H
and
1
H
1
H
1
|λ| 4 kL20 u1 kH ≤ |λ| 4 ku01 (1)kH + |λ| 4 kBu1 (0)kH .
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
For example, we have
3
|λ| 4 ku1 (0)kH ≤ C ku1 kW 2 (0,1;H(A),H) + |λ| ku1 kL2 (0,1;H)
2
≤ C kf kL2 (0,1;H) .
Similarly, we get the other bounds and by the same way the coerciveness estimate.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 17 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
4.
Solvability of the General Problem
Consider, now, the general problem with a parameter
(4.1) L0 (λ, D)u = λu(x)−u00 (x)+Au(x)+A(x)u(x) = f (x)
L10 u = δu(0)
(4.2)
x ∈ (0, 1),
= f1 ,
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
L20 u = u0 (1) + Bu(0) = f2 .
We have the result.
Theorem 4.1. Suppose the following conditions satisfied
Aissa Aibeche
1. A is a self-adjoint and positive-definite operator in H.
Title Page
2. The imbedding H(A) ⊂ H is compact.
3. B is continuous from H(A
1/2
) in H(A) and from H in H(A
Contents
1/2
).
4. δ 6= 0.
5. kA(.)ukL2 (0,1;H) ≤ kAukL2 (0,1;H) + C() kukL2 (0,1;H) .
Then the problem (4.1), (4.2), for f, f1 and f2 in L2 (0, 1; H), (H, H(A)) 3
4
and (H, H(A)) 1 respectively, and for λ such that |arg λ| ≤ φ < π, |λ| −→
4
JJ
J
II
I
Go Back
Close
Quit
Page 18 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
∞, has a unique solution belonging to the space Wp2 (0, 1; H(A), H), for p ∈
(1, ∞), and the following coercive estimate holds
(4.3)
ku00 kL2 (0,1;H) + kAukL2 (0,1;H) + |λ| kukL2 (0,1;H)
3 ≤ C kf kL2 (0,1;H) + A 4 f1 H1 3
1
+ |λ| 4 kf1 kH + A 4 f2 + |λ| 4 kf2 kH .
H
where C does not depend on u, f, f1, f2, and λ.
Proof. Let u be a solution of (4.1), (4.2) belonging to W22 (0, 1; H(A), H).
Then u is a solution of the problem

x ∈ (0, 1),

 L0 (λ, D)u = f (x) − A(x)u(x)
L
u
=
δu(0)
=
f1
10
(P0 )


L20 u = u0 (1) + Bu(0) = f2 .
From Theorem 3.2, we get the estimate
ku00 kL2 (0,1;H) + kAukL2 (0,1;H) + |λ| kukL2 (0,1;H)
3 ≤ C kf − A(.)ukL2 (0,1;H) + A 4 f1 1 H
3
1
+ |λ| 4 kf1 kH + A 4 f2 + |λ| 4 kf2 kH .
H
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 19 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
Using condition (5) of Theorem 3.2, we get
ku00 kL2 (0,1;H) + (1 − C) kAukL2 (0,1;H) + (|λ| − C.C()) kukL2 (0,1;H)
3 ≤ C kf kL2 (0,1;H) + A 4 f1 H1 3
1
4 4
4
+ |λ| kf1 kH + A f2 + |λ| kf2 kH .
H
Choosing such that C · < 1, the coerciveness estimates follows easily.
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 20 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
5.
Completeness of Root Functions
Let us define the operator L by
Lu ≡ −u00 + Au,
D (L) = W22 (0, 1; H(A), H, Lk u = 0, k = 0, 1) .
Lemma 5.1. Suppose that sj (I, H(A), H) ' Cj −q then
sj
I, W22
− 11
(0, 1; H(A), H) , L2 (0, 1; H) ' Cj 2 +q .
I (resp. I) is the imbedding of H(A) in H (resp. of W22 (0, 1; H(A), H) in
L2 (0, 1; H)) and sj (I, H(A), H) are the s-numbers of the operator I from
H(A) to H.
Proof. Let S1 be the operator defined in L2 (0, 1) such that S1 = S1∗ ≥ γ 2 I,
D(S1 ) = H(S1 ) = W22 (0, 1). From [17], we know that if H1 ⊂ H and H1 is
dense in H then there exists an operator S1 such that S1 = S1∗ and D(S1 ) = H.
Otherwise, let S2 be the operator defined by S2 = S2∗ ≥ γ 2 I, D(S2 ) = H(A). If
we define the operator S on L2 (0, 1)⊗H = L2 (0, 1; H) by S = S1 ⊗I2 +I1 ⊗S2 ,
where I1 (resp. I2 ) is the identity operator in L2 (0, 1) (resp. H). We have
sj S1−1 , L2 (0, 1), L2 (0, 1) ' sj (I, H(S1 ), L2 (0, 1)) ' Cj −2 ,
sj S2−1 , H, H ' sj (I, H(A), H) ' Cj −q .
Hence, from [11], we obtain sj (S −1 , L2 (0, 1; H), L2 (0, 1; H)) ' Cj
−
1
1 +q
2
.
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 21 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
Theorem 5.2. Let the conditions of Theorem 3.2 hold along with A−1 ∈ σq (H) ,
q > 0. Then, the system of root functions of the operator L is complete in
L2 (0, 1; H).
Proof. From Theorem 4.1, we have kR(λ, L)k ≤ C |λ|−1 for |arg λ| ≤ ϕ < π
and |λ| sufficiently large. Using Lemma 5.1, we have R(λ, L) ∈ σp (L2 (0, 1; H))
for p > 12 + 1q , so, for the operator L, all the conditions of [8, Theorem 126, 2.3,
p. 50], are fulfilled. This achieves the proof of the theorem.
Theorem 5.3. Suppose that the conditions of Theorem 5.2 are satisfied as well
as the condition D(A(x)) ⊂ D(A) and ∀ > 0, kA(·)ukH ≤ kAukH +
C() kukH . u ∈ D(A). Let A be the operator defined by (Au) (x) = A(x)u(x),
D (A) = L2 (0, 1; H). Then the system of root functions of L + A is complete in
L2 (0, 1; H).
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
Proof. It is clear that
Title Page
kAukL2 (0,1;H) ≤ kAukL2 (0,1;H) + C() kukL2 (0,1;H) .
Since by Theorem 4.1, we have
kAukL2 (0,1;H) ≤ C kf kL2 (0,1;H) = C k(L − λI) ukL2 (0,1;H) ,
Contents
JJ
J
II
I
Go Back
hence
kAukL2 (0,1;H) ≤ k(L − λI) ukL2 (0,1;H) + C() kukL2 (0,1;H)
and so, for |λ| sufficiently large and |arg λ| ≤ ϕ < π,
Close
Quit
Page 22 of 29
R(λ, L + A) ∈ σp (L2 (0, 1; H)) ,
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
and from Theorem 4.1 we have
kR(λ, L + A)k ≤ C |λ|−1
for |λ| sufficiently large and |arg λ| ≤ ϕ < π. Then the system of root functions
is complete in L2 (0, 1; H).
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 23 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
6.
Application
Let us consider, in the cylindrical domain Ω = [0, 1]×G the non local boundary
value problem for the Laplace equation with a parameter

L(λ)u = λu(x, y) − ∆u(x, y) + b(x, y)u(x, y)




= f (x, y),
(x, y) ∈ Ω;





y ∈ G;
(6.1) (P ) L1 u = δu(0, y) = f1 (y),


∂


L2 u = ∂x
u(1, y) + Bu(0, y) = f2 (y),
y ∈ G;





P u = u(x, y 0 ) = 0,
(x, y 0 ) ∈ Γ,
where Γ = [0, 1] × ∂G and ∂G is the boundary of G.
A number λ0 is called an eigenvalue of (P ) if the problem

L(λ0 )u = 0, (x, y) ∈ Ω;






 L1 u = 0,
y ∈ G;
0
(6.2)
(P )


L2 u = 0,
y ∈ G;





P u = 0,
(x, y 0 ) ∈ Γ,
has a non trivial solution that belongs to W22 (Ω). The non trivial solution u0 of
(P 0 ) that belongs to W22 (Ω) is called the eigenfunction of (P ) corresponding to
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 24 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
the eigenvalue λ0. Solutions uk of

L(λ0 )uk + uk−1 = 0,






 L1 uk = 0,
00
(6.3)
(P )


L2 uk = 0,





P uk = 0,
(x, y) ∈ Ω;
y ∈ G;
y ∈ G;
(x, y 0 ) ∈ Γ,
belonging to W22 (Ω) are associated functions of the k−th rank to the eigenvalue
u0 of (P ) . Eigenfunctions and associated functions of (P ) are gathered under
the general name of root functions of (P ) .
0,1
Theorem 6.1. Let b(x, y) ∈ W∞
(Ω), δ 6= 0, ∂G ∈ C 2 then
−
W2
mk
2
Aissa Aibeche
+ 34
(G, P u = 0) and for λ
1. (P ) , for f ∈ W20,1 (Ω, P u = 0), fk ∈
such that |λ| sufficiently large and |arg λ| ≤ ϕ < π, has a unique solution
that belongs to the space W22 (Ω), and for this solution we have the coercive
estimate
2
≤C
kf kL2 (Ω) +
k=1
Title Page
Contents
JJ
J
|λ| ku00 kL2 (Ω) + kukW 2 (Ω)
2
X
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
kfk k
II
I
Go Back
m
− 2k + 3
4 (G,P u=0)
W2
+
2
X
k=1
m
− 2k
|λ|
Close
!
+ 34
kfk kH
,
Quit
Page 25 of 29
where C does not depend on u and λ.
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
2. Root functions of (P ) are complete in L2 (Ω).
Proof. Consider in H = L2 (Ω) the operators A and A(x) defined by
Au = −∆u(y) + λ0 u(y),
A (x) u = b(x, y)u(y) − λ0 u(y),
D(A) = W22 (G, P u = 0)
D(A (x)) = W21 (G, P u = 0, m = 0).
Then the problem (P ) can be written in the form


λu(x) − u00 (x) + Au(x) + A(x)u(x) = 0 x ∈ (0, 1),



δu(0)
= f1




u0 (1) + Bu(0)
= f2 .
We have the compact imbedding W22 (Ω) ⊂ L2 (Ω). On the other hand
2
sj I, W22 (Ω), L2 (Ω) ' j − r+1 .
By virtue of Lemma 3.1 in [21, p. 60] we have
sj (I, H(A), L2 (Ω)) ' sj A−1 , L2 (Ω), L2 (Ω) .
W22 (Ω),
−1
−1
Since H(A) ⊂
then it follows that A ∈ σp (L2 (Ω), L2 (Ω)) , then
kR(λ, A)k ≤ C |λ| for |arg λ| ≤ ϕ < π and |λ| sufficiently large. Hence, all
conditions of Theorem 5.3 has been checked.
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 26 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
References
[1] S. AGMON, On the eigenfunctions and eigenvalues of general boundary
value problems, Comm. Pure Appl. Math. 15(1962), 119–147.
[2] S. AGMON AND L. NIRENBERG, Properties of solutions of ordinary differential equation in Banach spaces, Comm. Pure Appl. Math., 16 (1963),
121–239.
[3] M.S. AGRANOVIĆ AND M.L.VISIK, Elliptic problems with a parameter
and parabolic problems of general type, Russian Math. Surveys, 19 (1964),
53–161.
[4] A. AIBECHE, Coerciveness estimates for a class of elliptic problems, Diff.
Equ. Dynam. Sys., 4 (1993), 341–351.
[5] A. AIBECHE, Completeness of generalized eigenvectors for a class of
elliptic problems, Math. Results, 31 (1998), 1–8.
[6] F.E. BROWDER, On the spectral theory of elliptic differential operators I,
Math. Ann., 142 (1961), 22–130.
[7] M. DENECHE, Abstract differential equation with a spectral parameter in
the boundary conditions, Math. Results, 35 (1999), 217–227.
[8] N. DUNFORD
science, 1963.
AND
J.T. SCHWARTZ, Linear Operators Vol.II, Inter-
[9] H. O. FATTORINI, Second Order Linear Differential Equations in Banach
Spaces, North-Holland, Amsterdam, 1985.
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 27 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
[10] G. GEYMONAT AND P. GRISVARD, Alcuni risultati di teoria spettrale,
Rend. Sem. Mat. Univ. Padova, 38 (1967), 121–173.
[11] V.I. GORBACHUK AND M.I. GORBACHUK, Boundary Value Problems
for Operator Differential Equations, Kluwer Academic Publishers, 1991.
[12] P. GRISVARD, Caractérisation de quelques espaces d’interpolation, Arch.
Rat. Mech. Anal., 25 (1967), 40–63.
[13] T. KATO, Perturbation Theory for Linear Operators, Springer-Verlag,
New-York, 1966.
[14] M.V. KELDYSH, On the eigenvalues and eigenfunctions of certain class
of non self-adjoint equations, Dokl. Akad. Nauk. SSSR, 77 (1951), 11–14.
[15] H. KOMATSU, Fractional powers of operators II. Interpolation spaces,
Pacif. J. Math., 21 (1967), 89–111.
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
Title Page
[16] S.G. KREIN, Linear Differential Equations in Banach Spaces, Providence,
1971.
[17] J.L. LIONS AND E. MAGENES, Problèmes aux Limites non Homogènes
et Applications, Vol. I, Dunod, 1968.
[18] H. TANABE, Equations of Evolution, Pitman, London, 1979.
Contents
JJ
J
II
I
Go Back
[19] H. TRIEBEL, Interpolation Theory, Functions spaces, Differential Operators, North Holland, Amsterdam, 1978.
Close
[20] S.Y. YAKUBOV, Linear Differential Equations and Applications (in Russian), Baku, elm, 1985.
Page 28 of 29
Quit
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003
[21] S.Y. YAKUBOV, Completeness of Root Functions of Regular Differential
Operators, Longman, Scientific and Technical, New-York, 1994.
[22] S.Y. YAKUBOV AND Y.Y. YAKUBOV, Differential-Operator Equations.
Ordinary and Partial Differential Equations, CRC Press, New York, 1999.
Coerciveness Inequality for
Nonlocal Boundary value
Problems for Second Order
Abstract Elliptic Differential
Equations
Aissa Aibeche
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 29 of 29
J. Ineq. Pure and Appl. Math. 4(2) Art. 43, 2003