Journal of Inequalities in Pure and
Applied Mathematics
STRONGLY NONLINEAR ELLIPTIC UNILATERAL PROBLEMS
IN ORLICZ SPACE AND L1 DATA
volume 6, issue 2, article 54,
2005.
L. AHAROUCH AND M. RHOUDAF
Département de Mathématiques et Informatique
Faculté des Sciences Dhar-Mahraz
B.P. 1796 Atlas, Fès, Maroc.
Received 21 December, 2004;
accepted 06 April, 2005.
Communicated by: A. Fiorenza
EMail: l_aharouch@yahoo.fr
EMail: rhoudaf_mohamed@yahoo.fr
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
250-04
Quit
Abstract
In this paper, we shall be concerned with the existence result of Unilateral problem associated to the equations of the form,
Au + g(x, u, ∇u) = f,
where A is a Leray-Lions operator from its domain D(A) ⊂ W01 LM (Ω) into
W −1 EM (Ω). On the nonlinear lower order term g(x, u, ∇u), we assume that
it is a Carathéodory function having natural growth with respect to |∇u|, and
satisfies the sign condition. The right hand side f belongs to L1 (Ω).
2000 Mathematics Subject Classification: 35J60.
Key words: Orlicz Sobolev spaces, Boundary value problems, Truncations, Unilateral problems.
The authors would like to thank the referee for his comments.
Contents
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4
Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
References
Close
Quit
Page 2 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
1.
Introduction
Let Ω be an open bounded subset of RN , N ≥ 2, with segment property. Let us
consider the following nonlinear Dirichlet problem
(1.1)
− div(a(x, u, ∇u)) + g(x, u, ∇u) = f,
where f ∈ L1 (Ω), Au = − div a(x, u, ∇u) is a Leray-Lions operator defined
on its domain D(A) ⊂ W01 LM (Ω), with M an N -function and where g is a
nonlinearity with the "natural" growth condition:
|g(x, s, ξ)| ≤ b(|s|)(h(x) + M (|ξ|))
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
and which satisfies the classical sign condition
g(x, s, ξ) · s ≥ 0.
Title Page
Contents
−1
In the case where f ∈ W EM (Ω), an existence theorem has been proved in
[14] with the nonlinearities g depends only on x and u, and in [4] where g
depends also the ∇u.
For the case where f ∈ L1 (Ω), the authors in [5] studied the problem (1.1),
with the added assumption of exact natural growth
JJ
J
II
I
Go Back
Close
|g(x, s, ξ)| ≥ βM (|ξ|) for |s| ≥ µ
Quit
and in [6] no coercivity condition is assumed on g but the result is restricted
to the N -function, M satisfying a ∆2 -condition, while in [11] the authors were
concerned about the above problem without assuming a ∆2 -condition on M .
Page 3 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
The purpose of this paper is to prove an existence result for unilateral problems associated to (1.1) without assuming the ∆2 -condition in the setting of the
Orlicz-Sobolev space.
Further work for the equation (1.1) in the Lp case where there is no restriction
can be found in [17], and in [12, 9, 8] in the case of obstacle problems, see also
[18].
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
2.
Preliminaries
Let M : R+ → R+ be an N -function, i.e. M is continuous, convex, with
M (t) > 0 for t > 0, Mt(t) → 0 as t → 0 and Mt(t) → ∞ as t → ∞.
Rt
Equivalently, M admits the representation: M (t) = 0 a(s)ds where a :
R+ → R+ is nondecreasing, right continuous, with a(0) = 0, a(t) > 0 for
t > 0 and a(t) tends to ∞ as t → ∞.
Rt
The N -function M conjugate to M is defined by M = 0 ā(s)ds, where
ā : R+ → R+ is given by ā(t) = sup{s : a(s) ≤ t}.
The N -function M is said to satisfy the ∆2 -condition if, for some k
M (2t) ≤ kM (t),
(2.1)
∀t ≥ 0.
When (2.1) holds only for t ≥ some t0 > 0 then M is said to satisfy the ∆2 condition near infinity.
We will extend these N -functions to even functions on all R, i.e. M (t) =
M (|t|) if t ≤ 0.
Moreover, we have the following Young’s inequality
∀s, t ≥ 0,
st ≤ M (t) + M (s).
Let P and Q be two N -functions. P Q means that P grows essentially less
P (t)
rapidly than Q, i.e., for each > 0, Q(t)
→ 0 as t → ∞. This is the case if and
−1 (t)
limt→∞ Q
P −1 (t)
only if
= 0.
Let Ω be an open subset of RN . The Orlicz class KM (Ω) (resp. the Orlicz space LM (Ω) is defined as the set of (equivalence classes of) real valued
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
measurable functions u on Ω such that
Z
Z
u(x)
M (u(x))dx < +∞
resp.
M
dx < +∞ for some λ > 0 .
λ
Ω
Ω
LM (Ω) is a Banach space under the norm
Z
u(x)
kukM,Ω = inf λ > 0, M
dx ≤ 1
λ
Ω
and KM (Ω) is a convex subset of LM (Ω).
The closure in LM (Ω) of the set of bounded measurable functions with compact support in Ω is denoted by EM (Ω).
R The dual of EM (Ω) can be identified with LM (Ω) by means of the pairing
uv dx, and the dual norm of LM (Ω) is equivalent to k · kM ,Ω .
Ω
We now turn to the Orlicz-Sobolev space, W 1 LM (Ω) (resp. W 1 EM (Ω)) is
the space of all functions u such that u and its distributional derivatives of order
1 lie in LM (Ω) (resp. EM (Ω)). It is a Banach space under the norm
X
kuk1,M =
kDα ukM .
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
|α|≤1
Go Back
Thus, W 1 LM (Ω) and W 1 EM (Ω) can be identified with subspaces
of the prodQ
uct of N + 1 copies Q
of LM (Ω).
Q Denoting this
Q product
Q by LM , we will use the
weak topologies σ( LM , EM ) and σ( LM , LM ).
The space W01 EM (Ω) is defined as the (norm) closureQof the Q
Schwartz space
1
1
D(Ω) in W EM (Ω) and the space W0 LM (Ω) as the σ( LM , EM ) closure
of D(Ω) in W 1 LM (Ω).
Close
Quit
Page 6 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
Let W −1 LM (Ω) (resp. W −1 EM (Ω)) denote the space of distributions on Ω
which can be written as sums of derivatives of order ≤ 1 of functions in LM (Ω)
(resp. EM (Ω)). It is a Banach space under the usual quotient norm (for more
details see [1]).
We recall some lemmas introduced in [4] which will be used later.
Lemma 2.1. Let F : R → R be uniformly Lipschitzian, with F (0) = 0. Let
M be an N -function and let u ∈ W 1 LM (Ω) (resp. W 1 EM (Ω)). Then F (u) ∈
W 1 LM (Ω) ( resp. W 1 EM (Ω)). Moreover, if the set D of discontinuity points of
F 0 is finite, then
( 0
F (u) ∂x∂ i u a.e. in {x ∈ Ω : u(x) ∈
/ D},
∂
F (u) =
∂xi
0
a.e. in {x ∈ Ω : u(x) ∈ D}.
Lemma 2.2. Let F : R → R be uniformly Lipschitzian, with F (0) = 0. We assume that the set of discontinuity points of F 0 is finite. Let M be an N -function,
then the mapping F : W 1 LM (Ω)Q
→ W 1Q
LM (Ω) is sequentially continuous with
respect to the weak* topology σ( LM , EM ).
We give now the following lemma which concerns operators of Nemytskii
type in Orlicz spaces (see [4]).
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Lemma 2.3. Let Ω be an open subset of RN with finite measure. Let M, P and
Q be N -functions such that Q P , and let f : Ω × R → R be a Carathéodory
function such that, for a.e. x ∈ Ω and all s ∈ R:
Page 7 of 46
|f (x, s)| ≤ c(x) + k1 P −1 M (k2 |s|),
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
Quit
http://jipam.vu.edu.au
where k1 , k2 are real constants and c(x) ∈ EQ (Ω). Then the Nemytskii operator
Nf defined by Nf (u)(x) = f (x, u(x)) is strongly continuous from
1
1
= u ∈ LM (Ω) : d(u, EM (Ω)) <
P EM (Ω),
k2
k2
into EQ (Ω).
We define T01,M (Ω) to be the set of measurable functions u : Ω → R such
that Tk (u) ∈ W01 LM (Ω), where Tk (s) = max(−k, min(k, s)) for s ∈ R and
k ≥ 0. We give the following lemma which is a generalization of Lemma 2.1 of
[2] in Orlicz spaces.
Lemma 2.4. For every u ∈ T01,M (Ω), there exists a unique measurable function
v : Ω −→ RN such that
∇Tk (u) = vχ{|u|<k} , almost everywhere in Ω for every k > 0.
We will define the gradient of u as the function v, and we will denote it by
v = ∇u.
Lemma 2.5. Let λ ∈ R and let u and v be two measurable functions defined
on Ω which are finite almost everywhere, and which are such that Tk (u), Tk (v)
and Tk (u + λv) belong to W01 LM (Ω) for every k > 0 then
∇(u + λv) = ∇(u) + λ∇(v) a.e. in Ω
where ∇(u), ∇(v) and ∇(u+λv) are the gradients of u, v and u+λv introduced
in Lemma 2.4.
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
The proof of this lemma is similar to the proof of Lemma 2.12 in [10] for the
L case.
Below, we will use the following technical lemma.
p
Lemma 2.6. Let (fn ), f ∈ L1 (Ω) such that
(i) fn ≥ 0 a.e. in Ω
(ii) fn → f a.e. in Ω
R
R
(iii) Ω fn (x)dx → Ω f (x)dx
then fn → f strongly in L1 (Ω).
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
3.
Main Results
Let Ω be an open bounded subset of RN , N ≥ 2, with the segment property.
Given an obstacle function ψ : Ω → R, we consider
(3.1)
Kψ = {u ∈ W01 LM (Ω); u ≥ ψ a.e. in Ω},
this convex set is sequentially σ(ΠLM , ΠEM ) closed in W01 LM (Ω) (see [15]).
We now state conditions on the differential operator
(3.2)
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
Au = − div(a(x, u, ∇u))
(A1 ) a(x, s, ξ) : Ω × R × RN → RN is a Carathéodory function.
(A2 ) There exist two N -functions M and P with P M , function c(x) in
EM (Ω), constants k1 , k2 , k3 , k4 such that, for a.e. x in Ω and for all s ∈
R, ζ ∈ RN
|a(x, s, ζ)| ≤ c(x) + k1 P
−1
M (k2 |s|) + k3 M
−1
M (k4 |ζ|).
(A3 ) [a(x, s, ζ) − a(x, s, ζ 0 )](ζ − ζ 0 ) > 0 for a.e. x in Ω, s in R and ζ, ζ 0 in RN ,
with ζ 6= ζ 0 .
(A4 ) There exist δ(x) in L1 (Ω), strictly positive constant α such that, for some
fixed element v0 in Kψ ∩ W01 EM (Ω) ∩ L∞ (Ω),
a(x, s, ζ)(ζ − Dv0 ) ≥ αM (|ζ|) − δ(x)
for a.e. x in Ω, and all s ∈ R, ζ ∈ RN .
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
(A5 ) For each v ∈ Kψ ∩L∞ (Ω) there exists a sequence vn ∈ Kψ ∩W01 EM (Ω)∩
L∞ (Ω) such that vn → u for the modular convergence.
Furthermore let g : Ω × R × RN → R be a Carathéodory function such that
for a.e. x ∈ Ω and for all s ∈ R, ζ ∈ RN
(G1 ) g(x, s, ζ)s ≥ 0
(G2 ) |g(x, s, ζ)| ≤ b(|s|) (h(x) + M (|ζ|)),
where b : R+ → R+ is a continuous non decreasing function, and h is a
given nonegative function in L1 (Ω).
Consider the following Dirichlet problem:
A(u) + g(x, u, ∇u) = f in Ω.
(3.3)
Remark 1. The condition (A5 ) holds if one of the following conditions is verified.
1. There exist ψ ∈ Kψ such that ψ − ψ is continuous in Ω, (see [15, Proposition 9]).
2. ψ ∈
W01 EM (Ω),
(see [15, Proposition 10]).
We shall prove the following existence theorem.
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
Theorem 3.1. Assume that (A1 ) – (A5 ), (G1 ) and (G2 ) hold and f ∈ L1 (Ω).
Then there exists at least one solution of the following unilateral problem,


u ∈ T01,M (Ω), u ≥ ψ a.e. in Ω,





u, ∇u) ∈ L1 (Ω)

 g(x,
R
R
(P)
a(x, u, ∇u)∇Tk (u − v) dx + Ω g(x, u, ∇u)Tk (u − v) dx
Ω

R



≤
f Tk (u − v) dx,

Ω



∀ v ∈ Kψ ∩ L∞ (Ω), ∀k > 0.
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 12 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
4.
Proof of Theorem 3.1
To prove the existence theorem, we proceed by steps.
STEP 1: Approximate unilateral problems.
Let us define
g(x, s, ξ)
gn (x, s, ξ) =
1 + n1 |g(x, s, ξ)|
and let us consider the approximate unilateral problems:

un ∈ Kψ ∩ D(A),




 hAu , u − vi + R g (x, u , ∇u )(u − v)dx
n
n
n
n
n
Ω nR
(Pn )

≤ Ω fn (un − v)dx,



 ∀v ∈ K .
ψ
where fn is a regular function such that fn strongly converges to f in L1 (Ω).
From Gossez and Mustonen ([15, Proposition 5]), the problem (Pn ) has at
least one solution.
STEP 2: A priori estimates.
Let k ≥ kv0 k∞ and let ϕk (s) = se
It is well known that
(4.1)
ϕ0k (s) −
γs2
, where γ =
b(k)
α
2
b(k)
1
|ϕk (s)| ≥ , ∀s ∈ R (see [9]).
α
2
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
.
Close
Quit
Page 13 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
Taking un − ηϕk (Tl (un − v0 )) as test function in (Pn ), where l = k + kv0 k∞ ,
we obtain,
Z
a(x, un , ∇un )∇Tl (un − v0 )ϕ0k (Tl (un − v0 ))dx
Ω
Z
+ gn (x, un , ∇un )ϕk (Tl (un − v0 ))dx
Ω
Z
≤
fn ϕk (Tl (un − v0 ))dx.
Ω
Since
gn (x, un , ∇un )ϕk (Tl (un − v0 )) ≥ 0
on the subset {x ∈ Ω : |un (x)| > k}, then
Z
a(x, un , ∇un )∇(un − v0 )ϕ0k (Tl (un − v0 ))dx
{|un −v0 |≤l}
Z
≤
|gn (x, un , ∇un )||ϕk (Tl (un − v0 ))|dx
{|un |≤k}
Z
+ fn ϕk (Tl (un − v0 ))dx.
Ω
By using (A4 ) and (G1 ), we have
Z
α
M (|∇un |)ϕ0k (Tl (un − v0 )) dx
{|un −v0 |≤l}
Z
≤ b(|k|) (h(x) + M (∇Tk (un ))) |ϕk (Tl (un − v0 ))| dx
Ω
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
Z
+
δ(x)ϕ0k (Tl (un
Ω
Z
− v0 )) dx +
fn ϕk (Tl (un − v0 )) dx.
Ω
Since
{x ∈ Ω, |un (x)| ≤ k} ⊆ {x ∈ Ω : |un − v0 | ≤ l}
and the fact that h, δ ∈ L1 (Ω), further fn is bounded in L1 (Ω), then
Z
M (|∇Tk (un )|)ϕ0k (Tl (un − v0 )) dx
Ω
Z
b(k)
≤
M (|∇Tk (un )|)|ϕk (Tl (un − v0 ))| dx + ck ,
α Ω
where ck is a positive constant depending on k, which implies that
Z
b(k)
0
M (|∇Tk (un )|) ϕk (Tl (un − v0 )) −
|ϕk (Tl (un − v0 ))| dx ≤ ck .
α
Ω
By using (4.1), we deduce,
Z
(4.2)
M (|∇Tk (un )|) dx ≤ 2ck .
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Ω
Since Tk (un ) is bounded in W01 LM (Ω), there exists some vk ∈ W01 LM (Ω) such
that
Q
Q
Tk (un ) * vk weakly in W01 LM (Ω) for σ( LM , EM ),
(4.3)
Tk (un ) → vk strongly in EM (Ω) and a.e. in Ω.
Close
Quit
Page 15 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
STEP 3: Convergence in measure of un
Let k0 ≥ kv0 k∞ and k > k0 , taking v = un − Tk (un − v0 ) as test function in
(Pn ) gives,
Z
(4.4)
a(x, un , ∇un )∇Tk (un − v0 ) dx
Ω
Z
+ gn (x, un , ∇un )Tk (un − v0 ) dx
Ω
Z
fn Tk (un − v0 ) dx,
≤
Ω
since gn (x, un , ∇un )Tk (un − v0 ) ≥ 0 on the subset {x ∈ Ω, |un (x)| > k0 },
hence (4.4) implies that,
Z
Z
a(x, un , ∇un )∇Tk (un − v0 ) dx ≤ k
|gn (x, un , ∇un )| dx+kkf kL1 (Ω)
{|un |≤k0 }
Ω
which gives, by using (G1 ),
Z
(4.5)
a(x, un , ∇un )∇Tk (un − v0 ) dx
Ω
Z
Z
≤ kb(k0 )
|h(x)| dx + M (|∇Tk0 (un )|) dx + kc.
Ω
Ω
Combining (4.2) and (4.5), we have,
Z
a(x, un , ∇un )∇Tk (un − v0 ) dx ≤ k[ck0 + c].
Ω
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 16 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
By (A4 ), we obtain,
Z
M (|∇un |) dx ≤ kc1 ,
{|un −v0 |≤k}
where c1 is independent of k, since k is arbitrary, we have
Z
Z
M (|∇un |) dx ≤
M (|∇un |)dx ≤ kc2 ,
{|un |≤k}
{|un −v0 |≤k+kv0 k∞ }
i.e.,
Z
M (|∇Tk (un )|) dx ≤ kc2 .
(4.6)
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Ω
Now, we prove that un converges to some function u in measure (and therefore, we can always assume that the convergence is a.e. after passing to a suitable subsequence). We shall show that un is a Cauchy sequence in measure.
Let k > 0 large enough, by Lemma 5.7 of [13], there exist two positive
constants c3 and c4 such that
Z
Z
(4.7)
M (c3 Tk (un )) dx ≤ c4 M (|∇Tk (un )|) dx ≤ kc5 ,
Ω
Ω
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
then, we deduce, by using (4.7) that
Page 17 of 46
Z
M (c3 k) meas{|un | > k} =
M (c3 Tk (un )) dx ≤ c5 k,
{|un |>k}
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
hence
(4.8)
meas(|un | > k) ≤
c5 k
∀n, ∀k.
M (c3 k)
Letting k to infinity, we deduce that, meas{|un | > k} tends to 0 as k tends to
infinity.
For every λ > 0, we have
(4.9)
meas({|un − um | > λ}) ≤ meas({|un | > k}) + meas({|um | > k})
+ meas({|Tk (un ) − Tk (um )| > λ}).
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Consequently, by (4.3) we can assume that Tk (un ) is a Cauchy sequence in
measure in Ω.
Let > 0 then, by (4.9) there exists some k() > 0 such that meas({|un −
um | > λ}) < for all n, m ≥ h0 (k(), λ). This proves that (un ) is a Cauchy sequence in measure in Ω , thus converges almost everywhere to some measurable
function u. Then
Q
Q
Tk (un ) * Tk (u) weakly in W01 LM (Ω) for σ( LM , EM ),
(4.10)
Tk (un ) → Tk (u) strongly in EM (Ω) and a.e. in Ω.
Step 4: Boundedness of (a(x, Tk (un ), ∇Tk (un ))n in (LM (Ω))N .
Let w ∈ (EM (Ω))N be arbitrary, by (A3 ) we have
(a(x, un , ∇un ) − a(x, un , w))(∇un − w) ≥ 0,
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 18 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
which implies that
a(x, un , ∇un )(w −∇v0 ) ≤ a(x, un , ∇un )(∇un −∇v0 )−a(x, un , w)(∇un −w)
and integrating on the subset {x ∈ Ω, |un − v0 | ≤ k}, we obtain,
Z
a(x, un , ∇un )(w − ∇v0 ) dx
(4.11)
{|un −v0 |≤k}
Z
≤
a(x, un , ∇un )(∇un − ∇v0 ) dx
{|un −v0 |≤k}
Z
+
a(x, un , w)(w − ∇un ) dx.
{|un −v0 |≤k}
We claim that,
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Z
a(x, un , ∇un )(∇un − v0 ) dx ≤ c10 ,
(4.12)
{|un −v0 |≤k}
where c10 is a positive constant depending on k.
Indeed, if we take v = un − Tk (un − v0 ) as test function in (Pn ), we get,
Z
a(x, un , ∇un )(∇un − ∇v0 ) dx
{|un −v0 |≤k}
Z
+ gn (x, un , ∇un )Tk (un − v0 ) dx
Ω
Z
≤
fn Tk (un − v0 ) dx.
Ω
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 19 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
Since gn (x, un , ∇un )Tk (un − v0 ) ≥ 0 on the subset {x ∈ Ω, |un | ≥ kv0 k∞ },
which implies
Z
(4.13)
a(x, un , ∇un )(∇un − ∇v0 ) dx
Z
Z
≤ b(kv0 k∞ )
h(x) dx + M (∇Tkv0 k∞ (un ) dx + kkf kL1 (Ω) .
{|un −v0 |≤k}
Ω
Ω
Combining (4.2) and (4.13), we deduce (4.12).
On the other hand, for λ large enough, we have by using (A2 )
Z
M
(4.14)
{|un −v0 |≤k}
a(x, un , w)
λ
L. Aharouch and M. Rhoudaf
dx
≤M
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
c(x)
λ
+
k3
M (k4 |w|) + c ≤ c11 ,
λ
hence, |a(x, un , w)| bounded in LM (Ω), which implies that the second term of
the right hand side of (4.11) is bounded
Consequently, we obtain,
Z
(4.15)
a(x, un , ∇un )(w − ∇v0 ) dx ≤ c12 ,
{|un −v0 |≤k}
with c12 is positive constant depending of k.
Hence, by the Theorem of Banach–Steinhaus, the sequence (a(x, un ,
∇un ))χ{|un −v0 |≤k} )n remains bounded in (LM (Ω))N . Since k is arbitrary, we
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 20 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
deduce that (a(x, Tk (un ), ∇Tk (un )))n is also bounded in (LM (Ω))N , which implies that, for all k > 0 there exists a function hk ∈ (LM (Ω))N , such that,
(4.16) a(x, Tk (un ), ∇Tk (un )) * hk
weakly in (LM (Ω))N
for σ(ΠLM (Ω), ΠEM (Ω)).
STEP 5: Almost everywhere convergence of the gradient.
We fix k > kv0 k∞ . Let Ωr = {x ∈ Ω, |∇Tk (u(x))| ≤ r} and denote by χr
the characteristic function of Ωr . Clearly, Ωr ⊂ Ωr+1 and meas(Ω\Ωr ) −→ 0
as r −→ ∞.
Fix r and let s ≥ r, we have,
Z
(4.17)
0≤
[a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u))]
Ωr
× [∇Tk (un ) − ∇Tk (u)]dx
Z
≤
[a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u))]
Ωs
× [∇Tk (un ) − ∇Tk (u)]dx
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Z
[a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u)χs )]
=
Ωs
× [∇Tk (un ) − ∇Tk (u)χs ]dx
Z
≤
[a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u)χs )]
Close
Quit
Page 21 of 46
Ω
× [∇Tk (un ) − ∇Tk (u)χs ]dx.
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
By (A5 ) there exists a sequence vj ∈ Kψ ∩W01 EM (Ω)∩L∞ (Ω) which converges
to Tk (u) for the modular converge in W01 LM (Ω).
Here, we define
h
wn,j
= T2k (un − v0 − Th (un − v0 ) + Tk (un ) − Tk (vj )),
wjh = T2k (u − v0 − Th (u − v0 ) + Tk (u) − Tk (vj ))
and
wh = T2k (u − v0 − Th (u − v0 )),
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
where h > 2k > 0.
For η = exp(−4γk 2 ), we defined the following function as
h
h
(4.18)
vn,j
= un − ηϕk wn,j
.
We take
L. Aharouch and M. Rhoudaf
Title Page
h
vn,j
as test function in (Pn ), we obtain,
Z
Z
h
h
h
A(un ), ηϕk wn,j + gn (x, un , ∇un )ηϕk wn,j dx ≤
fn ηϕk wn,j
dx.
Ω
Ω
Which, implies that
(4.19)
A(un ), ϕk
Contents
JJ
J
II
I
Go Back
h
wn,j
Z
+
h
gn (x, un , ∇un )ϕk wn,j
dx
Ω
Z
h
≤
fn ϕk wn,j
dx.
Close
Quit
Page 22 of 46
Ω
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
It follows that
Z
h
h
(4.20)
a(x, un , ∇un )∇wn,j
ϕ0k wn,j
dx
Ω
Z
Z
h
h
+ gn (x, un , ∇un )ϕk wn,j dx ≤
fn ϕk wn,j
dx.
Ω
Ω
h
∇wn,j
Note that,
= 0 on the set where |un | > h + 5k, therefore, setting m =
5k + h, and denoting by (n, j, h) any quantity such that
lim
lim
lim (n, j, h) = 0.
h→+∞ j→+∞ n→+∞
If the quantity we consider does not depend on one parameter among n, j and
h, we will omit the dependence on the corresponding parameter: as an example,
(n, h) is any quantity such that
lim
L. Aharouch and M. Rhoudaf
Title Page
lim (n, h) = 0.
h→+∞ n→+∞
Contents
Finally, we will denote (for example) by h (n, j) a quantity that depends on
n, j, h and is such that
lim
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
lim h (n, j) = 0
JJ
J
II
I
j→+∞ n→+∞
for any fixed value of h.
We get, by (4.20),
Z
h
h
a(x, Tm (un ), ∇Tm (un ))∇wn,j
ϕ0k wn,j
dx
Ω
Z
Z
h
h
+ gn (x, un , ∇un )ϕk wn,j dx≤
fn ϕk wn,j
dx,
Ω
Ω
Go Back
Close
Quit
Page 23 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
h
In view of (4.10), we have ϕk wn,j
→ ϕk (wjh ) weakly∗ as n → +∞ in
L∞ (Ω), and then
Z
Z
h
fn ϕk wn,j dx →
f ϕk (wjh )dx as n → +∞.
Ω
Ω
Again tends j to infinity, we get
Z
Z
h
f ϕk (wj ) dx →
f ϕk (wh )dx
Ω
as j → +∞,
Ω
finally
letting h the infinity, we deduce by using the Lebesgue Theorem that
R
h
f
ϕ
(w
)dx → 0.
k
Ω
So that
Z
h
dx = (n, j, h).
fn ϕk wn,j
Ω
h
Since in the set {x ∈ Ω, |un (x)| > k}, we have g(x, un , ∇un )ϕk wn,j
≥ 0,
we deduce from (4.20) that
Z
(4.21)
h
h
a(x, Tm (un ), ∇Tm (un ))∇wn,j
ϕ0k wn,j
dx
Ω
Z
h
+
gn (x, un , ∇un )ϕk wn,j
dx ≤ (n, j, h).
{|un |≤k}
Splitting the first integral on the left hand side of (4.21) where |un | ≤ k and
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 24 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
|un | > k, we can write,
Z
(4.22)
h
h
a(x, Tm (un ), ∇Tm (un ))∇wn,j
ϕ0k wn,j
dx
Ω
Z
=
h
a(x, Tm (un ), ∇Tm (un ))[∇Tk (un ) − ∇Tk (vj )]ϕ0k wn,j
dx
{|un |≤k}
Z
h
h
+
a(x, Tm (un ), ∇Tm (un ))∇wn,j
ϕ0k wn,j
dx.
{|un |>k
The first term of the right hand side of the last inequality can write as
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Z
h
a(x, Tm (un ), ∇Tm (un ))[∇Tk (un ) − ∇Tk (vj )]ϕ0k wn,j
dx
(4.23)
{|un |≤k}
Z
≥
h
a(x, Tk (un ), ∇Tk (un ))[∇Tk (un ) − ∇Tk (vj )]ϕ0k wn,j
dx
Ω
Z
0
|a(x, Tk (un ), 0)||∇Tk (vj )|dx.
− ϕk (2k)
{|un |>k}
Recalling that, |a(x, Tk (un ), 0)|χ{|un |>k} converges to |a(x, Tk (u), 0)|χ{|u|>k}
strongly in LM (Ω), moreover, since |∇Tk (vj )| modular converges to |∇Tk (u)|,
then
Z
0
−ϕk (2k)
|a(x, Tk (un ), 0)||∇Tk (vj )|dx = (n, j).
{|un |>k}
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 25 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
For the second term of the right hand side of (4.14) we can write, using (A3 )
Z
h
h
(4.24)
a(x, Tm (un ), ∇Tm (un ))∇wn,j
ϕ0k wn,j
dx
{|un |>k}
Z
0
≥ −ϕk (2k)
|a(x, Tm (un ), ∇Tm (un ))|∇Tk (vj )|dx
{|un |>k}
Z
0
− ϕ (2k)
δ(x)dx.
{|un −v0 |>h
Since |a(x, Tm (un ), ∇Tm (un ))| is bounded in LM (Ω), we have, for a subsequence
|a(x, Tm (un ), ∇Tm (un ))| * lm
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
weakly in LM (Ω) in σ(LM , EM ) as n tends to infinity, and since
Title Page
|∇Tk (vj )|χ{|un |>k} → |∇Tk (vj )|χ{|u|>k}
strongly in EM (Ω) as n tends to infinity, we have
Z
0
− ϕk (2k)
|a(x, Tm (un ), ∇Tm (un ))|∇Tk (vj )|dx
{|un |>k}
Z
0
→ −ϕk (2k)
Contents
JJ
J
Go Back
lm |∇Tk (vj )|dx
{|u|>k}
as n tends to infinity.
Using now, the modular convergence of (vj ), we get
Z
Z
0
0
−ϕk (2k)
lm |∇Tk (vj )|dx → −ϕk (2k)
lm |∇Tk (u)|dx = 0
{|u|>k}
{|u|>k}
II
I
Close
Quit
Page 26 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
as j tends to infinity.
Finally
Z
0
(4.25) −ϕk (2k)
|a(x, Tm (un ), ∇Tm (un ))|∇Tk (vj )|dx = h (n, j).
{|un |>k}
On the other hand, since δ ∈ L1 (Ω) it is easy to see that
Z
0
δ(x)dx = (n, h).
(4.26)
−ϕk (2k)
{|un −v0 |>h
Combining (4.23) – (4.26), we deduce
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Z
(4.27)
h
h
a(x, Tm (un ), ∇Tm (un ))∇wn,j
ϕ0k wn,j
dx
ΩZ
h
≥
a(x, Tk (un ), ∇Tk (un ))[∇Tk (un ) − ∇Tk (vj )]ϕ0k wn,j
dx
Ω
+ (n, h) + (n, j) + h (n, j),
which implies that
Z
(4.28)
h
h
a(x, Tm (un ), ∇Tm (un ))∇wn,j
ϕ0k wn,j
dx
Ω Z
≥
a(x, Tk (un ), ∇Tk (un )) − a x, Tk (un ), ∇Tk (vj )χjs
Ω
h
dx
× ∇Tk (un ) − ∇Tk (vj )χjs ϕ0k wn,j
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 27 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
Z
+
h
a x, Tk (un ), ∇Tk (vj )χjs ∇Tk (un ) − ∇Tk (vj )χjs ϕ0k wn,j
dx
Ω
Z
h
dx
−
a(x, Tk (un ), ∇Tk (un ))∇Tk (vj )ϕ0k wn,j
Ω\Ωjs
+ (n, h) + (n, j) + h (n, j),
where χjs denotes the characteristic function of the subset Ωjs = {x ∈ Ω :
|∇Tk (vj )| ≤ s}.
h
By (4.16) and the fact that ∇Tk (vj )χΩ\Ωjs ϕ0k wn,j
tends to ∇Tk (vj )χΩ\Ωjs ϕ0k (wjh )
strongly in (EM (Ω))N , the third term on the right hand side of (4.28) tends to
the quantity
Z
hk ∇Tk (vj )ϕ0k wjh dx
Ω\Ωjs
as n tends to infinity.
Letting now j tend to infinity, by using the modular convergence of vj , we
have
Z
Z
0
h
hk ∇Tk (vj )χΩ\Ωjs ϕk wj dx →
hk ∇Tk (u)ϕ0k (wh )dx
Ω
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Ω\Ωs
Close
as j tends to infinity.
Quit
Page 28 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
Finally
Z
−
(4.29)
h
a(x, Tk (un ), ∇Tk (un ))∇Tk (vj )ϕ0k wn,j
dx
Ω\Ωjs
Z
=−
hk ∇Tk (u)ϕ0k (wh )dx + h (n, j).
Ω\Ωs
Concerning the second term on the right hand side of (4.28) we can write
Z
h
(4.30)
a x, Tk (un ), ∇Tk (vj )χjs ∇Tk (un ) − ∇Tk (vj )χjs ϕ0k wn,j
dx
ΩZ
=
a x, Tk (un ), ∇Tk (vj )χjs ∇Tk (un )ϕ0k (Tk (un ) − Tk (vj ))dx
Ω
Z
h
dx.
− a(x, Tk (un ), ∇Tk (vj )χjs )∇Tk (vj )χjs ϕ0k wn,j
Ω
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
The first term on the right hand side of (4.30) tends to the quantity
Z
a x, Tk (u), ∇Tk (vj )χjs ∇Tk (u)ϕ0k (Tk (u) − Tk (vj ))dx as n → ∞
JJ
J
II
I
Ω
Go Back
since
Close
a
x, Tk (un ), ∇Tk (vj )χjs
ϕ0k (Tk (un )
− Tk (vj ))
→ a x, Tk (u), ∇Tk (vj )χjs ϕ0k (Tk (u) − Tk (vj ))
Quit
Page 29 of 46
N
strongly in (EM (Ω))
by
Q
Q Lemma 2.3 and ∇Tk (un ) * ∇Tk (u) weakly in
(LM (Ω))N for σ ( LM , EM ).
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
For the second term on the right hand side of (4.30) it is easy to see that
Z
h
a x, Tk (un ), ∇Tk (vj )χjs ∇Tk (vj )χjs ϕ0k wn,j
dx
Ω
Z
−→
a x, Tk (u), ∇Tk (vj )χjs ∇Tk (vj )χjs ϕ0k wjh dx.
(4.31)
Ω
as n → ∞.
Consequently, we have
Z
a x, Tk (un ), ∇Tk (vj )χjs
(4.32)
ZΩ
=
h
∇Tk (un ) − ∇Tk (vj )χjs ) ϕ0k wn,j
dx
Title Page
+ j,h (n)
since
∇Tk (vj )χjs ϕ0k (wjh ) → ∇Tk (u)χs ϕ0k (wh )
strongly in (EM (Ω))N as j → +∞, it is easy to see that
Ω
L. Aharouch and M. Rhoudaf
a x, Tk (u), ∇Tk (vj )χjs ∇Tk (u) − ∇Tk (vj )χjs ) ϕ0k wjh dx
Ω
Z
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
a x, Tk (u), ∇Tk (vj )χjs
∇Tk (u) − ∇Tk (vj )χjs ) ϕ0k wjh dx
Z
−→
a(x, Tk (u), 0)∇Tk (u)ϕ0k (wh )dx
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 30 of 46
Ω\Ωs
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
as j → +∞, thus
Z
(4.33)
h
dx
a x, Tk (un ), ∇Tk (vj )χjs ∇Tk (un ) − ∇Tk (vj )χjs ϕ0k wn,j
Ω
Z
=
a(x, Tk (u), 0)∇Tk (u)ϕ0k (0)dx + (n, j).
Ω\Ωs
Combining (4.28), (4.29) and (4.32), we get
Z
h
a(x, Tm (un ), ∇Tm (un ))∇wn,j
ϕ0k
(4.34)
h
wn,j
dx
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
Ω
Z
≥
a(x, Tk (un ), ∇Tk (un )) − a x, Tk (un ), ∇Tk (vj )χjs
Ω
h
× ∇Tk (un ) − ∇Tk (vj )χjs ϕ0k wn,j
dx
Z
−
hk ∇Tk (u)ϕ0k (0)dx
Ω\Ωs
Z
+
a(x, Tk (u), 0)∇Tk (u)ϕ0k (0)dx + (n, j, h).
Ω\Ωs
We now, turn to the second term on the left hand side of (4.21), we have
Z
h
(4.35)
gn (x, un , ∇un )ϕk wn,j
dx
{|un |≤k}
Z
h
dx
≤ b(k) (h(x) + M (|∇Tk (un )|)) ϕk wn,j
Ω
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 31 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
Z
Z
b(k)
h
h
dx
≤ b(k) h(x)|ϕk wn,j |dx +
δ(x) ϕk wn,j
α
Ω
Ω
Z
b(k)
h
dx
+
a(x, Tk (un ), ∇Tk (un ))∇Tk (un ) ϕk wn,j
α Ω
Z
b(k)
h
dx
−
a(x, Tk (un ), ∇Tk (un ))∇v0 ϕk wn,j
α Ω
≤ (n, j, h)
Z
b(k)
h
dx.
+
a(x, Tk (un ), ∇Tk (un ))∇Tk (un ) ϕk wn,j
α Ω
The last term on the last side of this inequality reads as
Z
b(k) (4.36)
a(x, Tk (un ), ∇Tk (un )) − a x, Tk (un ), ∇Tk (vj )χjs
α Ω
h
dx
× ∇Tk (un ) − ∇Tk (vj )χjs ϕk wn,j
Z
b(k)
+
a x, Tk (un ), ∇Tk (vj )χjs
α Ω
h
dx
× ∇Tk (un ) − ∇Tk (vj )χjs ) ϕk wn,j
Z
b(k)
h
dx
+
a(x, Tk (un ), ∇Tk (un ))∇Tk (vj )χjs ϕk wn,j
α Ω
and reasoning as above, it is easy to see that
Z
b(k)
h
dx
a x, Tk (un ), ∇Tk (vj )χjs ∇Tk (un ) − ∇Tk (vj )χjs ϕk wn,j
α Ω
= (n, j)
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 32 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
and
b(k)
−
α
Z
h
dx = (n, j, h).
a(x, Tk (un ), ∇Tk (un ))∇Tk (vj )χjs ϕk wn,j
Ω
So that
(4.37)
Z
gn (x, un , ∇un )ϕk
{|un |≤k}
b(k)
≤
α
h
wn,j
dx
Z
a(x, Tk (un ), ∇Tk (un )) − a x, Tk (un ), ∇Tk (vj )χjs
Ω
h
dx + (n, j, h).
× ∇Tk (un ) − ∇Tk (vj )χjs ϕk wn,j
Combining (4.21), (4.34) and (4.37), we obtain
Z
(4.38)
a(x, Tk (un ), ∇Tk (un )) − a x, Tk (un ), ∇Tk (vj )χjs
Ω
b(k) j
0
h
h
× ∇Tk (un ) − ∇Tk (vj )χs ϕk wn,j −
ϕk wn,j
dx
α
Z
≤
hk ∇Tk (u)ϕ0k (0)dx
Ω\Ωs
Z
+
a(x, Tk (u), 0)∇Tk (u)ϕ0k (0)dx + (n, j, h),
Ω\Ωs
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 33 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
which implies that, by using (4.1)
Z
(4.39)
a(x, Tk (un ), ∇Tk (un )) − a x, Tk (un ), ∇Tk (vj )χjs
Ω
× ∇Tk (un ) − ∇Tk (vj )χjs dx
Z
≤2
hk ∇Tk (u)ϕ0k (0)dx
Ω\Ωs
Z
+2
a(x, Tk (u), 0)∇Tk (u)ϕ0k (0)dx + (n, j, h).
Ω\Ωs
Now, remark that
Z
(4.40)
[a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u)χs )]
Ω
× [∇Tk (un ) − ∇Tk (u)χs ]dx
Z
a(x, Tk (un ), ∇Tk (un )) − a x, Tk (un ), ∇Tk (vj )χjs
Ω
× ∇Tk (un ) − ∇Tk (vj )χjs dx
Z
+ a x, Tk (un ), ∇Tk (vj )χjs ∇Tk (un ) − ∇Tk (vj )χjs dx
ΩZ
− a(x, Tk (un ), ∇Tk (u)χs )[∇Tk (un ) − ∇Tk (u)χs ]dx
Ω
Z
+ a(x, Tk (un ), ∇Tk (un )) ∇Tk (vj )χjs − ∇Tk (u)χs dx.
=
Ω
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 34 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
We shall pass to the limit in n and j in the last three terms of the right hand side
of the last inequality, we get
Z
a x, Tk (un ), ∇Tk (vj )χjs ∇Tk (un ) − ∇Tk (vj )χjs dx
Ω
Z
=
a(x, Tk (u), 0)∇Tk (u)dx + (n, j),
Ω\Ωs
Z
Ω
a(x, Tk (un ), ∇Tk (u)χs )[∇Tk (un ) − ∇Tk (u)χs ]dx
Z
=
a(x, Tk (u), 0)∇Tk (u)dx + (n),
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Ω\Ωs
Title Page
and
Z
Contents
a(x, Tk (un ), ∇Tk (un )) ∇Tk (vj )χjs − ∇Tk (u)χs dx = (n, j),
Ω
which implies that
Z
(4.41)
[a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u)χs )]
Ω
× [∇Tk (un ) − ∇Tk (u)χs ]dx
=
a(x, Tk (un ), ∇Tk (un )) − a x, Tk (un ), ∇Tk (vj )χjs
Ω
× ∇Tk (un ) − ∇Tk (vj )χjs dx + (n, j).
Z
JJ
J
II
I
Go Back
Close
Quit
Page 35 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
Combining (4.17), (4.39) and (4.41), we have
Z
(4.42)
[a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u))]
Ωr
× [∇Tk (un ) − ∇Tk (u)]dx
Z
≤
[a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u)χs )]
Ω
× [∇Tk (un ) − ∇Tk (u)χs ]dx
Z
≤2
hk ∇Tk (u)ϕ0k (0)dx
Ω\Ωs
Z
+2
a(x, Tk (u), 0)∇Tk (u)ϕ0k (0)dx + (n, j, h).
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Ω\Ωs
By passing to the lim sup over n, and letting j, h, s tend to infinity, we obtain
Z
lim
[a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u))]
n→+∞
Ωr
× [∇Tk (un ) − ∇Tk (u)]dx = 0,
thus implies by the same method used in [5] that
Title Page
Contents
JJ
J
II
I
Go Back
Close
(4.43)
∇u → ∇un a.e. in Ω.
Quit
Page 36 of 46
Step 6: Modular convergence of the truncation:
By (4.16) and (4.43), we have hk = a(x, Tk (u), ∇Tk (u)), which implies by
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
using (4.42)
Z
(4.44)
[a(x, Tk (un ), ∇Tk (un ))(∇Tk (un ) − ∇v0 ) + δ(x)]dx
Z
≤ [a(x, Tk (un ), ∇Tk (un ))(∇Tk (u)χs − ∇v0 ) + δ(x)]dx
ZΩ
+ a(x, Tk (un ), ∇Tk (u)χs )(∇Tk (un ) − ∇Tk (u)χs )dx
Ω
Z
+2
a(x, Tk (u), ∇Tk (u))∇Tk (u)ϕ0k (0)dx
Ω\Ωs
Z
+2
a(x, Tk (u), 0)∇Tk (u)ϕ0k (0)dx + (n, j, h),
Ω
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Ω\Ωs
which implies, by using Fatou’s Lemma,
Z
(4.45)
[a(x, Tk (u), ∇Tk (u))(∇Tk (u) − ∇v0 ) + δ(x)]dx
Ω
Z
≤ lim inf [a(x, Tk (un ), ∇Tk (un ))(∇Tk (un ) − ∇v0 ) + δ(x)]dx
n→+∞ Ω
Z
≤ lim sup [a(x, Tk (un ), ∇Tk (un ))(∇Tk (un ) − ∇v0 ) + δ(x)]dx
n→+∞
ZΩ
≤ lim sup [a(x, Tk (un ), ∇Tk (un ))(∇Tk (u)χs − ∇v0 ) + δ(x)]dx
n→+∞
Ω
Z
+ lim sup a(x, Tk (un ), ∇Tk (u)χs )(∇Tk (un ) − ∇Tk (u)χs )dx
n→+∞
Ω
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 37 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
Z
+2
hk ∇Tk (u)ϕ0k (0)dx
ZΩ\Ωs
+2
a(x, Tk (u), 0)∇Tk (u)ϕ0k (0)dx + (n, j, h).
Ω\Ωs
Reasoning as above, we have
Z
(4.46) lim sup [a(x, Tk (un ), ∇Tk (un ))(∇Tk (u)χs − ∇v0 ) + δ(x)]dx
n→+∞
Ω
Z
= [a(x, Tk (u), ∇Tk (u))(∇Tk (u)χs − ∇v0 ) + δ(x)]dx,
Ω
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
and
Title Page
Z
lim sup
n→+∞
a(x, Tk (un ), ∇Tk (u)χs )(∇Tk (un ) − ∇Tk (u)χs )dx
Ω
Z
=
a(x, Tk (u), 0)∇Tk (u)dx.
Ω\Ωs
Which implies that
Z
(4.47)
[a(x, Tk (u), ∇Tk (u))(∇Tk (u) − ∇v0 ) + δ(x)]dx
Ω
Z
≤ lim inf [a(x, Tk (un ), ∇Tk (un ))(∇Tk (un ) − ∇v0 ) + δ(x)]dx
n→+∞ Ω
Z
≤ lim sup [a(x, Tk (un ), ∇Tk (un ))(∇Tk (un ) − ∇v0 ) + δ(x)]dx
n→+∞
Ω
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 38 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
Z
≤
[a(x, Tk (u), ∇Tk (u))(∇Tk (u)χs − ∇v0 ) + δ(x)]dx
Z
Z
0
+2
hk ∇Tk (u)ϕk (0)dx +
a(x, Tk (u), 0)∇Tk (u)dx
Ω\Ωs
Ω\Ωs
Z
a(x, Tk (u), 0)∇Tk (u)ϕ0k (0)dx.
+2
Ω
Ω\Ωs
Using the fact that
[a(x, Tk (u), ∇Tk (u))(∇Tk (u)χs − ∇v0 ) +
a(x, Tk (u), 0)∇Tk (u)ϕ0k (0)
in
δ(x)], hk ∇Tk (u)ϕ0k (0)
and
L1 (Ω)
and letting s → +∞, we get, since meas(Ω\Ωs ) → 0,
Z
(4.48)
[a(x, Tk (u), ∇Tk (u))(∇Tk (u) − ∇v0 ) + δ(x)]dx
Ω
Z
≤ lim inf [a(x, Tk (un ), ∇Tk (un ))(∇Tk (un ) − ∇v0 ) + δ(x)]dx
n→+∞ Ω
Z
≤ lim sup [a(x, Tk (un ), ∇Tk (un ))(∇Tk (un ) − ∇v0 ) + δ(x)]dx
n→+∞
Ω
Z
≤ [a(x, Tk (u), ∇Tk (u))(∇Tk (u) − ∇v0 ) + δ(x)]dx.
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Ω
Page 39 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
Finally, we have
Z
(4.49) lim
[a(x, Tk (un ), ∇Tk (un ))(∇Tk (un ) − ∇v0 ) + δ(x)]dx
n→+∞ Ω
Z
= [a(x, Tk (u), ∇Tk (u))(∇Tk (u) − ∇v0 ) + δ(x)]dx
Ω
and by using (A3 ), one obtains, by Lemma 2.6
(4.50)
M (∇Tk (un )) → M (∇Tk (u)) in L1 (Ω).
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Step 7: Equi-integrability of the nonlinearities.
We need to prove that
(4.51)
gn (x, un , ∇un ) → g(x, u, ∇u) strongly in L1 (Ω),
in particular it is enough to prove the equi-integrability of gn (x, un , ∇un ). To
this purpose. We take un − T1 (un − v0 − Th (un − v0 )) as test function in (Pn ),
we obtain
Z
Z
|gn (x, un , ∇un )|dx ≤
(|fn | + δ(x))dx.
{|un −v0 |>h+1}
Title Page
Contents
JJ
J
II
I
Go Back
Close
{|un −v0 |>h}
Quit
Let ε > 0, then there exists h(ε) ≥ 1 such that
Z
ε
(4.52)
|g(x, un , ∇un )|dx < .
2
{|un −v0 |>h(ε)}
Page 40 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
For any measurable subset E ⊂ Ω, we have
Z
|gn (x, un , ∇un )|dx
E
Z
≤ b(h(ε) + kv0 k∞ )(h(x) + M (∇Th(ε)+kv0 k∞ (un )))dx
E
Z
+
|g(x, un , ∇un )|dx.
{|un −v0 |>h(ε)}
In view of (4.50) there exists η(ε) > 0 such that
Z
ε
(4.53)
b(h(ε) + kv0 k∞ )(h(x) + M (∇Th(ε)+kv0 k∞ (un )))dx <
2
E
for all E such that meas(E) < η(ε).
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
Finally, combining (4.52) and (4.53), one easily has
Z
|gn (x, un , ∇un )| dx < ε for all E such that meas(E) < η(ε),
Title Page
E
which implies (4.51).
Step 8: Passing to the limit.
Let v ∈ Kψ ∩ W01 EM (Ω) ∩ L∞ (Ω), we take un − Tk (un − v) as test function
in (Pn ), we can write
Z
Z
(4.54)
a(x, un , ∇un )∇Tk (un − v)dx + g(x, un , ∇un )Tk (un − v)dx
Ω
Ω
Z
≤
fn Tk (un − v)dx,
Ω
L. Aharouch and M. Rhoudaf
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 41 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
which implies that
Z
(4.55)
a(x, un , ∇un )∇(un − v0 )dx
{|un −v|≤k}
Z
+
a(x, Tk+kvk∞ un , ∇Tk+kvk∞ (un ))∇(v0 − v)dx
{|un −v|≤k}
Z
Z
+ g(x, un , ∇un )Tk (un − v)dx ≤
fn Tk (un − v)dx.
Ω
Ω
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
By Fatou’s lemma and the fact that
a(x, Tk+kvk∞ (un ), ∇Tk+kvk∞ (un )) * a(x, Tk+kvk∞ (u), ∇Tk+kvk∞ (u))
weakly in (LM (Ω))N for σ(ΠLM , ΠEM ) one can easily see that
Z
(4.56)
a(x, u, ∇u)∇(u − v0 )dx
{|u−v|≤k}
Z
+
a(x, Tk+kvk∞ (u), ∇Tk+kvk∞ (u))∇(v0 − v)dx
{|u−v|≤k}
Z
Z
+ g(x, u, ∇u)Tk (u − v)dx ≤
f Tk (u − v)dx.
Ω
Ω
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Hence
Z
Z
a(x, u, ∇u)∇Tk (u − v)dx +
(4.57)
Ω
g(x, u, ∇u)Tk (u − v)dx
Ω
Z
≤
f Tk (u − v)dx.
Ω
Quit
Page 42 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
Now, let v ∈ Kψ ∩ L∞ (Ω), by the condition (A5 ) there exists vj ∈ Kψ ∩
W01 EM (Ω) ∩ L∞ (Ω) such that vj converges to v modular, let h > kv0 k∞ , taking
v = Th (vj ) in (4.57), we have
Z
(4.58)
a(x, u, ∇u)∇Tk (u − Th (vj ))dx
Ω
Z
+ g(x, u, ∇u)Tk (u − Th (vj ))dx
Ω
Z
≤
f Tk (u − Th (vj ))dx.
Ω
We can easily pass to the limit as j → +∞ to get
Z
(4.59)
a(x, u, ∇u)∇Tk (u − Th (v))dx
Ω
Z
+ g(x, u, ∇u)Tk (u − Th (v))dx
Ω
Z
≤
f Tk (u − Th (v))dx ∀ v ∈ Kψ ∩ L∞ (Ω),
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
Ω
the same, we pass to the limit as h → +∞, we deduce
Z
Z
(4.60)
a(x, u, ∇u)∇Tk (u − v)dx + g(x, u, ∇u)Tk (u − v)dx
Ω
Ω
Z
≤
f Tk (u − v)dx ∀v ∈ Kψ ∩ L∞ (Ω), ∀k > 0.
II
I
Go Back
Close
Quit
Page 43 of 46
Ω
Thus, the proof of the theorem is now complete.
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
References
[1] R. ADAMS, Sobolev Spaces, Academic Press, New York, (1975).
[2] P. BÉNILAN, L. BOCCARDO, T. GALLOUET, R. GARIEPY, M.
PIERRE AND J.L. VÁZQUEZ, An L1 -theory of existence and uniqueness
of nonlinear elliptic equations., Ann. Scuola Norm. Sup. Pisa, 22 (1995),
240–273.
[3] A. BENKIRANE, Approximation de type de Hedberg dans les espaces
W m LlogL(Ω) et application, Ann. Fac. Sci. Toulouse, 11(4) (1990), 67–
78.
[4] A. BENKIRANE AND A. ELMAHI, An existence theorem for a strongly
nonlinear elliptic problems in Orlicz spaces, Nonlinear Anal. T. M. A., 36
(1999), 11–24.
[5] A. BENKIRANE AND A. ELMAHI, A strongly nonlinear elliptic equation
having natural growth terms and L1 data, Nonlinear Anal. T. M. A., 39
(2000), 403–411.
[6] A. BENKIRANE, A. ELMAHI AND D. MESKINE, An existence theorem
for a class of elliptic problems in L1 , Applicationes Mathematicae, 29(4)
(2002), 439–457.
[7] A. BENKIRANE AND J.P. GOSSEZ, An approximation theorem for
higher order Orlicz-Sobolev spaces, Studia Math., 92 (1989), 231–255.
[8] L. BOCCARDO AND T. GALLOUET, Problémes unilatéraux avec données dans L1 , C.R. Acad. Sci. Paris, 311 (1990), 617–619.
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 44 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
[9] L. BOCCARDO, F. MURAT AND J.P. PUEL, Existence of bounded solutions for nonlinear ellipìtic unilateral problems, Ann. Mat. Pura e Appl.,
152 (1988), 183–196.
[10] G. DALMASO, F. MURAT, L. ORSINA AND A. PRIGNET, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola
Norm. Sup Pisa Cl. Sci., 12(4) (1999), 741–808.
[11] A. ELMAHI AND D. MESKINE, Nonlinear elliptic problems having natural growth and L1 data in Orlicz spaces, to appear in Anali di Mathimatica.
[12] A. ELMAHI AND D. MESKINE, Unilateral elleptic problems in L1 with
natural growth terms, J. Nonlinear and Convex Analysis, 5(1) (2004), 97–
112
[13] J.P. GOSSEZ, Nonlinear elliptic boundary value problems for equations
with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc.,
190 (1974), 163–205.
[14] J.P. GOSSEZ, A strongly nonlinear elliptic problem in Orlicz-Sobolev
spaces, Proc. AMS. symp. Pure. Math., 45 (1986), 163–205.
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
[15] J.P. GOSSEZ AND V. MUSTONEN, Variational inequalities in OrliczSobolev spaces, Nonlinear Anal., 11 (1987), 379–492.
Close
[16] M.A. KRASNOSEL’SKII AND Y.B. RUTIKII , Convex Functions and Orlicz Spaces, Noordhoff Groningen, (1969).
Page 45 of 46
Quit
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au
[17] A. PORRETTA, Existence for elliptic equations in L1 having lower order
terms with natural growth, Portugal. Math., 57 (2000), 179–190.
[18] R.M. POSTERARO, Estimates for solutions of nonlinear variational inequalities, Ann. Inst. H. Poincaré, 12 (1995), 577–597.
Strongly Nonlinear Elliptic
Unilateral Problems in Orlicz
Space and L1 Data
L. Aharouch and M. Rhoudaf
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 46 of 46
J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005
http://jipam.vu.edu.au