Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 219586, 18 pages
doi:10.1155/2009/219586
Research Article
Existence of Solution for
Nonlinear Elliptic Equations with
Unbounded Coefficients and L1 Data
Hicham Redwane
Faculté des Sciences Juridiques, Économiques et Sociales, Université Hassan 1,
B.P. 784, Settat 26 000, Morocco
Correspondence should be addressed to Hicham Redwane, redwane hicham@yahoo.fr
Received 20 July 2009; Accepted 17 November 2009
Recommended by Mónica Clapp
An existence result of a renormalized solution for a class of nonlinear elliptic equations is
N
established. The diffusion functions ax, u, ∇u may not be in L1loc Ω for a finite value of the
unknown and the data belong to L1 Ω.
Copyright q 2009 Hicham Redwane. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
In this paper we investigate the problem of existence of a renormalized solutions for elliptic
equations of the type
− divax, u, ∇u f
u0
on ∂Ω,
in Ω,
1.1
where Ω is an open bounded subset of RN , N ≥ 1, with the data f in L1 Ω. The
operator − divax, u, ∇u is a Leray-Lions operator defined on the weighted sobolev spaces
1,p
W0 Ω, w, but which is not restricted by any growth condition with respect to u see
assumptions 2.2, 2.4, and 2.5 of Section 3. The function ax, s, ξ is controlled by a real
function b : − ∞, m → R which blows up for a finite value m > 0 see 2.2, 2.3.
There are mainly two types of difficulties that are studying Problem 1.1. One
consists to give a sense to the flux ax, u, ∇u on the set {x ∈ Ω; ux m}. The second
one is that the data f only belong to L1 , so that proving existence of a weak solution
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i.e., in the distribution meaning seems to be an arduous task. To overcome this difficulty
we use in this paper the framework of renormalized solutions. This notion was introduced
by DiPerna and Lions 1 for the study of Boltzmann equation see also Lions 2 for a few
applications to fluid mechanics models. This notion was then adapted to elliptic vesion
of 1.1 in Boccardo et al. 3, and Murat 4, 5 see also 6, 7 for nonlinear parabolic
problems. At the same time the equivalent notion of entropy solutions has been developed
independently by Bénilan et al. 8 for the study of nonlinear elliptic problems.
In the case where ax, u, ∇u is replaced by du Au∇u problems with diffusion
matrices that have at least one diagonal coefficient that blows up for a finite value of the
unknown and f ∈ L2 Ω, existence and uniqueness has been established in Blanchard and
Redwane 9, 10.
As far as we have the stationary and evolution equations case 1.1, the existence and a
partial uniqueness of renormalized solutions have been proved in Blanchard et al. 11 in the
case where ax, u, ∇u is replaced by Ax, u∇u where Ax, s is a Carathéodory symmetric
matrices, such that Ax, s blows up as s → m− uniformly with respect to x. It has also
been applied to the study of linear and nonlinear elliptic and parabolic equations when the
diffusion coefficient has a singularity for a finite value of the unknown see Garcı́a Vázquez
and Ortegón Gallego 12, 13 and Orsina 14.
The paper is organized as follows. In Section 2 we will precise some basic properties of
weighted Sobolev spaces. Section 3 is devoted to specify the assumptions on ax, s, ξ, bs,
and f needed in the present study and gives the definition of a renormalized solution of 1.1.
In Section 4 Theorem 4.1 we establish the existence of such a solution.
2. Preliminaries
Throughout the paper, we assume that the following assumptions hold true. Ω is a bounded
open subset on RN , N ≥ 1. Let us suppose that 1 < p < ∞ is a real number, and ωx {ωi x}{0≤i≤N} is a vector of weight functions. Furthermore we suppose that every component
ωi x is a measurable function which is strictly positive and satisfies
ωi ∈ L1loc Ω,
ωi −1/p−1 ∈ L1loc Ω.
2.1
We define the weighted Lebesgue space Lp Ω, ω0 with weight ω0 , as the space of all realvalued measurable functions u for which
up,ω0 Ω
|ux|p ω0 xdx
1/p
< ∞.
2.2
In order to define the weighted Sobolev space of W 1,p Ω, ω, as the space of all real-valued
functions u ∈ Lp Ω, ω0 such that the derivaties in the sense of distributions satisfy ∂u/∂xi ∈
Lp Ω, ωi for all i 1, . . . , N. This set of functions forms a Banach space under the norm
u1,p,ω 1/p
N ∂u p
.
|ux| ω0 xdx ∂x ωi xdx
i
Ω
i1 Ω
p
2.3
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3
1,p
To deal with the Dirichlet problem, we use the space X W0 Ω, ω defined as the closure
1,p
of C0∞ Ω with respect to the norm · 1,p,ω . Note that, C0∞ Ω is dense in W0 Ω, ω and
1,p
W0 Ω, ω, · 1,p,ω is a reflexive Banach space. Note that the expression
1/p
N
∂u p
uX ∂x ωi xdx
i
i1 Ω
2.4
is a norm defined on X and is equivalent to the norm 2.3. Moreover X, · X is a reflexive
Banach space, and there exist a weight function σ on Ω and a parameter 1 < q < ∞ such that
the Hardy inequality
q
Ω
|u| σxdx
1/q
1/p
N ∂u p
≤C
∂x ωi xdx
i
i1 Ω
2.5
holds for every u ∈ X with a constant C > 0 independent of u. Moreover, the imbedding
X → Lq Ω, σ is compact.
1,p
We recall that the dual of the weighted Sobolev spaces W0 Ω, ω is equivalent to
1−p
W −1,p Ω, ω∗ , where ω∗ {ωi∗ ωi ; i 1, . . . , N} and p p/p − 1 is the conjugate of p.
For more details we refer the reader to 15 see also 16.
3. Assumptions on the Data and Definition of a Renormalized Solution
Throughout the paper, we assume that the following assumptions hold true. Ω is a bounded
open set on RN , N ≥ 1. Let 1 < p < ∞, and let ωx {ωi x}{0≤i≤N} be a vector of weight
functions.
1,p
Let now − divax, u, ∇u be a Leray-Lions operator defined on W0 Ω, ω into
−1,p
∗
Ω, ω and where
W
a : Ω × R × RN −→ RN is a Carathéodory function, such that
3.1
there exists a positive function b ∈ C0 −∞, m which satisfies
lim br ∞;
r → m−
m
bsds < ∞,
br ≥ α > 0 ∀r ∈ −∞, m,
0
ax, s, ξ · ξ ≥ bs
p−1
N
3.2
p
ωi x|ξi | ,
ax, s, 0 0,
i1
for almost every x ∈ Ω, for every s ∈ R and ξ ∈ RN .
For any i 1, . . . , N,
⎡
|ai x, s, ξ| ≤ ωi x1/p ⎣Lx σx1/p |s|q/p bs
N
p−1
⎤
1/p
p−1
ω xξj ⎦,
j
j1
3.3
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for almost every x ∈ Ω, for every s and ξ, and where Lx is a positive function in Lp Ω
ax, s, ξ − a x, s, ξ ξ − ξ ≥ 0,
3.4
for any ξ, ξ ∈ RN , for any s ∈ R and for almost every x ∈ Ω
f is an element of L1 Ω.
3.5
Remark 3.1. As already mentioned in the introduction, problem 1.1 does not admit a weak
solution under assumptions 3.1–3.5 since the growth of ax, u, ∇u is not controlled with
respect to u, the field ax, u, ∇u is not, in general, defined as a distribution because the
difficulty is defining the field ax, u, ∇u on the subset {x ∈ Ω; ux m} of Ω, since on
this set, bu ∞.
The following notations will be used throughout the paper. For any K ≥ 0, the
truncation at height K is defined by TK r max−K, minr, K, for any positive numbers l
and K, the functions TlK are defined by
⎧
⎪
−K,
⎪
⎪
⎨
TlK r r,
⎪
⎪
⎪
⎩
l,
if r ≤ −K,
if − K ≤ r ≤ l,
3.6
if r ≥ l.
We define for n ≥ 1 fixed
⎧
⎪
0,
if |r| ≤ n,
⎪
⎪
⎨
θn r T1 r − Tn r r − n sgs, if n ≤ |r| ≤ n 1,
⎪
⎪
⎪
⎩
sgs,
if |r| ≥ n 1,
3.7
and Sn r 1 − |θn r|, for all r ∈ R.
The definition of a renormalized solution for Problem 1.1 can be stated as follows.
Definition 3.2. A measurable function u defined on Ω is a renormalized solution of Problem
1.1 if
1,p
TK u ∈ W0 Ω, ω
ux ≤ m
∀K ≥ 0,
for almost every x ∈ Ω,
N
1−p
K
K
,
Lp Ω, wi
a x, Tm
u, ∇Tm
u χ{u<m} ∈
i1
3.8
3.9
3.10
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ax, u, ∇u∇udx −→ 0 as n −→ ∞,
1
δ
{−n−1≤ux≤−n}
{m−2δ≤ux≤m−δ}
5
3.11
ax, u, ∇u∇udx −→
{um}
fdx
as δ −→ 0,
3.12
and if, for every function S in W 1,∞ R such that suppS is compact and Sm 0, u satisfies
Ω
ax, u, ∇u∇ Suϕ dx Ω
fSuϕdx,
1,p
∀ϕ ∈ W0 Ω, ω ∩ L∞ Ω.
3.13
The following remarks are concerned with Definition 3.2.
Remark 3.3. Notice that, thanks to our regularity assumptions 3.8, 3.9, 3.10 and the
choice of S, all terms in 3.13 are well defined.
The following two identifications are made in 3.13:
K
K
u, ∇Tm
u∇Suϕ for almost every
i ax, u, ∇u∇Suϕ identifies with ax, Tm
x ∈ Ω, where K > 0 and suppS ⊂ −K, K. As a consequence of 3.8, 3.9, and
1,p
3.10, and of S ∈ W 1,∞ R, ϕ ∈ W0 Ω, ω ∩ L∞ Ω, it follows that
K
K
a x, Tm
u, ∇Tm
u ∇ S uϕ ∈ L1 Ω,
3.14
ii fSuϕ ∈ L1 Ω, because f ∈ L1 Ω and Suϕ ∈ L∞ Ω.
4. Existence Result
This section is devoted to establish the following existence theorem.
Theorem 4.1. Under assumptions 3.1–3.5 there exists a renormalized solution u of Problem 1.1.
Proof. The proof is divided into 7 steps. In Step 1, we introduce an approximate problem.
Step 2 is devoted to establish a few a priori estimates, the limit u of the approximate solutions
uε is introduced and it is shown that u satisfies 3.8 and 3.9. Step 3 is devoted to prove an
energy estimate Lemma 4.2 which is a key point for the monotonicity arguments that are
developed in Step 4. Step 5 is devoted to prove that u satisfies 3.11. In Step 6 we prove that
u satisfies 3.12. Finally, Step 7 is devoted to prove that u satisfies 3.13 of Definition 3.2.
Step 1. Let us introduce the following regularization of the data:
1/ε
bε r b Tm−ε
r
1/ε
aε x, s, ξ a x, Tm−ε
s, ξ
f ε ∈ Lp Ω;
∀r ∈ R for ε > 0,
a.e. in Ω, ∀s ∈ R, ∀ξ ∈ RN ,
ε
f 1
≤ f L1 Ω : f ε −→ f strongly in L1 Ω as ε tends to 0.
L Ω
4.1
4.2
4.3
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Let us now consider the following regularized problem:
− divaε x, uε , ∇uε f ε
in Ω,
4.4
uε 0 on ∂Ω.
4.5
In view of 3.3, 4.1, and 4.2, aε satisfy. For i 1, . . . , N
⎡
⎤
N
q/p
p−1
ε
1/ε
1/p
1/p
1/p
p−1
ε
a x, s, ξ ≤ ωi x ⎣Lx σx Tm−ε s
b s
ωj xξj ⎦
i
4.6
j1
a.e. x ∈ Ω, for all s ∈ R, ξ ∈ RN . And
α ≤ bε r ≤
max
br Cε
{−1/ε≤r≤m−ε}
∀r ∈ R.
4.7
1,p
As a consequence, proving existence of a weak solution uε ∈ W0 Ω, ω of 4.4 and
4.5 is an easy task see, e.g., Theorem 2.1 and Remark 2.1 in Chapter 2 of 17 and see also
18.
Step 2. A priori estimates and pointwise convergence of uε .
Using TK uε as a test function in 4.4 leads to
a x, u , ∇u ∇TK u dx ε
Ω
ε
ε
ε
Ω
f ε TK uε dx ≤ K f L1 Ω .
4.8
Since aε satisfies 3.2, 4.2, and owing to 4.8 we have
Ω
ε p−1
b u ε
αp−1
N ∂TK uε p
∂x ωi xdx ≤ K f L1 Ω ,
i1
i
N ∂TK uε p
∂x ωi xdx ≤ K f L1 Ω .
i
Ω i1
4.9
4.10
From 4.10 we deduce with a classical argument see, e.g., 18 that, for a subsequence still
indexed by ε,
uε −→ u
a.e. in Ω,
1,p
TK uε −→ TK u weakly in W0 Ω, ω and strongly in Lq Ω, σ,
as ε tends to 0, where u is a measurable function defined on Ω which is finite a.e. in Ω.
4.11
4.12
International Journal of Mathematics and Mathematical Sciences
Taking now Zε TmK uε 0
7
bε sds as a test function in 4.4 gives
Ω
aε x, uε , ∇uε ∇Zε dx Ω
4.13
f ε Zε dx.
Since aε satisfies 3.2 and b satisfies 3.1, permit to deduce from 4.13 that
N ∂Zε p
∂x ωi xdx ≤ CK f L1 Ω ,
i
Ω i1
4.14
m
where |Zε | ≤ −K bsds CK is a constant independent of ε.
Now for a fixed K > 0, assumption 3.3 gives for i 1, . . . , N,
ε
K
K
uε , ∇Tm
uε ai x, Tm
⎡
1/p ⎣
≤ ωi x
Lx σx
1/p
max K, m
q/p
⎤
N
∂Zε p−1
1/p
⎦
.
ωj x
∂x
j j1
4.15
In view of 4.14 and 4.15, we deduce that
N
1−p
K
K
aε x, Tm
,
Lp Ω, wi
uε , ∇Tm
uε is bounded in
4.16
i1
then there exists a function XK ∈
N
i1 L
p
1−p
Ω, wi
such that
N
1−p
K
K
aε x, Tm
Lp Ω, wi
uε , ∇Tm
uε XK weakly in
as ε −→ 0.
4.17
i1
To prove that u is less or equal to m is an easy task which is performed exactly as in 10, 11.
uε − Tm
uε as a test function in 4.4 leads to
Using T2m
ε
a x, u , ∇u ∇ T2m
u − Tm
uε dx ε
f ε T2m
u − Tm
uε dx,
4.18
ε
ε
aε x, uε , ∇ T2m
u − Tm
uε ∇ T2m
u − Tm
uε dx ≤ mf L1 Ω .
4.19
ε
Ω
ε
ε
Ω
which implies easily that
Ω
Then 3.2, 4.1, and 4.2 yield
p−1
bm − ε
N ∂ T uε − T uε p
m
2m
ωi xdx ≤ mf L1 Ω .
∂xi
Ω i1
4.20
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International Journal of Mathematics and Mathematical Sciences
With the help of Poincaré’s inequality, we have
Ω
p
ε
T u − Tm
uε ω0 xdx ≤
2m
Cm
bm − εp−1 f ,
L1 Ω
4.21
where C does not depend on ε. Then in view of 3.1, 4.11, and ω0 > 0, we can pass to the
limit in 4.21 as ε tends to 0, to deduce that
T2m
u − Tm
u 0
u≤m
a.e. in Ω,
Let us now take TK vε as a test function in 4.4, where vε Ω
4.22
a.e. in Ω.
aε x, uε , ∇uε ∇TK vε dx Ω
uε
0
bε sds. We obtain
f ε TK vε dx ≤ K f L1 Ω .
4.23
Then 3.2 yields
N ∂TK vε p
∂x ωi xdx ≤ K f L1 Ω .
i
Ω i1
4.24
We deduce with a classical argument that, for a subsequence still indexed by ε,
vε −→ v
a.e. in Ω,
4.25
1,p
TK vε −→ TK v weakly in W0 Ω, ω,
4.26
as ε tends to 0, where v is a measurable function defined on Ω which is finite a.e. in Ω.
Using the admissible test function θn vε in 4.4 leads to
Ω
aε x, uε , ∇uε ∇θn vε dx Ω
f ε θn vε dx.
4.27
As a consequence of the previous convergence results, we are in a position to pass to the limit
as ε tends to 0 in 4.27
lim
ε→0 Ω
a x, u , ∇u ∇θn v dx ε
ε
ε
ε
Ω
fθn vdx.
4.28
International Journal of Mathematics and Mathematical Sciences
9
Using the pointwise convergence of θn u to 0 as n tends to ∞ and |θn u| ≤ 1 a.e.
1
in
Ω independently of n, since f ∈ L Ω, Lebesgue’s convergence theorem shows that
fθn vdx → 0, as n tends to ∞. Passing to the limit in 4.28 we obtain
Ω
lim lim
n → ∞ε → 0 {n≤|vε |≤n
1}
aε x, uε , ∇uε ∇vε dx 0.
4.29
Step 3. In this step we prove the following monotonicity estimate.
Lemma 4.2. The subsequence of uε defined in Step 1 satisfies for any K ≥ 0
lim
K ε
K
aε Tm
u , ∇Tm
u K ε
K
dx 0.
∇T
−
−
∇T
u
u
m
m
bε uε p−1
K ε
K
aε Tm
u , ∇Tm
uε bε uε p−1
ε→0 Ω
4.30
Proof. Let K ≥ 0 be fixed. Equality 4.30 is split into
K ε
K
aε Tm
u , ∇Tm
uε bε uε p−1
Ω
−
K ε
K
aε Tm
u , ∇Tm
u bε uε p−1
K
K
∇Tm
uε − ∇Tm
u dx
4.31
Aε1 Aε2 Aε3 ,
where
K ε
K
aε Tm
u , ∇Tm
uε Aε1
bε uε p−1
Ω
K ε
K
aε Tm
u , ∇Tm
uε Aε2
Aε3 −
−
bε uε p−1
ε K ε
K
a Tm u , ∇Tm
u Ω
4.32
K
∇Tm
udx ds dt,
K
K
∇Tm
uε − ∇Tm
u dx ds dt.
bε uε p−1
Ω
K
∇Tm
uε dx ds dt,
In the sequel we pass to the limit in 4.31 when ε tends to 0.
Limit of Aε1
Using the admissible test function Sn vε Sn v a x, u , ∇u ε
Ω
ε
ε
Ω
where vε f ε Sn vε ε
0
bup−1
TmK u
0
uε
K
∇Tm
u
TmK u
0
1/bsp−1 ds in 4.4 leads to
dx 1
bsp−1
a u , ∇u ∇Sn v ·
ε
Ω
ε
ε
ε
T K u
dsdx,
bε sds, pass to the limit as ε tends to 0 in 4.33.
m
0
1
bsp−1
ds dx
4.33
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International Journal of Mathematics and Mathematical Sciences
Since suppSn ⊂ −n 1, n 1 and {x ∈ Ω; |vε | ≤ n 1} ⊂ {x ∈ Ω; |uε | ≤ n 1/α},
we have for i 1, . . . , N and ε ≤ α/n 1
ε
a x, uε , ∇uε Sn vε aε x, Tn
1/α uε , ∇Tn
1/α uε Sn vε i
i
⎡
q/p
1/p ⎣
≤ Sn L∞ R ωi x
Lx σx1/p Tn
1/α uε 4.34
⎤
N
∂T vε p−1
1/p
n
1
⎦
.
ωj x
∂x
j
j1
In view of 4.24, 4.34 we deduce that for fixed n ≥ 1:
N
1−p
aε x, Tn
1/α uε , ∇Tn
1/α uε Sn vε is bounded in
,
Lp Ω, wi
4.35
i1
independently of ε ≤ α/n 1. Then there exists a function Yn ∈
for fixed n ≥ 1:
N
i1 L
p
N
1−p
Sn vε aε x, Tn
1/α uε , ∇Tn
1/α uε Yn weakly in
Lp Ω, wi
i1
1−p
Ω, wi
such that
as ε −→ 0.
4.36
Now for maxK, m ≤ n/α, we have
Sn vε aε x, Tn
1/α uε , ∇Tn
1/α uε χ{−K<uε <m}
Sn vε aε x, TKm uε , ∇TKm uε χ{−K<uε <m}
4.37
a.e. in Ω, which implies that, through the use of 4.17, 4.25, and 4.36 and passing to the
limit as ε tends to 0,
Yn χ{−K<u<m} Sn vXK χ{−K<u<m}
4.38
a.e. in Ω − {{u −K} ∪ {u m}} for maxK, m ≤ n/α. As a consequence of 4.38 we have
for maxK, m ≤ n/α
K
Yn ∇Tm
u Sn vXK ∇K
m u
a.e. in Ω.
4.39
International Journal of Mathematics and Mathematical Sciences
11
We are now in a position to exploit 4.33, which gives together with 4.36 and 4.39
lim
ε→0 Ω
Sn vε aε x, uε , ∇uε lim
ε→0
Ω
K
∇Tm
u
Yn
bup−1
Ω
bup−1
dx
K
∇Tm
u
Sn vε aε x, Tn
1/α uε , ∇Tn
1/α uε dx
bup−1
Ω
K
∇Tm
u
Sn vXK
4.40
dx
K
∇Tm
u
bup−1
dx.
Passing to the limit as n tends to ∞ in 4.40 leads to
Sn v a x, u , ∇u ε
lim lim
n → ∞ε → 0 Ω
ε
ε
ε
K
∇Tm
u
bup−1
dx Ω
XK
K
∇Tm
u
bup−1
dx.
4.41
The second term of 4.33
T K
m u
1
ε
ε
ε
ε
ds
dx
a x, u , ∇u ∇Sn v .
p−1
Ω
bs
0
maxm, K
≤
aε x, uε , ∇uε ∇vε dx.
αp−1
{n≤|vε |≤n
1}
4.42
Then 4.29 implies that
a x, u , ∇u ∇Sn v .
ε
lim lim
n → ∞ε → 0 Ω
ε
ε
ε
T K u
m
1
bsp−1
0
ds dx 0.
4.43
In view 4.41 and 4.43, passing to the limit as ε tends to 0 and as n tends to ∞ in 4.33 is
an easy task and leads to
Ω
XK
K
∇Tm
u
bu
p−1
dx TmK u
Ω
f
1
bsp−1
0
4.44
ds dx.
We are now in a position to exploit 4.44.
T K uε The use of the test function 0 m 1/bε sp−1 ds in 4.4, yields
Ω
aε x, uε , ∇uε K
∇Tm
uε bε uε dx p−1
Ω
fε
TmK uε 0
1
bε sp−1
ds dx.
4.45
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International Journal of Mathematics and Mathematical Sciences
Passing to the limit as ε tends to 0 in 4.45, in view 4.44, we have
lim Aε
ε→0 1
lim
ε→0 Ω
a x, u , ∇u ε
ε
ε
K
∇Tm
uε dx bε uε p−1
Ω
XK
K
∇Tm
u
bup−1
dx.
4.46
Limit of Aε2
In view of 4.12, 4.17 and since 1/bε uε p−1 converges to 1/bup−1 a.e. in Ω and due to the
bound 1/bε uε p−1 ≤ 1/αp−1 a.e. in Ω, we have
lim Aε
ε→0 2
−
Ω
XK
K
∇Tm
u
bup−1
4.47
dx.
Limit of Aε3
Let us remark that 3.1, 4.1, and 4.11 imply that
K
K
aε x, Tm
uε , ∇Tm
u
bε uε p−1
−→
K
K
a x, Tm
u, DTm
u
bup−1
a.e. in Ω,
4.48
as ε tends to 0, and that for i 1, . . . , N
ε
a T K uε , ∇T K u m
i m
bε uε p−1
⎡
p−1 ⎤
q/p
N
∂T K u
1
max
m
K,
1/p
1/p ⎦
m
1/p
≤ wi x⎣ p−1 Lx σ
w
x
j
∂x
α
αp−1
j
j1
4.49
a.e. in Ω, uniformly with respect to ε.
It follows that when ε tends to 0
K
K
a x, Tm
uε , ∇Tm
u
bε uε p−1
−→
K
a x, Tm
u, ∇TK u
bup−1
strongly in
N
1−p
.
Lp Ω, wi
4.50
i1
In view of 4.12, we conclude that
N
K
K
∇Tm
Lp Ω, wi , as ε goes to 0.
uε − ∇Tm
u 0 weakly in
4.51
i1
As a consequence of 4.50 and 4.51 we have for all K > 0
lim Aε3 0.
ε→0
4.52
International Journal of Mathematics and Mathematical Sciences
13
Equations 4.46, 4.46, 4.47, and 4.46 allow to pass to the limit as ε tends to zero in 4.31
and to obtain 4.30 of Lemma 4.2.
Step 4. In this step we identify the weak limit XK and we prove the weak L1 convergence of
K
K
K
uε , ∇Tm
uε /bε uε p−1 ∇Tm
uε as ε tends to 0.
the “truncated” energy aε x, Tm
Lemma 4.3. For fixed K ≥ 0, one has
K
K
XK a x, Tm
u, ∇Tm
u
a.e. in {x ∈ Ω; ux < m}.
4.53
And as ε tends to 0
K
K
aε x, Tm
uε , ∇Tm
uε bε uε p−1
K
∇Tm
uε K
K
a x, Tm
u, ∇Tm
u
p−1
bu
K
∇Tm
u weakly in L1 Ω. 4.54
Proof. Let K ≥ 0 be fixed. From 4.11 and 4.50 together with 4.30 of Lemma 4.2, we obtain
K
K
aε x, Tm
uε , ∇Tm
uε XK
K
ε
∇T
lim
u
dx
m
K ε p−1
K p−1 ∇TK udx.
ε→0 Ω
Ω
b Tm u b Tm u
4.55
We remark the monotone character a with respect to ξ and since 1/bε uε p−1 converges to
1/bup−1 a.e. in Ω and due to the bound 1/bε uε p−1 ≤ 1/αp−1 a.e. in Ω, we conclude that for
p
all ψ ∈ N
i1 L Ω, wi we have
0 ≤ lim
K
K
aε x, Tm
uε , ∇Tm
uε ε→0 Ω
bε uε p−1
K
K
aε x, Tm
uε , ∇Tm
uε lim
ε→0 Ω
− lim
ε→0 Ω
bε uε p−1
K
aε x, Tm
uε , ψ bε uε p−1
K
aε x, Tm
uε , ψ K ε
∇Tm u − ψ dx
−
p−1
bε uε K
∇Tm
uε − ψ dx
K
∇Tm
uε − ψ dx
4.56
K
a x, Tm
u, ψ K
K
∇T
∇T
−
ψ
dx
−
−
ψ
dx
u
u
m
m
p−1
bup−1
Ω bu
Ω
K
a x, Tm
u, ψ K
XK
∇T
−
−
ψ
dx.
×
u
m
p−1
bup−1
Ω bu
XK
The usual Minty’s argument applies in view of 4.56. It follows that
XK
bup−1
K
K
a x, Tm
u, ∇Tm
u
bup−1
which together with 4.20 yields 4.53 of Lemma 4.3.
a.e. in Ω
4.57
14
International Journal of Mathematics and Mathematical Sciences
In order to prove 4.54, we observe that the monotone character of a with respect to
ξ and 4.30 give
K
K
aε x, Tm
uε , ∇Tm
uε bε uε p−1
K
K
aε x, Tm
uε , ∇Tm
u K ε
K
∇Tm u − ∇Tm
−
u −→ 0
p−1
bε uε 4.58
strongly in L1 Ω as ε tends to 0. Moreover 4.12, 4.17, 4.50, and 4.53 imply that
K
K
aε x, Tm
uε , ∇Tm
uε K
∇Tm
u
bε uε p−1
K
K
a x, Tm
u, ∇Tm
u
bup−1
K
∇Tm
u
4.59
K
∇Tm
u
4.60
weakly in L1 Ω as ε tends to 0
K
K
aε x, Tm
uε , ∇Tm
u
bε uε p−1
K
∇Tm
uε K
K
a x, Tm
u, ∇Tm
u
bup−1
weakly in L1 Ω as ε tends to 0, and
K
K
aε x, Tm
uε , ∇Tm
u
bε uε p−1
K
∇Tm
u
K
K
a x, Tm
u, ∇Tm
u
bup−1
K
∇Tm
u
4.61
strongly in L1 Ω as ε tends to 0.
Using the above convergence results 4.59, 4.60, and 4.61 in 4.58 we obtain that
for any K ≥ 0
K
K
aε x, Tm
uε , ∇Tm
uε bε uε p−1
K
∇Tm
uε K
K
a x, Tm
u, ∇Tm
u
bu
p−1
K
∇Tm
u
4.62
weakly in L1 Ω as ε tends to 0.
Step 5. In this step we prove that u satisfies 3.11.
n
1 ε
n
u − Tm
uε as a test function in 4.4 leads to
Using Tm
n
1 ε
n
ε
n
1 ε
n
a x, u , ∇u ∇ Tm u − Tm u dx f ε Tm
u − Tm
uε dx.
ε
Ω
ε
ε
Ω
4.63
International Journal of Mathematics and Mathematical Sciences
15
n
1
n
· − Tm
· ⊂ −n 1, −n, we have
Since suppTm
{−n−1≤uε x≤−n}
Ω
aε x, uε , ∇uε ∇uε dx
n
1 ε
n
aε x, uε , ∇uε ∇ Tm
u − Tm
uε dx
aε x, uε , ∇uε p−1
n
1 ε
n
n
1
∇ Tm
uε dx
u − Tm
uε b Tm−1
bε uε p−1
ε
n
1 ε
n
1 ε
a x, Tm
u , ∇Tm
u Ω
bε uε p−1
Ω
−
n
n
aε x, Tm
uε , ∇Tm
uε bε uε p−1
Ω
4.64
p−1
n
1 ε
n
1
∇Tm
uε dx
u b Tm−1
p−1
n
n
1
∇Tm
dx.
uε b Tm−1
uε n
1
n
1
uε p−1 converges to bTm−1
up−1 a.e. in Ω
In view of 4.54 of Lemma 4.3 and since bTm−1
p−1
n
1
ε p−1
and due to the bound bTm−1 u ≤ maxs∈−n−1,m−1 bs
a.e. in Ω, we can pass to the
limit as ε tends to 0 for fixed n ≥ 0 to obtain
lim
ε → 0 {−n−1≤uε x≤−n}
n
1
n
1
a x, Tm
u, ∇Tm
u
bu
Ω
−
Ω
aε x, uε , ∇uε ∇uε dx
p−1
p−1
n
1
n
1
∇Tm
u
dx
ub Tm−1
n
n
p−1
ax, Tm
u, ∇Tm
u
n
n
1
∇T
u
dx
T
ub
m
m−1
p−1
4.65
bu
{−n−1≤ux≤−n}
ax, u, ∇u∇udx.
Taking the limit as ε tends to 0 and n tends to ∞ in 4.63 and using the estimate 4.64 and
4.65 show that
lim
n → ∞ {−n−1≤ux≤−n}
ax, u, ∇u∇udx ≤ lim
n → ∞ {u≤−n}
f dx 0.
4.66
Step 6. In this step we prove that u satisfies 3.12.
Using Sn vε 1/δTm−δ
u − Tm−2δ
u as a test function in 4.4 leads to
1
δ
Ω
Sn vε aε x, uε , ∇uε ∇ Tm−δ
u − Tm−2δ
u dx
Ω
Sn vε f ε
Tm−δ
u − Tm−2δ
u
δ
4.67
dx,
16
International Journal of Mathematics and Mathematical Sciences
uε
where vε 0 bε sds. Since suppSn ⊂ −n 1, n 1 and {x ∈ Ω; |vε | ≤ n 1} ⊂ {x ∈
Ω; |uε | ≤ n 1/α} we have
1
δ
Ω
Sn vε aε x, uε , ∇uε ∇ Tm−δ
u − Tm−2δ
u dx
1
δ
Sn v a x, Tn
1/α u , ∇Tn
1/α u ∇ Tm−δ
u − Tm−2δ
u dx.
ε
Ω
ε
ε
4.68
ε
In view of 4.22, 4.36, 4.39, and 4.53, passing to the limit as ε tends to 0 and n tends to
∞
1
n → ∞ ε → 0 δ
lim lim
Sn vε aε x, Tn
1/α uε , ∇Tn
1/α uε ∇ Tm−δ
u − Tm−2δ
u dx
Ω
1
n → ∞ δ
lim
Ω
Sn va x, Tn
1/α u, ∇Tn
1/α u ∇ Tm−δ
u − Tm−2δ
u dx
1
Sn vax, Tm−δ u, ∇Tm−δ u∇ Tm−δ
u − Tm−2δ
u dx
n → ∞ δ
Ω
1
ax, Tm−δ u, ∇Tm−δ u∇ Tm−δ
u − Tm−2δ
u dx
δ Ω
1
ax, u, ∇u∇udx.
δ {m−2δ≤u≤m−δ}
lim
4.69
Taking the limit as ε tends to 0, n tends to ∞ and δ tends to 0 in 4.67 and using the estimate
4.68 and 4.69 show that
1
lim
δ→0δ
{m−2δ≤u≤m−δ}
ax, u, ∇u∇udx lim
δ→0 Ω
f
Tm−δ
u − Tm−2δ
u
δ
dx
{um}
4.70
fxdx.
1,p
Step 7. In this step, u is shown to satisfy 3.13. Let ϕ ∈ W0 Ω, ω ∩ L∞ Ω and let S be a
function in W 1,∞ R such that S has a compact support and Sm 0. Let K be a positive
uε
real number such that suppS ⊂ −K, K and vε 0 bε sds. Using SuSn vε ϕ as a test
function in 4.4 leads to
Sn v a x, u , ∇u ∇ Suϕ dx ε
Ω
ε
Ω
Suϕaε x, uε , ∇uε ∇Sn vε dx
4.71
f Sn v Suϕdx.
ε
Ω
ε
4.71.
ε
ε
In what follows we pass to the limit as ε tends to 0 and n tends to ∞ in each term of
International Journal of Mathematics and Mathematical Sciences
17
Limit of First Term in 4.71
Since supp Sn ⊂ −n 1, n 1 and {x ∈ Ω; |vε | ≤ n 1} ⊂ {x ∈ Ω; |uε | ≤ n 1/α}, we have
Sn vε aε x, uε , ∇uε Sn vε aε x, Tn
1/α uε , ∇Tn
1/α uε a.e. in Ω.
4.72
In view of 4.22, 4.36, 4.39, and 4.53, passing to the limit as ε tends to 0
lim
ε→0 Ω
Sn vε aε x, uε , ∇uε ∇ Suϕ dx
lim
ε→0 Ω
Ω
Ω
Sn vε aε x, Tn
1/α uε , ∇Tn
1/α uε ∇ Suϕ dx
Sn va x, Tn
1/α u, ∇Tn
1/α u ∇ Suϕ dx
K
K
Sn va x, Tm
u, ∇Tm
u ∇ Suϕ dx,
lim lim
n → ∞ε → 0 Ω
lim
Sn vε aε x, uε , ∇uε ∇ Suϕ dx
n → ∞ Ω
Ω
4.73
K
K
Sn va x, Tm
u, ∇Tm
u ∇ Suϕ dx
K
K
a x, Tm
ax, u, ∇u∇ Suϕ dx.
u, ∇Tm
u ∇ Suϕ dx Ω
Limit of Second Term in 4.71
Since suppSn ⊂ −n 1, −n ∪ n 1, n for any n ≥ 1. As a consequence
Suϕaε x, uε , ∇uε ∇Sn vε dx ≤ S ∞ ϕ ∞
L Ω
L Ω
Ω
{n≤|vε |≤n
1}
aε x, uε , ∇uε ∇vε dx.
4.74
Taking the limit as ε tends to 0 and n tends to ∞ in 4.74 and using the estimate 4.29 show
that
lim lim Suϕaε x, uε , ∇uε ∇Sn vε dx 0.
n → ∞ε → 0
Ω
4.75
Limit of the Right-Hand Side of 4.71
Due to 4.3 and 4.25, we have
lim lim
n → ∞ε → 0 Ω
f Sn v Suϕdx ε
ε
Ω
fSuϕdx.
4.76
18
International Journal of Mathematics and Mathematical Sciences
As a consequence of the previous convergence results, we are in a position to pass to the limit
as ε tends to 0 in 4.71 and to conclude that u satisfies 3.13. The proof of Theorem 4.1 is
achieved.
Acknowledgment
The author would like to thank the anonymous referees for interesting remarks.
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