On the L^p-theory of Anisotropic singular perturbations of elliptic problems

0.28 MB·English

by  Chokri Ogabi

ON THE Lp−THEORY OF ANISOTROPIC SINGULAR
PERTURBATIONS OF ELLIPTIC PROBLEMS
5
1
CHOKRIOGABI
0
2
Academie de Grenoble, 38300 France
b
e
F

7
2 Abstract. InthisarticlewegiveanextentionoftheL2−theoryofanisotropic
singularperturbationsforellipticproblems. Westudyalinearandsomenon-
] linear problems involving Lp data (1 < p < 2). Convergences in pseudo
P Sobolevspacesareprovedforweakandentropysolutions,andrateofconver-
A gence isgivenincylindricaldomains
.
h
t 1. Introduction
a
m 1.1. Preliminaries. In this article we shall give an extension of the L2−theory
[ ofthe asymptotic behavior ofelliptic, anisotropicsingular perturbations problems.

ThiskindofsingularperturbationshasbeenintroducedbyM.Chipot[6]. Fromthe
3
v physicalpointof view,these problemscanmodelize diffusion phenomena whenthe
4 diffusion coefficients in certain directions are going toward zero. The L2 theory of
3 theasymptoticbehavioroftheseproblemshasbeenstudiedbyM.Chipotandmany
4
co-authors. First of all, let us begin by a brief discussion on the uniqueness of the
7
weaksolution(byweakasolutionwemeanasolutioninthesenseofdistributions)
0
. to the problem
1
0 −div(A∇u)=f
(1)
5 u=0 on ∂Ω
1 (cid:26)
: where Ω ⊂ RN, N ≥ 2 is a bounded Lipschitz domain, we suppose that f ∈
v
i Lp(Ω) (1<p<2). The diffusion matrix A=(aij) is supposed to be bounded and
X
satisfiesthe ellipticity assumptiononΩ (see assumptions(2)and(3)insubsection
r 1.2). It is well known that (1) has at least a weak solution in W1,p(Ω). Moreover,
a 0
ifAissymmetricandcontinuousand∂Ω∈C2 [2]then(1)hasaunique solutionin
W1,p(Ω). If A is discontinuous the uniqueness assertion is false, in [15] Serrin has
0
given a counterexample when N ≥ 3. However, if N = 2 and if ∂Ω is sufficiently
smoothandwithoutanycontinuityassumptiononA,(1)hasauniqueweaksolution
in W1,p(Ω). The proof is based on the Meyers regularitytheorem (see for instance
0
[13]). To treat this pathology, Benilin, Boccardo,Gallouet, and al have introduced
the concept of the entropy solution [4] for problems involving L1 data (or more
generally a Radon measure).
Date:28,January2015.
1991 Mathematics Subject Classification. 35J15,35B60,35B25.
Key words and phrases. Anisotropicsingular perturbations, ellipticproblem, Lp−theory, en-
tropysolutions. Asymptoticbehaviour. rateofconvergence.
1
2 CHOKRIOGABI
For every k>0 We define the function T :R→R by
k
s ,|s|≤k
T (s)=
k ksgn(x) |s|≥k
(cid:26)
And we define the space T1,2 introduced in [4].
0
u :Ω→R measurable such that for any k >0 there exists
T1,2(Ω)= (φ )⊂H1(Ω):φ →T (u) a.e in Ω
0  n 0 n k 
and (∇φ ) is bounded in L2(Ω)
 n n∈N 
This definition of T01,2 is equivalent to the original one given in [4].In fact, this
is a characterizationof this space [4]. Now, more generally, for f ∈L1(Ω) we have
the following definition of entropy solution [4].
Definition 1. A function u∈T1,2(Ω) is said to be an entropy solution to (1) if
0
A∇u·∇T (u−ϕ)dx≤ fT (u−ϕ)dx, ϕ∈D(Ω), k>0
k k
ZΩ ZΩ
We refer the reader to [4] for more details about the sense of this formulation.
The main results of [4] show that (1) has a unique entropy solution which is also
a weak solution of (1) moreover since Ω is bounded then this solution belongs to
W1,r(Ω).
0
1≤r\<NN−1
1.2. Description of the problem and functional setting. Throughout this
articlewewillsupposethatf ∈Lp(Ω),1<p<2,(wecansupposethatf ∈/ L2(Ω)).
Wegiveadescriptionofthelinearproblem(somenonlinearproblemswillbestudied
later). Consider the following singular perturbations problem
−div(A ∇u )=f
ǫ ǫ , (2)
u =0 on ∂Ω
ǫ
(cid:26)
where Ω is a bounded Lipschitz domain of RN. Let q ∈N∗, N −q ≥2. We denote
by x= (x ,...,x )=(X ,X )∈Rq ×RN−q i.e. we split the coordinates into two
1 N 1 2
parts. With this notation we set
∇
∇=(∂ ,...,∂ )T = X1 ,
x1 xN ∇
(cid:18) X2(cid:19)
where
∇ =(∂ ,...,∂ )T and ∇ =(∂ ,...,∂ )T
X1 x1 xq X2 xq+1 xN
Let A=(a (x)) be a N ×N matrix which satisfies the ellipticity assumption
ij
∃λ>0:Aξ·ξ ≥λ|ξ|2 ∀ξ ∈RN for a.e x∈Ω, (3)
and
a (x)∈L∞(Ω),∀i,j =1,2,....,N, (4)
ij
We have decomposed A into four blocks
A A
A= 11 12 ,
A A
21 22
(cid:18) (cid:19)
ON THE Lp−THEORY OF ANISOTROPIC SINGULAR... 3
where A , A are respectively q × q and (N − q)× (N −q) matrices. For
11 22
0<ǫ≤1 we have set
ǫ2A ǫA
A = 11 12
ǫ ǫA A
21 22
(cid:18) (cid:19)
WedenoteΩ = X ∈RN−q :(X ,X )∈Ω andΩ1 =P ΩwhereP :RN →
X1 2 1 2 1 1
Rp is the usual projector. We introduce the space
(cid:8) (cid:9)
u∈Lp(Ω)|∇ u∈Lp(Ω),
V = X2
p and for a.e X ∈Ω1,u(X ,·)∈W1,p(Ω )
(cid:26) 1 1 0 X1 (cid:27)
We equip V with the norm
p
1
kuk = kukp +k∇ ukp p ,
Vp Lp(Ω) X2 Lp(Ω)
(cid:16) (cid:17)
then one can show easily that (V ,k·k ) is a separable reflexive Banach space.
p Vp
The passage to the limit (formally) in (2) gives the limit problem
−div (A ∇ u (X ,·))=f (X ,·)
X2 22 X2 0 1 1 (5)
u (X ,·)=0 on ∂Ω X ∈Ω1
(cid:26) 0 1 X1 1
TheL2-theory(whenf ∈L2)ofproblem(2)hasbeentreatedin[8],convergence
hasbeenprovedinV andrateofconvergenceinthe L2−normhasbeengiven. For
2
the L2−theory of several nonlinear problems we refer the reader to [9],[10],[14].
This article is mainly devoted to study the Lp−theory of the asymptotic behavior
of linear and nonlinear singularly perturbed problems. In other words, we shall
study the convergenceu →u inV (Notice that in [9], authors have treated some
ǫ 0 p
problems involving Lp data where some others data of the equations depend on p,
onecancheckeasilythatitisnottheLptheorywhichweexposeinthismanuscript).
Let us briefly summarize the content of the paper:
• In section 2: We study the linear problem, we proveconvergencesfor weak
and entropy solutions.
• In section3: We give the rate of convergencein a cylindrical domainwhen
the data is independent of X .
1
• In section 4: We treat some nonlinear problems.
2. The Linear Problem
The main results in this section are the following
Theorem 1. Assume (3), (4) then there exists a sequence (u ) ⊂ W1,p(Ω)
ǫ 0<ǫ≤1 0
of weak solutions to (2) and u ∈V such that ǫ∇ u →0 in Lp(Ω), u →u in
0 p X1 ǫ ǫ 0
V where u satisfies (5) for a.e X ∈Ω1.
p 0 1
Corollary1. Assume(3),(4)thenifAissymmetricandcontinuousand∂Ω∈C2,
then there exists a unique u ∈V such that u (X ;·) is the unique solution to (5)
0 p 0 1
in W1,p(Ω ) for a.e X . Moreover the sequence (u ) of the unique solutions
0 X1 1 ǫ 0<ǫ≤1
(in W1,p(Ω)) to (2) converges in V to u and ǫ∇ u →0 in Lp(Ω).
0 p 0 X1 ǫ
Proof. This corollary follows immediately from Theorem 1 and uniqueness of the
solutionsof(2)and(5)asmentionedinsubsection1.1(Noticethat∂Ω ∈C2). (cid:3)
X1
4 CHOKRIOGABI
Theorem2. Assume(3),(4)thenthereexistsauniqueu ∈V suchthatu (X ,·)
0 p 0 1
is the unique entropy solution of (5). Moreover, the sequence of the entropy solu-
tions (u ) of (2) converges to u in V and ǫ∇ u →0 in Lp(Ω).
ǫ 0<ǫ≤1 0 p X1 ǫ
2.1. Weak convergence. Let us prove the following primary result
Theorem 3. Assume (3), (4) then there exists a sequence (u ) ⊂W1,p(Ω) of
ǫk k∈N 0
weak solutions to (2) (ǫ →0 as k →∞) and u ∈V such that ∇ u ⇀∇ u ,
k 0 p X2 ǫk X2 0
ǫ ∇ un ⇀0, u ⇀u in Lp(Ω−weak. and u satisfies (5) for a.e X ∈Ω1.
k X1 ǫk ǫk 0 0 1
Proof. By density let (f ) ⊂ L2(Ω) be a sequence such that f → f in Lp(Ω),
n n∈N n
we can suppose that ∀n ∈ N :kf k ≤ M, M ≥ 0. Consider the regularized
n Lp
problem
un ∈H1(Ω), A ∇un·∇ϕdx= f ϕdx , ϕ∈D(Ω) (6)
ǫ 0 ǫ ǫ n
ZΩ ZΩ
Assumptions (2) and (3) shows that un exists and it is unique by the Lax-
ǫ
Milgram theorem. (Notice that un also belongs to W1,p(Ω)). We introduce the
ǫ 0
function
t
θ(t)= (1+|s|)p−2ds, t∈R
Z
0
This kind of function has been used in [3]. We have θ′(t) = (1+|t|)p−2 ≤ 1 and
θ(0)=0,therefore we have θ(u)∈H1(Ω) for everyu∈H1(Ω). Testing with θ(un)
0 0 ǫ
in (6) and using the ellipticity assumption we deduce
λǫ2 (1+|un|)p−2|∇ un|2dx+λ (1+|un|)p−2|∇ un|2dx
ǫ X1 ǫ ǫ X2 ǫ
ZΩ ZΩ
2
≤ f θ(un)dx ≤ |f |(1+|un|)p−1dx,
n ǫ p−1 n ǫ
ZΩ ZΩ
where we have used |θ(t)| ≤ 2(1+|t|)p−1. In the other hand, by Ho¨lder’s inequality
p−1
we have
p 1−p
|∇ un|pdx≤ (1+|un|)p−2|∇ un|2dx 2 (1+|un|)pdx 2
X2 ǫ ǫ X2 ǫ ǫ
ZΩ (cid:18)ZΩ (cid:19) (cid:18)ZΩ (cid:19)
From the two previous integral inequalities we deduce
p
|∇ un|pdx≤ 2 |f |(1+ un )p−1dx 2 ×
X2 ǫ λ(p−1) n ǫk
ZΩ (cid:18) ZΩ (cid:19)
(cid:12) (cid:12) 1−p
(cid:12) (cid:12) 2
(1+ un )pdx
ǫk
(cid:18)ZΩ (cid:19)
(cid:12) (cid:12)
By Ho¨lder’s inequality we get (cid:12) (cid:12)
1 1
k∇ unk ≤ 2kfnkLp 2 (1+|un|)pdx 2p (7)
X2 ǫ Lp(Ω) λ(p−1) ǫ
(cid:18) (cid:19) (cid:18)ZΩ (cid:19)
Using Minkowki inequality we get
k∇ unk2 ≤C(1+kunk ),
X2 ǫ Lp(Ω) ǫ Lp(Ω)
ON THE Lp−THEORY OF ANISOTROPIC SINGULAR... 5
Thanks to Poincar´e’sinequality kunk ≤C k∇ unk we obtain
ǫ Lp(Ω) Ω X2 ǫ Lp(Ω)
k∇ unk2 ≤C′(1+k∇ unk ),
X2 ǫ Lp(Ω) X2 ǫ Lp(Ω)
wheretheconstantC′ dependsonp,λ,mes(Ω),M andC . Whence,wededuce
Ω
kunk , k∇ unk ≤C′′ (8)
ǫ Lp(Ω) X2 ǫ Lp(Ω)
Similarly we obtain
kǫ∇ unk ≤C′′′, (9)
X1 ǫ Lp(Ω)
where the constants C′′, C′′′ are independent of n and ǫ, so
Const
kunk ≤ (10)
ǫ W1,p(Ω) ǫ
Fixǫ,sinceW1,p(Ω)isreflexivethen(10)impliesthatthereexistsasubsequence
(unl(ǫ)) and u ∈ W1,p(Ω) such that unl(ǫ) ⇀ u ∈ W1,p(Ω) (as l → ∞) in
ǫk l∈N ǫ 0 ǫ ǫ 0
W1,p(Ω)−weak. Now, passing to the limit in (6) as l→∞ we deduce
A ∇u ·∇ϕdx= fϕdx , ϕ∈D(Ω) (11)
ǫ ǫ
ZΩ ZΩ
Whence u is a weak solution of (2) (u = 0 on ∂Ω in the trace sense of
ǫ ǫ
W1,p−functions, indeed the trace operator is well defined since ∂Ω is Lipschitz).
Now, from (8) and (9) we deduce
kuǫkLp(Ω) ≤liminf uǫnl(ǫ) ≤C′
l→∞ Lp(Ω)
(cid:13) (cid:13)
and similarly we obtain (cid:13) (cid:13)
(cid:13) (cid:13)
kǫ∇ u k , k∇ u k ≤C′
X1 ǫ Lp(Ω) X2 ǫ Lp(Ω)
Using reflexivity and continuity of the derivation operator on D′(Ω) one can
extractasubsequence(u ) suchthat∇ u ⇀∇ u ,ǫ ∇ un ⇀0,u ⇀
ǫk k∈N X2 ǫk X2 0 k X1 ǫk ǫk
u in Lp(Ω)−weak. Passing to the limit in (11) we get
0
A ∇ u ·∇ ϕdx= fϕdx , ϕ∈D(Ω) (12)
22 X2 0 X2
ZΩ ZΩ
Now, we will prove that u ∈ V . Since ∇ u ⇀ ∇ u and u ⇀ u in
0 p X2 ǫk X2 0 ǫk 0
Lp(Ω)−weak then there exists a sequence (U ) ⊂ conv({u } ) such that
n n∈N ǫk k∈N
∇ U → ∇ u in Lp(Ω)−strong, where conv({u } ) is the convex hull of
X2 n X2 0 ǫk k∈N
the set {u } . Notice that we have U ∈ W1,p(Ω) then -up to a subsequence-
ǫk k∈N n 0
we have U (X ,·) ∈ W1,p(Ω ), a.e X ∈ Ω1. And we also have -up to a
n 1 0 X1 1
subsequence- ∇ U (X ,·) → ∇ u (X ,.) in Lp(Ω )− strong a.e X ∈ Ω1.
X2 n 1 X2 0 1 X1 1
Whence u (X ,.)∈W1,p(Ω ) for a.e X ∈Ω1, so u ∈V .
0 1 0 X1 1 0 p
Finally, we will prove that u is a solution of (5). Let E be a Banach space, a
0
familyofvectors{e } inE issaidtobeaBanachbasisoraSchauderbasisofE
n n∈N
∞
if for every x∈E there exists a family of scalars (α ) such that x= α e ,
n n∈N n n
n=0
X
where the series converges in the norm of E. Notice that Schauder basis does not
always exist. In [11] P. Enflo has constructed a separable reflexive Banach space
without Schauder basis!. However, the Sobolev space W1,r ( 1 < r < ∞) has a
0
Schauder basis whenever the boundary of the domain is sufficiently smooth [12].
Now,wearereadytofinishtheproof. Let(U ×V ) beacountablecoveringofΩ
i i i∈N
6 CHOKRIOGABI
such that U ×V ⊂Ω where U ⊂Rq,V ⊂RN−q are two bounded open domains,
i i i i
where ∂V is smooth (V are Euclidian balls for example), such a covering always
i i
exists. Now, fix ψ ∈ D(V ) then it follows from (12) that for every ϕ ∈ D(U ) we
i i
have
ϕdX A ∇ u ·∇ ψdX = ϕdX fψdX
1 22 X2 0 X2 2 1 2
ZUi ZVi ZUi ZVi
Whence for a.e X ∈U we have
1 i
A (X ,·)∇ u (X ,·)·∇ ψdX = f(X ,·)ψdX
22 1 X2 0 1 X2 2 1 2
ZVi ZVi
Notice that by density we can take ψ ∈ W1,p′(V ) where p′ is the conjugate of
0 i
p. Using the same techniques as in [8], where we use a Schauder basis of W1,p′(V )
0 i
and a partition of the unity, one can easily obtain
A (X ,·)∇ u (X ,·)·∇ ϕdx= f(X ,·)ϕdx, ϕ∈D(Ω),
22 1 X2 0 1 X2 1
ZΩX1 ZΩX1
for a.e X ∈ Ω1. Finally, since u (X ,·) ∈ W1,p(Ω ) (as proved above) then
1 0 1 0 X1
u (X ,·) is a solution of (5) (Notice that Ω is also a Lipschitz domain so the
0 1 X1
trace operator is well defined). (cid:3)
2.2. Strong convergence. Theorem 1 will be proved in three steps. the proof
is based on the use of the approximated problem (6). In the first step, we shall
construct the solution of the limit problem
Step1:Letun ∈H1(Ω)be the unique solutionto(6), existenceanduniqueness
ǫ 0
of un follows from assumptions (3), (4) as mentioned previously. One have the
ǫ
following
Proposition1. Assume(3), (4) then there exists (un) ⊂V suchthat ǫun →0
0 n∈N 2 ǫ
in L2(Ω), un →un in V for every n∈N, in particular the two convergences holds
ǫ 0 2
in Lp(Ω) and V respectively. And un is the unique weak solution in V to the
p 0 2
problem
div (A (X ,·)∇ un(X ,·))=f (X ,·), X ∈Ω1
X2 22 1 X2 0 1 n 1 1 (13)
un(X ,·)=0 on ∂Ω
(cid:26) 0 1 X1
Proof. ThisresultfollowsfromtheL2−theory(Theorem1in[8]),Theconvergences
in V and Lp(Ω) follow from the continuous embedding V ֒→V , L2(Ω)֒→Lp(Ω)
p 2 p
(p<2). (cid:3)
Now, we construct u the solution of the limit problem (5). Testing with
0
ϕ = θ(un(X ,·)) in the weak formulation of (13) (θ is the function introduced
0 1
insubsection2.1)andestimatinglikeinthe proofofTheorem3weobtainasin(7)
k∇ un(X ,·)k
X2 0 1 Lp(ΩX1)
1 1
≤ kfn(X1,·)kLp(ΩX1) 2 × (1+|un(X ,·)|)pdX 2p (14)
λ(p−1) 0 1 2
! ZΩX1 !
ON THE Lp−THEORY OF ANISOTROPIC SINGULAR... 7
IntegratingoverΩ1 andusingCauchy-Schwaz’sinequalityinthe righthandside
we get
1
k∇ unkp ≤Ckf kp2 (1+|un|)pdx 2
X2 0 Lp(Ω) n Lp(Ω) 0
(cid:18)ZΩ (cid:19)
and therefore
k∇ unk2 ≤C′(1+kunk )
X2 0 Lp(Ω) 0 Lp(Ω)
Using Poincar´e’s inequality kunk ≤ C k∇ unk ( which holds since
0 Lp(Ω) Ω X2 0 Lp(Ω)
un(X ,·)∈W1,p(Ω ) a.e X ∈Ω1), one can obtain the estimate
0 1 0 X1 1
kunk ≤C′′ for every n∈N, (15)
0 Lp(Ω)
where C′′ is independent of n. Now, using the linearity of the problem and (13)
withthetestfunctionθ(un(X ,·)−um(X ,·)), m,n∈Nonecanobtainlikein(14)
0 1 0 1
k∇ (un(X ,·)−um(X ,·))k
X2 0 1 0 1 Lp(ΩX1)
1
≤ kfn(X1,·)−fm(X1,·)kLp(ΩX1) 2 ×
λ(p−1)
!
1
2p
(1+|un(X ,·)−um(X ,·)|)pdX
0 1 0 1 2
ZΩX1 !
integrating over Ω1 and using Cauchy-Schwarzand (15) yields
k∇X2(un0 −um0 )kLp(Ω) ≤Ckfn−fmkL21p(Ω),
where C is independent of m and n. The Poincar´e’sinequality shows that
kun0 −um0 kVp ≤C′kfn−fmkL12p(Ω)
Since (f ) is a converging sequence in Lp(Ω) then this last inequality shows
n n∈N
that (un) is a Cauchy sequence in V , consequently there exists u ∈ V such
0 n∈N p 0 p
that un →u in V . Now, passing to the limit in (6) as ǫ→0 we get
0 0 p
A ∇ un·∇ ϕdX = f ϕdX , ϕ∈D(Ω)
22 X2 0 X2 2 n 2
Z Z
Ω Ω
Passing to the limit as n→∞ we deduce
A ∇ u ·∇ ϕdX = fϕdX , ϕ∈D(Ω)
22 X2 0 X2 2 2
ZΩ ZΩ
Then it follows as proved in Theorem 3 that u satisfies (5). Whence we have
0
proved the following
Proposition 2. Under assumption of Proposition 1 there exists u ∈ V solution
0 p
to (5) such that un →u in V where (un) is the sequence given in Proposition
0 0 p 0 n∈N
1
Step2: In this second step we will construct the sequence (u ) solutions
ǫ 0<ǫ≤1
of (2), one can prove the following
8 CHOKRIOGABI
Proposition 3. There exists a sequence (u ) ⊂W1,p(Ω) of weak solutions to
ǫ 0<ǫ≤1 0
(2) such that un →u in W1,p(Ω) for every ǫ fixed. Moreover, un →u in V and
ǫ ǫ ǫ ǫ p
ǫ∇ un →ǫ∇ u , uniformly in ǫ.
X2 ǫ X2 ǫ
Proof. Using the linearity of(6) testing with θ(un−um), m,n∈N we obtain as in
ǫ ǫ
(7)
1 1
k∇ un−umk ≤ kfn−fmkLp 2 (1+|un−um|)p 2p
X2 ǫ ǫ Lp(Ω) λ(p−1) ǫ ǫ
(cid:18) (cid:19) (cid:18)ZΩ (cid:19)
And (8) gives
k∇X2(unǫ −umǫ )kLp(Ω) ≤Ckfn−fmkL21p
where C is independent of ǫ and n, whence Poincar´e’sinequality implies
kun−umk ≤C′kf −f k12 (16)
ǫ ǫ Vp n m Lp
Similarly we obtain
kǫ∇ (un−um)k ≤C′′kf −f k21 (17)
X2 ǫ ǫ Lp(Ω) n m Lp
its follows that
kunǫ −umǫ kW1,p(Ω) ≤ Cǫ kfn−fmkL21p
Thelastinequalityimpliesthatforeveryǫfixed(un) isaCauchysequencein
ǫ n∈N
W1,p(Ω), Then there exists u ∈W1,p(Ω) such that un →u in W1,p(Ω), then the
0 ǫ 0 ǫ ǫ
passageto the limit in (6) showsthat u is a weaksolutionof(2). Finally (16) and
ǫ
(17)showthatun →u (respǫ∇ un →ǫ∇ u )inV ( respinLp(Ω)) uniformly
ǫ ǫ X2 ǫ X2 ǫ p
in ǫ. (cid:3)
Step3: Now, we are ready to conclude. Proposition 1, 2 and 3 combined with
the triangular inequality show that u →u in V and ǫ∇ u →0 in Lp(Ω), and
ǫ 0 p X2 ǫ
the proof of Theorem 1 is finished.
2.3. Convergence of the entropy solutions. As mentioned in section 1 the
entropy solution u of (2) exists and it is unique. We shall construct this entropy
ǫ
solution. Usingtheapproximatedproblem(6),onehasaW1,p−stronglyconverging
sequence un → u ∈ W1,p(Ω) as shown in Proposition 3. We will show that
ǫ ǫ 0
u ∈ T1,2(Ω). Clearly we haveT (un) ∈ H1(Ω) for every k > 0. Now testing with
ǫ 0 k ǫ 0
T (un) in (6) we obtain
k ǫ
A ∇un·∇T (un)dx= f T (un)dx
ǫ ǫ k ǫ n k ǫ
ZΩ ZΩ
Using the ellipticity assumption we get
Mk
|∇T (un)|2 ≤ (18)
k ǫ λ(1+ǫ2)
ZΩ
Fixǫ,k,wehaveunǫ →uǫ inLp(Ω)thenthereexistsasubsequence(unǫl)l∈N such
that unǫl → uǫ a.e x ∈ Ω and since Tk is bounded then it follows that Tk(unǫl) →
T (u ) a.e in Ω and strongly in L2(Ω) whence u ∈T1,2(Ω).
k ǫ ǫ 0
It follows by (18) that there exists a subsequence still labelled Tk(unǫl) such
that ∇Tk(unǫl) → vǫ,k ∈ L2(Ω).The continuity of ∇ on D′(Ω) implies that vǫ,k =
ON THE Lp−THEORY OF ANISOTROPIC SINGULAR... 9
∇Tk(uǫ), whence Tk(unǫl) → Tk(uǫ) in H1(Ω). Now, since Tk(unǫl) ∈ H01(Ω) then
we deduce that T (u )∈H1(Ω).
k ǫ 0
It follows [4] that
A ∇u ·∇T (u −ϕ)dx≤ fT (u −ϕ)dx
ǫ ǫ k ǫ k ǫ
ZΩ ZΩ
Whence u is the entropy solution of (2). Similarly the function u (constructed
ǫ 0
in Proposition 2) is the entropy solution to (5) for a.e X The uniqueness of u in
1 0
V follows from the uniqueness of the entropy solution of problem (5). Finally, the
p
convergences given in Theorem 2 follows from Theorem 1.
Remark 1. Uniqueness of the entropy solutions implies that it does not depend on
the choice of the approximated sequence (f ) .
n n
2.4. A regularity result for the entropy solution of the limit problem.
In this subsection we assume that Ω = ω ×ω where ω , ω are two bounded
1 2 1 2
Lipschitz domains of Rq, RN−q respectively. We introduce the space
W ={u∈Lp(Ω)|∇ u∈Lp(Ω)}
p X1
We suppose the following
f ∈W and A (x)=A (X ) i.e A is independent of X (19)
p 22 22 2 22 1
Theorem 4. Assume (3), (4), (19) then u ∈ W1,p(Ω), where u is the entropy
0 0
solution of (5).
Proof. Let (un) the sequence constructed in subsection 2.2, we have un → u in
0 0 0
V , where u is the entropy solution of (5) as mentioned in the above subsection.
p 0
Let ω′ ⊂⊂ ω be an open subset, for 0 < h < d(∂ω ,ω′) and for X ∈ ω′ we
1 1 1 1 1 1
set τiun =un(X +he ,X ) where e =(0,..,1,..,0) then we have by (13)
h 0 0 1 i 2 i
A ∇ (τiun−un)·∇ ϕdX = (τif −f )ϕdX , ϕ∈D(ω )
22 X2 h 0 0 X2 2 h n n 2 2
Zω2 Zω2
where we have used A (x)=A (X ).
22 22 2
t
We introduce the function θ (t) = (δ+|s|)p−2ds, δ > 0, t ∈ R we have
δ
Z
0
0<θ′(t)=(δ+|t|)p−2 ≤δp−2 and |θ (t)|≤ 2(δ+|t|)p−1
δ δ p−1
Testingwithϕ= h1θδ(τihun0h−un0)∈H01(ω2). Tomakethenotationslessheavywe
set
τiun−un (τif −f )
U = h 0 0, h n n =F
h h
Then we get
θ′(U)A ∇ U ·∇ UdX = Fθ (U)dX
δ 22 X2 X2 2 δ 2
Zω2 Zω2
Using the ellipticity assumption for the left hand side and Ho¨lder’s inequality
for the right hand side of the previous inequality we deduce
p−1
λ θ′(U)|∇ U|2dX ≤ 2 kFk (δ+|U|)pdX p
δ X2 2 p−1 Lp(ω2) 2
Zω2 (cid:18)Zω2 (cid:19)
10 CHOKRIOGABI
Using Ho¨lder’s inequality we derive
p 2−p
k∇ Ukp ≤ θ′(U)|∇ U|2dX 2 θ′(U)p−p2dX 2
X2 Lp(ω2) δ X2 2 δ 2
(cid:18)Zω2 (cid:19) (cid:18)Zω2 (cid:19)
p
p−1 2
2 p p
≤ kFk (δ+|U|) dX ×
λ(p−1) Lp(ω2)(cid:18)Zω2 2(cid:19) !
2−p
θ′(U)p−p2dX 2
δ 2
(cid:18)Zω2 (cid:19)
Then we deduce
1
k∇ Uk2 ≤ 2 kFk (δ+|U|)pdX p
X2 Lp(ω2) λ(p−1) Lp(ω2) 2
(cid:18)Zω2 (cid:19)
Now passing to the limit as δ →0 using the Lebesgue theorem we deduce
1
k∇ Uk2 ≤ 2 kFk (|U|)pdX p ,
X2 Lp(ω2) λ(p−1) Lp(ω2) 2
(cid:18)Zω2 (cid:19)
and Poincar´e’sinequality gives
2C
k∇ Uk ≤ ω2 kFk
X2 Lp(ω2) λ(p−1) Lp(ω2)
Now, integrating over ω′ yields
1
τiun−un 2C (τif −f )
h 0 0 ≤ ω2 h n n
h λ(p−1) h
(cid:13)(cid:13) (cid:13)(cid:13)Lp(ω′1×ω2) (cid:13)(cid:13) (cid:13)(cid:13)Lp(ω′1×ω2)
Passin(cid:13)g to the lim(cid:13)it as n → ∞ using th(cid:13)e invariance o(cid:13)f the Lebesgue measure
(cid:13) (cid:13) (cid:13) (cid:13)
under translations we get
τiu −u 2C (τif −f)
h 0 0 ≤ ω2 h
h λ(p−1) h
(cid:13)(cid:13) (cid:13)(cid:13)Lp(ω′1×ω2) (cid:13)(cid:13) (cid:13)(cid:13)Lp(ω′1×ω2)
When(cid:13)ce, since f (cid:13)∈W then (cid:13) (cid:13)
(cid:13) (cid:13) p (cid:13) (cid:13)
τiu −u
h 0 0 ≤C,
h
(cid:13)(cid:13) (cid:13)(cid:13)Lp(ω′1×ω2)
where(cid:13)C isindepe(cid:13)ndentofh,thereforewehave∇ u ∈Lp(Ω). Combiningthis
(cid:13) (cid:13) X1 0
with u ∈V we get the desired result. (cid:3)
0 p
3. The Rate of convergence Theorem
In this section we suppose that Ω = ω × ω where ω ,ω are two bounded
1 2 1 2
Lipschitz domains of Rq and RN−q respectively. We suppose that A , A and f
12 22
depend on X only i.e A (x) = A (X ), A (x) = A (X ) and f(x) = f(X ) ∈
2 12 12 2 22 22 2 2
Lp(ω ) (1<p<2), f ∈/ L2(ω ).
2 2
Letu , u betheuniqueentropysolutionsof(2),(5)respectivelythenunderthe
ǫ 0
above assumptions we have the following
Theorem 5. For every ω′ ⊂⊂ ω and m ∈ N∗ there exists C ≥ 0 independent of
1 1
ǫ such that
ku −u k ≤Cǫm
ǫ 0 Wp(ω′1×ω2)