OS1.dvi


CUBO A Mathematical Journal
Vol.12, No¯ 01, (133–148). March 2010

Well-Posedness Results for Anisotropic Nonlinear Elliptic

Equations with Variable Exponent and L1-Data

Stanislas OUARO

Laboratoire d’Analyse Mathématique des Equations (LAME),

UFR. Sciences Exactes et Appliquées, Université de Ouagadougou

03 BP 7021 Ouaga 03

Ouagadougou, Burkina Faso

email : souaro@univ-ouaga.bf

ABSTRACT

We study the anisotropic boundary value problem −

N
∑

i=1

∂

∂xi
ai

(

x,
∂

∂xi
u

)

= f in Ω, u = 0 on

∂Ω, where Ω is a smooth open bounded domain in RN (N ≥ 3) and f ∈ L1(Ω). We prove the
existence and uniqueness of an entropy solution for this problem.

RESUMEN

Estudiamos el problema de valores en la frontera anisotropico −
N
∑

i=1

∂

∂xi
ai

(

x,
∂

∂xi
u

)

= f

en Ω, u = 0 sobre ∂Ω, donde Ω es un dominio abierto suave do RN (N ≥ 3) y f ∈ L1(Ω).
Proveamos la existencia y unicidad de una solución de entroṕıa para este problema.

Key words and phrases: Anisotropic; variable exponent; entropy solution; electrorheological

fluids.

Math. Subj. Class.: 35J20, 35J25, 35D30, 35B38, 35J60.



134 Stanislas OUARO CUBO
12, 1 (2010)

1 Introduction

Let Ω be an open bounded domain of RN (N ≥ 3) with smooth boundary. Our aim is to prove

the existence and uniqueness of an entropy solution to the anisotropic nonlinear elliptic problem






















−
N
∑

i=1

∂

∂xi
ai

(

x,
∂

∂xi
u

)

= f in Ω

u = 0 on ∂Ω,

(1.1)

where the right-hand side f ∈ L1(Ω).

We assume that for i = 1, ...,N the function ai : Ω × R → R is Carathéodory and satisfies

the following conditions: ai(x,ξ) is the continuous derivative with respect to ξ of the mapping

Ai : Ω × R → R, Ai = Ai(x,ξ), i.e. ai(x,ξ) =
∂

∂ξ
Ai(x,ξ) such that

the following equality and inequalities holds

Ai(x, 0) = 0, (1.2)

for almost every x ∈ Ω.

There exists a positive constant C1 such that

|ai(x,ξ)| ≤ C1(ji(x) + |ξ|
pi(x)−1) (1.3)

for almost every x ∈ Ω and for every ξ ∈ R, where ji is a nonnegative function in L
p′i(.)(Ω), with

1/pi(x) + 1/p
′
i(x) = 1.

There exists a positive constant C2 such that

(ai(x,ξ) − ai(x,η)) . (ξ − η) ≥

{

C2 |ξ − η|
pi(x) if |ξ − η| ≥ 1

C2 |ξ − η|
p
−

i if |ξ − η| < 1,
(1.4)

for almost every x ∈ Ω and for every ξ,η ∈ R, with ξ 6= η,

and

|ξ|
pi(x) ≤ ai(x,ξ).ξ ≤ pi(x)Ai(x,ξ) (1.5)

for almost every x ∈ Ω and for every ξ ∈ R.

We also assume that the variable exponent pi(.) : Ω → [2,N) are continuous functions for all

i = 1, ...,N such that:

p̄(N − 1)

N(p̄ − 1)
< p−i <

p̄(N − 1)

N − p̄
,

N
∑

i=1

1

p−i
> 1 and

p+i − p
−
i − 1

p−i
<

p̄ − N

p̄(N − 1)
, (1.6)

where
1

p̄
=

1

N

N
∑

i=1

1

p−i
, p−i := ess inf

x∈Ω
pi(x) and p

+
i := ess sup

x∈Ω
pi(x).

We introduce the numbers

q =
N(p̄ − 1)

N − 1
, q∗ =

Nq

N − q
=
N(p̄ − 1)

N − p̄
. (1.7)



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Well-Posedness Results for Anisotropic ... 135

A prototype example that is covered by our assumptions is the following anisotropic (p1(.), ...,pN (.))-

harmonic equation:

Set Ai(x,ξ) = (1/pi(x)) |ξ|
pi(x), ai(x,ξ) = |ξ|

pi(x)−2 ξ where pi(x) ≥ 2. Then we get the following

equation:

−
N
∑

i=1

∂

∂xi

(

∣

∣

∣

∣

∂

∂xi
u

∣

∣

∣

∣

pi(x)−2 ∂

∂xi
u

)

= f

which, in the particular case when pi = p for any i ∈ {1, ...,N} is the anisotropic p(.)-Laplace

equation.

For the proof of existence of entropy solutions of (1.1), we follow [2] and derive a priori estimates

for the approximated solutions un and the partial derivatives
∂un
∂xi

in the Marcinkiewicz spaces Mq
∗

and Mp
−

i
q/p̄ respectively (see section 2 or [2] for definition of Marcinkiewicz spaces).

The study of nonlinear elliptic equations involving the p−Laplace operator is based on the theory

of standard Sobolev spaces W m,p(Ω) in order to find weak solutions. For the nonhomogeneous

p(.)-Laplace operators, the natural setting for this approach is the use of the variable exponent

Lebesgue and Sobolev spaces Lp(.)(Ω) and W m,p(.)(Ω).

Variable exponent Lebesgue spaces appeared in the literature for the first time in a 1931 article

by Orlicz[21]. After [21], Orlicz abandoned the study of variable exponent spaces to concentrate

on the theory of function spaces that now bears his name (Orlicz spaces). After Orlicz’s work (cf.

[21]), H. Hudzik [14], and J. Musielak [20] investigated the variable exponent Sobolev spaces.

Variable exponent Lebesgue spaces on the real line have been independently developed by Russian

researchers, notably Sharapudinov[26] and Tsenov [27]. The next major step in the investigation

of variable exponent Lebesgue and Sobolev spaces was the comprehensive paper by O. Kovacik

and J. Rakosnik in the early 90’s [16]. This paper established many of basic properties of Lebesgue

and Sobolev spaces with variables exponent. Variable Sobolev spaces have been used in the last

decades to model various phenomena. In [5], Chen, Levine and Rao proposed a framework for

image restoration based on a Laplacian variable exponent. Another application which uses non-

homogeneous Laplace operators is related to the modelling of electrorheological fluids. The first

major discovery in electrorheological fluids was due to Willis Winslow in 1949 (cf. [28]). These

fluids have the interesting property that their viscosity depends on the electric field in the fluid.

They can raise the viscosity by as much as five orders of magnitude. This phenomenon is known

as the Winslow effect. For some technical applications, consult Pfeiffer et al [22]. Electrorheolog-

ical fluids have been used in robotics and space technology. The experimental research has been

done mainly in the USA, for instance in NASA laboratories. For more information on properties,

modelling and the application of variable exponent spaces to these fluids, we refer to Diening [6],

Rajagopal and Ruzicka [22], and Ruzicka [24].

In this paper, the operator involved in (1.1) is more general than the p(.)−Laplace operator. Thus,

the variable exponent Sobolev space W 1,p(.)(Ω) is not adequate to study nonlinear problems of

this type. This leads us to seek entropy solutions for problems (1.1) in a more general variable

exponent Sobolev space which was introduced for the first time by Mihäılescu et al [18].



136 Stanislas OUARO CUBO
12, 1 (2010)

As the right-hand side in (1.1) is in L1(Ω), the suitable notion of solution is a notion of entropy

solution (cf. [3]).

The remaining part of this paper is organized as follows: Section 2 is devoted to mathematical pre-

liminaries including, among other things, a brief discussion of variable exponent Lebesgue, Sobolev,

anisotropic spaces and Marcinkiewicz spaces. Existence of weak energy solution for (1.1) where

f ∈ L∞(Ω) was proved in [14]; We will also briefly recall the results of [14] in section 2. The main

existence and uniqueness result is stated and proved in section 3.

2 Mathematical Preliminaries

In this section, we define Lebesgue, Sobolev and anisotropic spaces with variable exponent and give

some of their properties. Roughly speaking, anistropic Lebesgue and Sobolev spaces are functional

spaces of Lebesgue’s and Sobolev’s type in which different space directions have different roles.

Given a measurable function p(.) : Ω → [1,∞). We define the Lebesgue space with variable

exponent Lp(.)(Ω) as the set of all measurable function u : Ω → R for which the convex modular

ρp(.)(u) :=

∫

Ω

|u|
p(x)

dx

is finite. If the exponent is bounded, i.e., if p+ < ∞, then the expression

|u|p(.) := inf
{

λ > 0 : ρp(.)(u/λ) ≤ 1
}

defines a norm in Lp(.)(Ω), called the Luxembourg norm. The space (Lp(.)(Ω), |.|p(.)) is a separable

Banach space. Moreover, if p− > 1, then L
p(.)(Ω) is uniformly convex, hence reflexive, and its

dual space is isomorphic to Lp
′(.)(Ω), where 1

p(x)
+ 1

p′(x)
= 1. Finally, we have the Hölder type

inequality:
∣

∣

∣

∣

∫

Ω

uvdx

∣

∣

∣

∣

≤

(

1

p−
+

1

p′−

)

|u|p(.) |v|p′(.) , (2.1)

for all u ∈ Lp(.)(Ω) and v ∈ Lp
′(.)(Ω).

Now, let

W 1,p(.)(Ω) :=
{

u ∈ Lp(.)(Ω) : |∇u| ∈ Lp(.)(Ω)
}

,

which is a Banach space equipped with the norm

‖u‖1,p(.) := |u|p(.) + |∇u|p(.) .

An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the mod-

ular ρp(.) of the space L
p(.)(Ω).

Next, we define W
1,p(.)
0 (Ω) as the closure of C

∞
0 (Ω) in W

1,p(.)(Ω) under the norm ‖u‖1,p(.).

Set C+(Ω) =

{

p(.) ∈ C(Ω) : min
x∈Ω

p(x) > 1

}

.

Furthermore, if p(.) ∈ C+(Ω) is logarithmic Hölder continuous, then C
∞
0 (Ω) is dense in W

1,p(.)
0 (Ω),



CUBO
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Well-Posedness Results for Anisotropic ... 137

that is H
1,p(.)
0 (Ω) = W

1,p(.)
0 (Ω) (cf. [13]). Since Ω is an open bounded set and p(.) ∈ C+(Ω) is

logarithmic Hölder, the p(.)−Poincaré inequality

|u|p ≤ C |∇u|p(.)

holds for all u ∈ W
1,p(.)
0 (Ω), where C depends on p, |Ω|, diam(Ω) and N (see [13]), and so

‖u‖ := |∇u|p(.) ,

is an equivalent norm in W
1,p(.)
0 (Ω). Of course also the norm

‖u‖p(.) :=

N
∑

i=1

∣

∣

∣

∣

∂

∂xi
u

∣

∣

∣

∣

p(.)

is an equivalent norm in W
1,p(.)
0 (Ω). Hence the space W

1,p(.)
0 (Ω) is a separable and reflexive Banach

space.

Let us present a natural generalization of the variable exponent Sobolev space W
1,p(.)
0 (Ω) (cf. [18])

that will enable us to study the problem (1.1) with sufficient accuracy.

First of all, we denote by −→p (.) : Ω → RN the vectorial function −→p = (p1(.), ...,pN (.)). The

anisotropic variable exponent Sobolev space W
1,−→p (.)
0 (Ω) is defined as the closure of C

∞
0 (Ω) with

respect to the norm

‖u‖−→p (.) :=
N
∑

i=1

∣

∣

∣

∣

∂

∂xi
u

∣

∣

∣

∣

pi(.)

.

The space
(

W
1,−→p (.)
0 (Ω),‖u‖−→p (.)

)

is a reflexive Banach space (cf. [18]).

Let us introduce the following notations:

−→
P + = (p

+
1 , ...,p

+
N ),

−→
P − = (p

−
1 , ...,p

−
N ),

P ++ = max
{

p+1 , ...,p
+
N

}

,P +− = max
{

p−1 , ...,p
−
N

}

,P−− = min
{

p−1 , ...,p
−
N

}

,

and

P∗− =
N

N
∑

i=1

1

p−i
− 1

, P−,∞ = max
{

P +− ,P
∗
−

}

.

We have the following result (cf. [18]):

Theorem 2.1. Assume Ω ⊂ RN (N ≥ 3) is a bounded domain with smooth boundary. Assume

relation (1.6) is fulfilled. For any q ∈ C(Ω) verifying

1 < q(x) < P−,∞ for all x ∈ Ω,



138 Stanislas OUARO CUBO
12, 1 (2010)

then the embedding

W
1,−→p (.)
0 (Ω) →֒ L

q(.)(Ω)

is continuous and compact.

We Also recall the result of the study of problem (1.1) for the right-hand side f ∈ L∞(Ω) (cf. [15]).

We first recall the definition of weak energy solution of (1.1) for the right-hand side more regular

i.e f ∈ L∞(Ω).

Definition 2.2. Let f ∈ L∞(Ω); a weak energy solution of (1.1) is a function u ∈ W
1,−→p (.)
0 (Ω)

such that
∫

Ω

N
∑

i=1

ai(x,
∂

∂xi
u).

∂

∂xi
ϕdx =

∫

Ω

f(x)ϕdx, for all ϕ ∈ W
1,−→p (.)
0 (Ω). (2.2)

We have proved among other results in [15] the following Theorem

Theorem 2.3. Assume (1.2)-(1.6) and f ∈ L∞(Ω). Then there exists a unique weak energy

solution of (1.1).

Finally, in this paper, we will use the Marcinkiewicz spaces Mq(Ω)(1 < q < ∞) with constant

exponent. Note that the Marcinkiewicz spaces Mq(.)(Ω) in the variable exponent setting was

introduced for the first time by Sanchon and Urbano (see [25]).

Marcinkiewicz spaces Mq(Ω)(1 < q < ∞) contain the measurable functions g : Ω → R for which

the distribution function

λg(γ) = |{x ∈ Ω : |g(x)| > γ}| , γ ≥ 0,

satisfies an estimate of the form

λg(γ) ≤ Cγ
−q, for some finite constant C > 0.

The space Mq(Ω) is a Banach space under the norm

‖g‖
∗
Mq (Ω) = sup

t>0
t1/q

(

1

t

∫ t

0

g∗(s)ds

)

,

where g∗ denotes the nonincreasing rearrangement of f:

g∗(t) = inf {γ > 0 : λg (γ) ≤ t} .

We will use the following pseudo norm

‖g‖Mq (Ω) = inf
{

C : λg(γ) ≤ Cγ
−q, ∀γ > 0

}

,

which is equivalent to the norm ‖g‖
∗
Mq (Ω) (see [2]).

We have the following Lemma (for the proof, see [2, proof of Lemma 2.2]).

Lemma 2.4. Let g be a nonnegative function in W
1,
−→
P −

0 (Ω). Assume p̄ < N, and that there exists

a constant c such that

N
∑

i=1

∫

{|g|≤γ}

∣

∣

∣

∣

∂g

∂xi

∣

∣

∣

∣

p
−

i

dx ≤ c(γ + 1),∀γ > 0. (2.3)



CUBO
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Well-Posedness Results for Anisotropic ... 139

Then there exists a constant C, depending on c, such that

‖g‖
M

N (p̄−1)
N −p̄ (Ω)

≤ C.

3 Existence and Uniqueness of Entropy Solution

In this section, we study the problem (1.1) for a right-hand side f ∈ L1(Ω). In the L1 setting, the

suitable notion of solution for the study of (1.1) is the notion of entropy solution (cf. [3]).

We first define the troncation function Tt by Tt(s) := max{−t,min{t,s}}.

Definition 3.1. A measurable function u is an entropy solution to problem (1.1) if, for every

t > 0, Tt(u) ∈ W
1,−→p (.)
0 (Ω) and

∫

Ω

N
∑

i=1

ai(x,
∂

∂xi
u).

∂

∂xi
Tt(u − ϕ)dx ≤

∫

Ω

f(x)Tt(u − ϕ)dx, (3.1)

for all ϕ ∈ W
1,−→p (.)
0 (Ω) ∩ L

∞(Ω).

Remark 3.2. A function u such that Tt(u) ∈ W
1,−→p (.)
0 (Ω) for all t > 0 does not necessarily belong

in W
1,1
0 (Ω). However, it is possible to define its weak gradient, still denoted by ∇u.

Our main result in this section is the following:

Theorem 3.3. Assume (1.2)-(1.6) and f ∈ L1(Ω). Then there exists a unique entropy solution

u to problem (1.1).

Proof.

The proof of this Theorem will be done in three steps.

∗Step 1. A priori estimates. We start with the existence of the weak gradient for every

measurable function u such that Tt(u) ∈ W
1,−→p (.)
0 (Ω) for all t > 0.

Proposition 3.4. If u is a measurable function such that Tt(u) ∈ W
1,−→p (.)
0 (Ω) for all t > 0, then

there exists a unique measurable function v : Ω → RN such that

vχ{|u|<t} = ∇Tt(u) for a.e. x ∈ Ω, and for all t > 0,

where χA denotes the characteristic function of a measurable set A. Moreover, if u belongs to

W
1,1
0 (Ω), then v coincides with the standard distributional gradient of u.

Proof. As Tt(u) ∈ W
1,−→p (.)
0 (Ω) →֒ W

1,
−→
P −

0 (Ω) ⊂ W
1,1
0 (Ω) for all t > 0 since 1 < p

−
i for all

i = 1, ...,N, then by Theorem 1.5 in [1], the result follows.

Proposition 3.5. Assume (1.2)-(1.6) and f ∈ L1(Ω). Let u be an entropy solution of (1.1). If

there exists a positive constant M such that

N
∑

i=1

∫

{|u|>t}

tqi(x)dx ≤ M, for all t > 0, (3.2)

then
N
∑

i=1

∫

{

∣

∣

∣

∂
∂xi

u
∣

∣

∣

αi(.)
>t

} tqi(x)dx ≤ ‖f‖1 + M for all t > 0,



140 Stanislas OUARO CUBO
12, 1 (2010)

where αi(.) = pi(.)/(qi(.) + 1)., for all i = 1, ...,N.

Proof. Take ϕ = 0 in (3.1), we have

N
∑

i=1

∫

Ω

ai(x,
∂

∂xi
Tt(u)).

∂

∂xi
Tt(u)dx ≤

∫

Ω

f(x)Tt(u)dx.

We deduces from inequality above that

N
∑

i=1

∫

Ω

∣

∣

∣

∣

∂

∂xi
Tt(u)

∣

∣

∣

∣

pi(x)

dx ≤ t‖f‖1 , for all t > 0.

Therefore, defining ψ := Tt(u)/t, we have, for all t > 0,

N
∑

i=1

∫

Ω

tpi(x)−1
∣

∣

∣

∣

∂

∂xi
ψ

∣

∣

∣

∣

pi(x)

dx =
1

t

∫

Ω

∣

∣

∣

∣

∂

∂xi
Tt(u)

∣

∣

∣

∣

pi(x)

dx ≤ ‖f‖1 .

From the above inequality, the definition of αi(.) and (3.2), we have

N
∑

i=1

∫

{

∣

∣

∣

∂
∂xi

u
∣

∣

∣

αi(.)
>t

} tqi(x)dx ≤

∫

{

∣

∣

∣

∂
∂xi

u
∣

∣

∣

αi(.)
>t

}

∩{|u|≤t}

tqi(x)dx +

∫

{|u|>t}

tqi(x)dx

≤

∫

{|u|≤t}

tqi(x)







∣

∣

∣

∂
∂xi

u
∣

∣

∣

αi(x)

t







pi(x)

αi(x)

dx + M ≤ ‖f‖1 + M, for all t > 0.

Proposition 3.6. Assume (1.2)-(1.6) and f ∈ L1(Ω). Let u be an entropy solution of (1.1), then

1

h

N
∑

i=1

∫

{|u|≤h}

∣

∣

∣

∣

∂

∂xi
Th(u)

∣

∣

∣

∣

pi(x)

dx ≤ M

for every h > 0, with M a positive constant. More precisely, there exists D > 0 such that

meas{|u| > h} ≤ DP
−

−

1 + h

hP
−

−

.

Proof. Taking ϕ = 0 in the entropy inequality (3.1) and using (1.5), we obtain

N
∑

i=1

∫

{|u|≤h}

∣

∣

∣

∣

∂

∂xi
Th(u)

∣

∣

∣

∣

pi(x)

dx ≤ h‖f‖1 ≤ Mh

for all h > 0.

Next,

N
∑

i=1

∫

{|u|≤h}

∣

∣

∣

∣

∂

∂xi
Th(u)

∣

∣

∣

∣

pi(x)

dx ≤ Mh ⇒

N
∑

i=1

∫

{|u|≤h}

∣

∣

∣

∣

∂

∂xi
Th(u)

∣

∣

∣

∣

P
−

−

dx ≤ C(1 + h).



CUBO
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Well-Posedness Results for Anisotropic ... 141

We can write the above inequality as

N
∑

i=1

∥

∥

∥

∥

∂

∂xi
Th(u)

∥

∥

∥

∥

P
−

−

P
−

−

≤ C(1 + h) or ‖Th(u)‖
W

1,P
−

−

0 (Ω)
≤ [C(1 + h)]

1

P
−

− .

By the Poincaré inequality in constant exponent, we obtain

‖Th(u)‖
L

P
−

− (Ω)
≤ D(1 + h)

1

P
−

− .

The above inequality imply that

∫

Ω

|Th(u)|
P

−

− dx ≤ DP
−

− (1 + h),

from which we obtain

meas{|u| > h} ≤ DP
−

−

1 + h

hP
−

−

.

∗ Step 2. Uniqueness of entropy solution. The proof of uniqueness of entropy solutions follow

the technics by Bénilan et al [3] (see also [25]). Let h > 0 and u,v two entropy solutions of (1.1).

We write the entropy inequality (3.1) corresponding to the solution u, with Thv as test function,

and to the solution v, with Thu as test function. Upon addition, we get



























































∫

{|u−Thv|≤t}

N
∑

i=1

ai(x,
∂

∂xi
u).

∂

∂xi
(u − Thv)dx+

∫

{|v−Thu|≤t}

N
∑

i=1

a(x,
∂

∂xi
v).

∂

∂xi
(v − Thu)dx

≤

∫

Ω

f(x)(Tt(u − Thv) + Tt(v − Thu))dx.

(3.3)

Define

E1 := {|u − v| ≤ t, |v| ≤ h} , E2 := E1 ∩ {|u| ≤ h} , and E3 := E1 ∩ {|u| > h} .



142 Stanislas OUARO CUBO
12, 1 (2010)

We start with the first integral in (3.3). By (1.5), we have



























































































































































































































































































∫

{|u−Thv|≤t}

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
(u − Thv)dx =

∫

{|u−Thv|≤t}∩({|v|≤h}∪{|v|>h})

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
(u − Thv)dx =

∫

{|u−Thv|≤t,|v|≤h}

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
(u − Thv)dx+

∫

{|u−Thv|≤t,|v|>h}

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
(u − Thv)dx

=

∫

{|u−v|≤t,|v|≤h}

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
(u − v)dx +

∫

{|u−Thv|≤t,|v|>h}

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
udx

≥

∫

{|u−v|≤t,|v|≤h}

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
(u − v)dx =

∫

E1

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
(u − v)dx

=

∫

E1∩({|u|≤h}∪{|u|>h})

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
(u − v)dx =

∫

E2

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
(u − v)dx

+

∫

E3

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
(u − v)dx =

∫

E2

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
(u − v)dx+

∫

E3

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
udx −

∫

E3

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
vdx

≥

∫

E2

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
(u − v)dx −

∫

E3

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
vdx.

(3.4)



CUBO
12, 1 (2010)

Well-Posedness Results for Anisotropic ... 143

Using (1.3) and (2.1), we estimate the last integral in (3.4) as follows



































∣

∣

∣

∣

∣

∫

E3

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
vdx

∣

∣

∣

∣

∣

≤ C1

∫

E3

N
∑

i=1

(

ji(x) +

∣

∣

∣

∣

∂

∂xi
u

∣

∣

∣

∣

pi(x)−1
)

∣

∣

∣

∣

∂

∂xi
v

∣

∣

∣

∣

dx

≤ C

N
∑

i=1



|j|p′
i
(.) +

∣

∣

∣

∣

∣

∣

∣

∣

∣

∂

∂xi
u

∣

∣

∣

∣

pi(x)−1
∣

∣

∣

∣

∣

p′
i
(.),{h<|u|≤h+t}





∣

∣

∣

∣

∂

∂xi
v

∣

∣

∣

∣

pi(.),{h−t<|v|≤h}

,

(3.5)

where

∣

∣

∣

∣

∣

∣

∣

∂
∂xi

u
∣

∣

∣

pi(x)−1
∣

∣

∣

∣

p′
i
(.),{h<|u|≤h+t}

=

∥

∥

∥

∥

∣

∣

∣

∂
∂xi

u
∣

∣

∣

pi(x)−1
∥

∥

∥

∥

L
p′

i
(.)

({h<|u|≤h+t})

.

For all i = 1, ...,N, the quantity

(

|ji|p′
i
(.) +

∣

∣

∣

∣

∣

∣

∣

∂
∂xi

u
∣

∣

∣

pi(x)−1
∣

∣

∣

∣

p′
i
(.),{h<|u|≤h+t}

)

is finite, since

u ∈ W
1,−→p (.)
0 (Ω) and ji ∈ L

p′i(.)(Ω); then by Proposition 3.6, the last expression converges to zero

as h tends to infinity. Therefore, from (3.4) and (3.5), we obtain

∫

{|u−Thv|≤t}

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
(u − Thv)dx ≥ Ih +

∫

E2

N
∑

i=1

a(x,
∂

∂xi
u).

∂

∂xi
(u − v)dx, (3.6)

where Ih converges to zero as h tends to infinity. We may adopt the same procedure to treat the

second term in (3.3) to obtain

∫

{|v−Thu|≤t}

N
∑

i=1

a(x,
∂

∂xi
v).

∂

∂xi
(v − Thu)dx ≥ Jh −

∫

E2

N
∑

i=1

a(x,
∂

∂xi
v).

∂

∂xi
(u − v)dx, (3.7)

where Jh converges to zero as h tends to infinity.

Next, consider the right-hand side of inequality (3.3). Noting that

Tt(u − Thv) + Tt(v − Thu) = 0 in {|u| ≤ h, |v| ≤ h} ;

we obtain
∣

∣

∣

∣

∫

Ω

f(x)(Tt(u − Thv) + Tt(v − Thu))dx

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

∫

{|u|>h}

f(x)(Tt(u − Thv) + Tt(v − Thu))dx +

∫

{|u|≤h}

f(x)(Tt(u − Thv) + Tt(v − Thu))dx

∣

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

∫

{|u|>h}

f(x)(Tt(u − Thv) + Tt(v − Thu))dx +

∫

{|u|≤h,|v|>h}

f(x)(Tt(u − Thv) + Tt(v − Thu))dx

∣

∣

∣

∣

∣

≤ 2t

(

∫

{|u|>h}

|f|dx +

∫

{|v|>h}

|f|dx

)

.

According to Proposition 3.6, both meas{|u| > h} and meas{|v| > h} tend to zero as h goes to

infinity, then by the inequality above, the right-hand side of inequality (3.3) tends to zero as h



144 Stanislas OUARO CUBO
12, 1 (2010)

goes to infinity. From this assertion, (3.3), (3.6) and (3.7), we obtain, letting h → +∞,

∫

{|u−v|≤t}

N
∑

i=1

(a(x,
∂

∂xi
u) − a(x,

∂

∂xi
v)).

∂

∂xi
(u − v)dx ≤ 0, for all t > 0.

By assertion (1.4), we conclude that
∂

∂xi
u =

∂

∂xi
v, for all i = 1, ..,N a.e. in Ω.

We deduce that

‖u − v‖−→p (.) =

N
∑

i=1

∣

∣

∣

∣

∂

∂xi
u −

∂

∂xi
v

∣

∣

∣

∣

pi(.)

= 0,

and hence u = v, a.e. in Ω.

∗ Step 3. Existence of entropy solutions. Let (fn)n be a sequence of bounded functions,

strongly converging to f ∈ L1(Ω) and such that

‖fn‖1 ≤ ‖f‖1 , for all n. (3.8)

We consider the problem























−

N
∑

i=1

∂

∂xi
ai(x,

∂

∂xi
un) = fn in Ω

un = 0 on ∂Ω.

(3.9)

It follows from Theorem 2.3 (cf. [15]) that problem (3.9) has a unique weak energy solution

un ∈ W
1,−→p (.)
0 (Ω) since fn ∈ L

∞(Ω). Our interest is to prove that these approximated solutions

un tend, as n goes to infinity, to a measurable function u which is an entropy solution of the limit

problem (1.1). To this end, we derive a priori estimates for un and
∂

∂xi
un in the Marcinkiewicz

spaces Mq
∗

and Mp
−

i
q/p̄ where q∗ and p̄ are defined in (1.6) and (1.7).

Let γ > 0, denote Tγ the corresponding truncation function. Note that

DTγ (r) =















1 if |r| < γ

0 if |r| > γ.

In particular, we have

ai(x,ξ)DTγ (r)ξ ≥ ai(x,ξ)ξχ{|r|<γ}, for all i = 1, ...,N. (3.10)

Lemma 3.7. There exists a constant c, not depending on n, such that

N
∑

i=1

∫

{|un|≤γ}

∣

∣

∣

∣

∂un
∂xi

∣

∣

∣

∣

p
−

i

dx ≤ c(γ + 1), for all γ > 0. (3.11)



CUBO
12, 1 (2010)

Well-Posedness Results for Anisotropic ... 145

Proof. Inserting ϕ = Tγ (un) into (2.2), we have

∫

Ω

N
∑

i=1

ai(x,
∂

∂xi
un)DTγ (un)

∂

∂xi
undx =

∫

Ω

fnTγ (un)dx.

Using (3.10) and the coercivity condition (1.5), we obtain (3.11).

Lemma 3.8. There exists a constant C, not depending on n, such that

‖un‖Mq∗ (Ω) ≤ C

and
∥

∥

∥

∥

∂un
∂xi

∥

∥

∥

∥

M
p
−

i
q/p̄

(Ω)

≤ C, for all i = 1, ...,N.

Proof. The result of Lemma 3.8 is a direct consequence of Lemma 3.7 (cf. [2, proof of Lemma 3.3]).

In view of Lemma 3.8 and following [2] ( see also [4, Lemma A.2]), we deduce that un is uni-

formly bounded in Lk0 (Ω) for some k0 < q
∗ with k0 > p

−
i q/p̄ and

∂un
∂xi

is uniformly bounded in

Lki (Ω) for some ki > 1 with p
−
i − 1 < ki < p

−
i q/p̄, for all i = 1, ...,N. From this, we get that un

is uniformly bounded in the isotropic Sobolev space W
1,kmin
0 (Ω), where kmin = min(k1, ...,kN ).

Consequently, we can assume without loss of generality (see also [2, Lemma 3.4]) that as n → 0,































































un → u a.e in Ω and in L
kmin (Ω)

un ⇀ u in W
1,kmin
0 (Ω)

∂un
∂xi

→
∂u

∂xi
in L1(Ω), for all i = 1, ...,N.

ai(x,
∂un
∂xi

) → ai(x,
∂u

∂xi
) a.e in Ω and in L1(Ω), for all i = 1, ...,N.

(3.12)

Now, fix t > 0, ϕ ∈ W
1,−→p (.)
0 (Ω) ∩ L

∞(Ω), and choose Tt(un − ϕ) as a test function in (2.2), with

u replaced by un to obtain

∫

Ω

N
∑

i=1

ai(x,
∂

∂xi
un).

∂

∂xi
Tt(un − ϕ)dx =

∫

Ω

fn(x)Tt(un − ϕ)dx.

Note that this choice can be made using a standard density argument. We now pass to the limit

in the previous identity. For the right-hand side, the convergence is obvious since fn converges

strongly in L1 to f and Tt(un − ϕ) converges weakly-* in L
∞, and a.e., to Tt(u − ϕ).

Next, we write the left hand side as

∫

{|un−ϕ|≤t}

N
∑

i=1

ai(x,
∂

∂xi
un).

∂

∂xi
undx −

∫

{|un−ϕ|≤t}

N
∑

i=1

ai(x,
∂

∂xi
un).

∂

∂xi
ϕdx (3.13)



146 Stanislas OUARO CUBO
12, 1 (2010)

By (3.12), the second integral of (3.13) converges to

∫

{|u−ϕ|≤t}

N
∑

i=1

ai(x,
∂

∂xi
u).

∂

∂xi
ϕdx.

For the first integral in (3.13), as

N
∑

i=1

ai(x,
∂

∂xi
un).

∂

∂xi
un is nonnegative by (1.5), we obtain by

using (3.12) and Fatou’s Lemma

∫

{|u−ϕ|≤t}

N
∑

i=1

ai(x,
∂

∂xi
u).

∂

∂xi
udx ≤ lim

n→∞
inf

∫

{|un−ϕ|≤t}

N
∑

i=1

ai(x,
∂

∂xi
un).

∂

∂xi
undx.

Gathering results, we obtain

∫

Ω

N
∑

i=1

ai(x,
∂

∂xi
u).

∂

∂xi
Tt(u − ϕ)dx ≤

∫

Ω

f(x)Tt(u − ϕ)dx,

i.e., u is an entropy solution of (1.1).

Received: June, 2008. Revised: november, 2008.

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