ANISOTROPIC EQUATIONS:
UNIQUENESS AND EXISTENCE RESULTS
STANISLAV ANTONTSEV† , MICHEL CHIPOT††
Abstract. We study uniqueness of weak solutions for elliptic equations of the following type
p −2
−∂xi ai (x, u) |∂xi u| i ∂xi u + b(x, u) = f (x)
in a bounded domain Ω ⊂ Rn with Lipschitz continuous boundary Γ = ∂Ω. We
consider in particular mixed boundary conditions -i.e., Dirichlet condition on one
part of the boundary and Neumann condition on the other part. We study also
uniqueness of weak solutions for the parabolic equations


pi −2


∂
u
=
∂
a
(x,
t,
u)
|∂
u|
∂
u
+f
in Ω × (0, T ),

t
x
i
x
x
i
i
i



u=0
on Γ × (0, T ) = ∂Ω × (0, T ),







u(x, 0) = u0
x ∈ Ω.
It is assumed that the constant exponents pi satisfy 1 < pi < ∞ and the coefficients
ai are such that 0 < λ ≤ λi ≤ ai (x, u) < ∞, ∀i, a.e. x ∈ Ω, (a.e. t ∈ (0, T )), ∀u ∈ R.
We indicate also conditions which guarantee existence of solutions.
1991 Mathematics Subject Classification. 35K65, 35J70, 35K60, 35K65, 46E35.
Key words and phrases.
Anisotropic equations, nonlinear elliptic equations, nonlinear parabolic
equations.
The first author was partially supported by the research project POCI/MAT/61576/2004,
FCT/MCES, Portugal. The second author was supported by the Swiss National Science Foundation under the contract # 20-113287/1 .
1
1. Introduction
In recent years an increasing interest has turned towards anisotropic elliptic and
parabolic equations. A special interest in the study of such equations is motivated by
their applications to the mathematical modeling of physical and mechanical processes
in anisotropic continuous medium. We refer to the recent works [2, 3],[6]-[13],[17]-[19],
[24, 25] where it is possible to find some references. In these papers various types of
nonlinear anisotropic elliptic and parabolic equations are studied from the point of
view of existence and nonexistence, regularity and qualitative properties with respect
to the data.
The main aim of this note is to prove uniqueness of weak solutions to the above
mentioned equations under weak feeble restrictions on the coefficients ai . To prove it
we will follow some of the ideas of [1, 15, 16].
We do not discuss the existence and the regularity of such solutions under generic
conditions. We just underline some scheme to prove it. In fact, existence results are
technical improvements of classical results stated in [21]-[23]. For more exotic equations
existence results can be found in [2]-[8],[10]-[13], [17]-[20],[24, 25].
The paper is organized as follows.
In Section 2 we consider elliptic equations and prove uniqueness results when at
least one pi ≤ 2, and when a lower order monotone term b(x, u) exists and pi > 2, ∀i.
We give also an existence theorem and consider more general equations.
The similar scheme is developed in Section 3 for parabolic equations.
2. Elliptic equations
2.1. Uniqueness result when at least one pi ≤ 2. We consider the Dirichlet problem
−∂xi ai (x, u) |∂xi u|pi −2 ∂xi u = f
in Ω,
(2.1)
u=0
on Γ = ∂Ω,
in a bounded domain Ω ⊂ Rn with Lipschitz continuous boundary Γ = ∂Ω.
We assume that the components of the constant vector p~ = (p1 , p2 , ..., pn ) satisfy
1 < p1 ≤ p2 ≤ ... ≤ pn < ∞.
(2.2)
We define
W01, p~ (Ω) = u ∈ W01,
p1
(Ω) : ∂xi u ∈ Lpi (Ω) , i = 1, ..., n
and we equip this space with the norm
kukW 1,
0
p
~
(Ω)
=
n
X
i=1
2
k∂xi ukLpi (Ω) .
0
By W0−1, p~ (Ω) we denote the dual space of W01, p~ (Ω) . A weak solution u ∈ W01, p~ (Ω) of
problem (2.1) is a function u such that
Z
ai (x, u) |∂xi u|pi −2 ∂xi u ∂xi v =< f, v >, ∀v ∈ W01, p~ (Ω) .
(2.3)
Ω
We are going to prove uniqueness and existence of a solution u ∈ W01, p~ (Ω) to (2.3).
We will suppose
0 < λ ≤ λi ≤ ai (x, u) ≤ Λ < ∞, ∀i, a.e. x ∈ Ω, ∀u ∈ R,
(2.4)
|ai (x, u) − ai (x, v)| ≤ ω(|u − v|), ∀i, a.e. x ∈ Ω, ∀u, v ∈ R,
(2.5)
where
Z
ds
= +∞.
0+ ω(s)
We will use the Young inequality (q ∈ (1, ∞), 0 ≤ a, b, δ < ∞)
(2.6)
q0
(qδ)− q q0 1
1
q
ab ≤ δa +
b , + 0 = 1.
0
q
q q
and the coerciveness inequality
γ |ξi − ηi |2+σ {|ξi | + |ηi |}pi −2−σ ≤ |ξi |pi −2 ξi − |ηi |pi −2 ηi ) · (ξi − ηi ).
(2.7)
(2.8)
which holds for some positive constant γ and any nonnegative constant σ, for every ξi , ηi , i =
1, ..., n, (see Lemma 2.2, [5]).
Theorem 2.1. Under the conditions (2.4), (2.5), any weak solution u ∈ W01, p~ (Ω) to
(2.1)(or (2.3)) is unique if at least one pi ≤ 2.
Proof. We define
Fε (x) =

R


 x
ds
ε ω 2 (s)
x ≥ ε,
0
x ≤ ε.



(2.9)
Let u, v be two solutions to (2.3). Set w = u − v. We take as test function in (2.3)
Fε (w) (for u, v). We get by subtraction
Z
∂xi w
ai (x, u) |∂xi u|pi −2 ∂xi u − |∂xi v|pi −2 ∂xi v
=
ω2
Ωε
Z
∂x w
{ai (x, v) − ai (x, u)} |∂xi v|pi −2 ∂xi v i2 ,
ω
Ωε
3
(2.10)
where
Ωε = {x ∈ Ω | w(x) > ε} .
(2.11)
(In the above we have a summation in i). Using the ellipticity conditions (2.4), (2.5)
and (2.8), we get with summation in i and for µ = γλ .
Z
∂xi w 2
∂xi w p
−2
p
−1
i
i
≤
|∂xi v|
µ
ω (|∂xi u| + |∂xi v|)
ω .
Ωε
Ωε
Z
(2.12)
· Suppose that pi ≥ 2. Then
∂xi w pi −1
Ii :=
≤
ω |∂xi v|
Ωε
Z
(2.13)
Z
∂xi w 2
pi
pi −2
≤δ
|∂xi v|
+ C(δ, pi )
ω |∂xi v|
Ωε
Ωε
2
Z
Z ∂xi w pi
p
−2
i
(|∂x u| + |∂x v|)
|∂
v|
,
+
C(δ,
p
)
≤δ
x
i
i
i
i
ω Ωε
Ωε
Z
for every δ > 0 by the Young inequality (2.7) with q = 2.
· Suppose that pi ≤ 2. Then
Z
pi −1
|∂xi v|
Ii =
Ωε
Z Z
∂xi w ∂xi w pi
pi
|∂xi v| ,
ω ≤δ
ω + C(δ, pi )
Ωε
Ωε
(2.14)
for any δ > 0 we used (2.7) with q = pi . Moreover we have
Z p −2
p −2
∂xi w pi
∂xi w pi
=
(|∂x u| + |∂x v|) i2 pi (|∂x u| + |∂x v|)− i2 pi
i
i
i
i
ω ω Ωε
Ωε
Z
(2.15)
Z
∂xi w 2
pi −2
≤δ
+ C(δ, pi )
(|∂xi u| + |∂xi v|)pi
ω (|∂xi u| + |∂xi v|)
Ωε
Ωε
Z
for any δ > 0 (we used (2.7) with q = 2/pi with a slight modification if pi = 2).
Combining (2.14), (2.15), we come to the same estimate which was obtained for pi ≥
4
2 (compare with (2.13))
Z
∂xi w 2
pi
pi −2
Ii ≤ δ
+ C(δ, pi )
(|∂xi u| + |∂xi v|) .
ω (|∂xi u| + |∂xi v|)
Ωε
Ωε
Z
(2.16)
Finally we get for every i
Z
∂xi w 2
pi
pi −2
Ii ≤ δ
+ C(δ, pi )
(|∂xi u| + |∂xi v|) ,
ω (|∂xi u| + |∂xi v|)
Ωε
Ωε
Z
(2.17)
for every δ > 0. We collect now (2.12)-(2.16) with a suitable choice of δ > 0. Then
we obtain
Z
∂xi w 2
p
−2
i
≤C
(|∂xi u| + |∂xi v|)pi ≤ C 0 ,
ω (|∂xi u| + |∂xi v|)
Ωε
Ωε
Z
(2.18)
with some constant C 0 = C 0 (δ, pi ) independent of ε. If there is one pi say pk < 2 ,
applying the Hölder inequality with q = 2/pk , q 0 = 2/(2 − pk ) we obtain
Z pk −2
2−pk
∂xk w pk
∂xi w pk
p
p
=
(|∂x u| + |∂x v|) 2 k (|∂x u| + |∂x v|) 2 k
k
k
k
k
ω ω Ωε
Ωε
Z
(2.19)
! p2k Z
2−p
k
2
∂xi w 2
p
−2
p
k
(|∂x u| + |∂x v|) k
(|∂xk u| + |∂xk v|)
.
k
k
ω Ωε
Ωε
Z
≤
By (2.18), (2.19) we deduce
∂xk w pk
0
ω ≤C.
Ωε
Z
(2.20)
We introduce the function Gε defined by
 Z x

ds


x>
ω(s)
,
G (x) =


 0
x≤
which satisfies from above
Z
0
|∂xk G (w)|pk ≤ C .
Ωε
5
(2.21)
By the Poincaré inequality it follows that
Z
00
|G (w)|pk ≤ C .
(2.22)
Ωε
If {w = u − v > } has a positive measure, letting → 0, we obtain in accord to (2.6)
that
R
Ω
|G (w)|pk → ∞, which contradicts the above inequality. Hence we have u ≥ v.
By changing the rôle of u and v this completes the proof.
Remark 2.1. With the same proof as above one could allow in (2.1) a monotone lower
order term b(x, u) where u 7→ b(x, u) is nonincreasing (see also below).
2.2. Uniqueness result when a lower order term exists and pi > 2, ∀i. Let us
consider the problem
−∂xi ai (x, u) |∂xi u|pi −2 ∂xi u + b(x, u) = f
in Ω,
(2.23)
u=0
on Γ = ∂Ω.
That is to say u ∈ W01, p~ (Ω) is a weak solution in the sense that
Z
ai (x, u) |∂xi u|pi −2 ∂xi u ∂xi v + b(x, u)v =< f, v > ∀v ∈ W01, p~ (Ω) .
(2.24)
Ω
Let us denote by m(t) the uniform modulus of continuity of the ai ’s i.e.
m(t) = max
i
|ai (x, u) − ai (x, v)| .
sup
(2.25)
x∈Ω, |u−v|≤t
If the ai (x, u) are Carathéodory functions uniformly continuous with respect to x we
have
m(t) → 0 as t → 0.
(2.26)
Moreover m(t) is nondecreasing with t. We denote by ω the function
ω(t) = min(m(t), 1).
We assume that (2.4) holds and since the ai ’s are bounded for some constant C we
have
|ai (x, u) − ai (x, v)| ≤ Cω(|u − v|) ∀i, a.e. x ∈ Ω, ∀u, v ∈ R.
(2.27)
We suppose that there exists α such that
Z
ds
pn
pi
= +∞, 1 < α ≤
≤
, ∀i.
(2.28)
α
pn − 1
pi − 1
0+ ω (s)
6
Then, for every ε > 0, there exists a δ(ε) such that
Z ε
ds
= 1.
α
δ(ε) ω (s)
(2.29)
We then define Fε (x) as
Fε (x) =

 R 0
x
ds
δ(ε) ω α (s)

1
x ≤ δ(ε),
δ(ε) ≤ x ≤ ε,
ε ≤ x.
(2.30)
Clearly this function is Lipschitz continuous. We assume also
u → b(x, u) is increasing.
(2.31)
Theorem 2.2. Under the conditions (2.4), (2.27), (2.28), (2.31) the weak solution
u ∈ W01, p~ (Ω) to (2.24) is unique.
Proof. Let u, v be two solutions to (2.23). Setting w = u − v, we take as test function
in (2.24) Fε (w). Thus we have, dropping x in b(x, u)
Z
Z
pi −2
pi −2
0
ai (x, u) |∂xi u|
∂xi u − |∂xi v|
∂xi v ∂xi w Fε (w) + (b(u) − b(v)) Fε (w)
Ωε
Ω
(2.32)
Z
=
{ai (x, v) − ai (x, u)} |∂xi v|pi −2 ∂xi v ∂xi w Fε0 (w).
Ωε
The integration in the integral containing Fε0 is taking place only on the set
Ωε = {x | δ(ε) < w(x) < ε}
(2.33)
when some derivative occurs. Using (2.4), (2.27), we derive for some constant µ > 0
Z
X Z |∂x w|pi
i
µ
dx + (b(u) − b(v)) Fε (w)
(2.34)
ωα
Ωε
Ω
i
XZ
|∂xi w|
≤
|ai (x, v) − ai (x, u)| |∂xi v|pi −1
ωα
Ωε
i
X Z |∂x w|
i
≤C
|∂xi v|pi −1 .
(α−1)
Ωε ω
i
7
Next we use the Young inequality (2.7) with q = pi to evaluate the left hand side. From
(2.34) we get
µ
|∂xi w|pi
dx +
ωα
XZ
Ωε
i
Ωε
i
(b(u) − b(v)) Fε (w)
(2.35)
Ω
X
|∂xi w|pi
+
C
ω (α−1)pi
i
XZ
≤δ
Z
Z
|∂xi v|pi
Ωε
for any δ > 0 and for some constant C = C(δ, pi ). We know that ω ≤ 1 and thus
ω (α−1)pi ≥ ω α
since
α ≥ (α − 1)pi ⇔ α ≤
pi
.
pi − 1
(2.36)
Then from (2.35) we obtain
µ
|∂xi w|pi
dx +
ωα
XZ
Ωε
i
≤δ
XZ
Ωε
i
Z
(b(u) − b(v)) Fε (w)
Ω
X
|∂xi w|pi
+
C
ωα
i
Z
|∂xi v|pi .
Ωε
Let us choose δ small enough such that δ < µ, (δ is now fixed). We obtain
Z
(b(x, u) − b(x, v)) Fε (w) ≤ C
Ω
0
XZ
i
|∂xi v|pi
Ωε
for some constant C 0 . Note that
χΩε → 0 a.e., Fε (w) → 1 on w > 0.
Passing to the limit we get
Z
(b(x, u) − b(x, v)) ≤ 0,
u−v>0
8
and thus u − v ≤ 0 if b is monotone increasing in u. Exchanging the rôle of u, v, we
get u = v provided α is chosen such that
α≤
pi
∀i.
pi − 1
Remark 2.2. If
β = inf
i
pn
pi
=
>1
pi − 1
pn − 1
one can take α = β and (2.28) holds for
1
m(t) ≤ Ct β
i.e. for ai ’s Hölder continuous in u of exponent β1 .
2.3. Existence of solutions. In this section we discuss the existence of weak solutions
for the problem (2.24). We do not consider this question under the weakest possible
assumptions, our aim in this note being uniqueness.
To prove the desired existence results we follow [1, 2, 3], in which the Dirichlet
problem was studied.
Considering the equation
Z
ai (x, u) |∂xi u|pi −2 ∂xi u ∂xi v + b(x, u)v =< f, v > ∀v ∈ W01, p~ (Ω) ,
(2.37)
Ω
we assume that:
ai , b : Ω × R → R are Carathéodory functions,
(2.38)
0 < λ ≤ λi ≤ ai (x, u) ≤ Λ < ∞, ∀i, a.e. x ∈ Ω, ∀u ∈ R,
(2.39)
and that
β
|b(x, u)| ≤ C0 |u|β−1 + h(x), 0 ≤ h(x) ∈ L β−1 (Ω), 1 ≤ β ≤ pn ,
(recall that pn is the largest pi )
f (x) ∈ W0−1,
9
p
~
0
(Ω) .
(2.40)
(2.41)
Theorem 2.3. Under the conditions (2.2), (2.38)-(2.41) the problem (2.37) has at
least one weak solution u ∈ W01,
p
~
(Ω).
Proof. Let v(x) be any given function such that v(x) ∈ Lpn (Ω) . We define functions
Ai by
Ai (x) = ai (x, v(x)), ∀i.
We then introduce the operator Λ : W01, p~ (Ω) 7→ W0−1,
Z
p
~
0
(Ω) ,
Ai (x) |∂xi u|pi −2 ∂xi u ∂xi ϕ + b(x, u)ϕ.
(Λu, ϕ) =
Ω
It is obvious that the mapping Λ : W01, p~ (Ω) 7→ W0−1,
0
p
~
(Ω) is continuous and mono-
tone. Let us verify that it is coercive. Using (2.4), (2.40) we get
λ
XZ
pi
|∂xi u| −
Z (2.42)
Ω
Ω
i
C0 |u|β + h|u| ≤ (Λu, u) .
Next applying the Young, Hölder and Poincaré inequalities, we obtain
Z Z β
0
C0 |u| + h|u| ≤ C
|u|β + h β−1
β
Ω
≤C
00
Z
(2.43)
Ω
pn
pβ
Z
n
|∂xn u|
+
Ω
h
β
β−1
!
Z
pn
≤δ
|∂xn u|
+C
000
Z
1+
Ω
Ω
h
β
β−1
.
Ω
Combining (2.42), (2.43) with 2δ = λ, we come to
λX
2 i
Z
pi
Z
|∂xi u| − C 1 +
Ω
h
β
β−1
≤ (Λu, u) ,
(2.44)
Ω
which guarantee that the operator Λ is coercive. The space W01, p~ (Ω) is reflexive. By
the Browder-Minty theorem ([14], Theorem 7.3.2) the equation
Λu = f
10
(2.45)
has at least one weak solution u ∈ W01, p~ (Ω) for every f ∈ W0−1,
p
~
0
(Ω) . Thus the
mapping
Φ : v → u,
(2.46)
defines an operator from Lpn (Ω) into itself. Moreover if u is solution to (2.45) by (2.44)
we have
Z
β
λX
pi
β−1
k∂xi ukLpi (Ω) ≤ C 1 + h
+ kf kW −1,
0
2 i
Ω
where kf kW −1,
p
~0
0
(Ω)
p
~0
X
(Ω)
k∂xi ukLpi (Ω) .
(2.47)
i
denotes the strong dual norm of f . It follows by Young’s inequality
that
X
λX
k∂xi ukpLipi (Ω) + C(h, f ).
k∂xi ukpLipi (Ω) ≤ δ
2 i
i
Choosing δ =
λ
4
(2.48)
we derive that if u is solution to (2.45)
λX
k∂xi ukLpi (Ω) ≤ C(λ, h, f ).
2 i
(2.49)
Due to the Poincaré inequality
kukLpn (Ω) ≤ C k∂xn ukLpn (Ω)
(2.50)
we obtain for some new constant
kukLpn (Ω) ≤ C(λ, h, f ) = R.
(2.51)
It is easy to verify that the operator Φ is compact and continuous and transforms the
ball {u ∈ Lpn (Ω) | kukLpn (Ω) ≤ R} into itself. According to the Schauder fixed point
theorem the operator Φ has at least one fixed point, which defines a weak solution of
problem (2.37). This completes the proof of the theorem.
11
2.4. Generalizations. Let us consider more generally the problem
−∂xi (ai (x, u, ∇u)) + b(x, u) = f
in Ω,
u=0
on Γ = ∂Ω.
(2.52)
A solution u is an element u ∈ W01, p~ (Ω) such that
Z
Z
ai (x, u, ∇u) ∂xi v + b(x, u)v =< f, v > ∀v ∈ W01, p~ (Ω) .
Ω
(2.53)
Ω
Assume that
ai (x, u, ∇u) ∈ Lpi (Ω), b(x, u) ∈ W0−1,
p
~0
(Ω) , ∀u ∈ W01, p~ (Ω) ; f ∈ W0−1,
p
~0
(Ω) .
(2.54)
First we consider the case
2 < pi < ∞, i = 1, ..., n.
(2.55)
We assume that for some positive constantµ > 0, a.e. x ∈ Ω, ∀u, v ∈ R ∀η, ξ ∈ Rn we
have
(2.56)
µ |ηi − ξi |pi − ω(|u − v|) (|ηi | + |ξi |)pi −1 |ηi − ξi |
≤ (ai (x, u, η) − ai (x, v, ξ), ηi − ξi ) ,
where the function ω(s), α satisfy (2.28), (2.36). Let u, v be two solutions of (2.52)
and w = u − v. Then taking as test function in (2.53) Fε (w) defined by (2.30), we come
to an analog to (2.32). Indeed we have
Z
Z
0
(2.57)
{ai (x, u, ∇u) − ai (x, v, ∇v)} ∂xi w Fε (w) + (b(u) − b(v)) Fε (w) = 0.
Ω
Ωε
Using (2.56), we get
Z
X Z |∂x w|pi
X Z |∂x w|
i
i
µ
dx + (b(u) − b(v)) Fε (w) ≤
(|∂xi u| + |∂xi v|)pi −1 ,
α
α−1
ω
Ωε
Ω
Ωε ω
i
i
(2.58)
(compare with (2.34)). Next repeating the arguments of the proof of the Theorem 2.2
we have
Theorem 2.4. Under the conditions (2.54)-(2.56), there exists at most one solution
u ∈ W01, p~ (Ω) to (2.53).
Now we consider the case when at least one pi is less or equal to 2. We assume
that for some positive constant µ > 0, a.e. x ∈ Ω, ∀u, v ∈ R ∀η, ξ ∈ Rn
µ |ηi − ξi |2 (|ηi | + |ξi |)pi −2 − ω(|u − v|) (|ηi | + |ξi |)pi −1 |ηi − ξi |
(2.59)
≤ (ai (x, u, η) − ai (x, v, ξ) , ηi − ξi ),
where the function ω(s) defined by (2.9). Repeating the arguments of the Theorem
2.1, we obtain
12
Theorem 2.5. Under the conditions (2.54), (2.59) there exists at most one solution
u ∈ W01, p~ (Ω) to (2.53) if at least one pi is less or equal to 2.
Remark 2.3. The theorems 2.1-2.5 remain valid for mixed boundary conditions
u = 0 on ΓD , ai (x, u, ∇u)νi = 0 on ΓN ,
(2.60)
where ~ν = (ν1 , ..., ν1 ) is the unit normal vector to ΓN , Γ = ∂Ω = ΓD ∪ ΓN and
mes ΓD > 0. In this case we consider weak solutions u ∈ W01, p~ (Ω, ΓD ) where
W01, p~ (Ω, ΓD ) = u ∈ W01,
p1
(Ω, ΓD ) : ∂xi u ∈ Lpi (Ω) , i = 1, ..., n .
3. Parabolic equations
3.1. Uniqueness of solution. We consider the problem

 ∂t u = ∂xi ai (x, t, u) |∂xi u|pi −2 ∂xi u + f in QT = Ω × (0, T ),
u=0
on ΓT = ∂Ω × (0, T ),

u(x, 0) = u0
x ∈ Ω.
(3.1)
We define
V0 p~ (QT ) = L∞ (0, T ; L2 (Ω)) ∩ Lp~ (0, T ; W01, p~ (Ω))
and we equip it with the norm
p1
n Z T
X
i
pi
k∂xi ukLpi (Ω)
kukV p~ (QT ) = sup kukL2 (Ω) +
.
0
0≤t≤T
0
i=1
0
(Recall that W01, p~ (Ω) was defined in the section (2.1).) By V0−1,~p (QT ) we denote the
dual space of V01, p~ (QT ) . We define
n
o
0
V (QT ) = u ∈ V01, p~ (QT ) , ut ∈ V0−1, p~ (QT ) .
By a solution to (3.1) we mean a u ∈ V (QT ) such that
(ut , ϕ)QT +
n
X
ai (x, t, u) |∂xi u|pi −2 ∂xi u, ∂xi ϕ
QT
= (f, ϕ)Q , u(x, 0) = u0 ,
T
i=1
Z
p
~
∀ϕ ∈ V0 (QT ) , a.e. t ∈ (0, T ), (u, v)Qt =
uv.
Qt
13
(3.2)
0
Note that, according to (3.2), u satisfies ut ∈ V0−1, p~ (Ω) and
0
(ut , ϕ)QT ≤ C kϕkV p~ (QT ) kukV p~ (QT ) + kf kV −1, p~ (Q ) .
0
0
0
T
(3.3)
We suppose that
0 < λ ≤ λi ≤ ai (x, t, u) ≤ Λ ∀i, ∀u ∈ R, a.e. (x, t) ∈ Ω × (0, T ).
(3.4)
Denote by m(t) the uniform modulus of continuity of the ai in u i.e. set
m(s) =
|ai (x, t, u) − ai (x, t, v)| .
sup
(3.5)
i, x, t, |u−v|≤s
Modifying eventually m(t) for t ≥ 1 one can find a function ω which coincides with
m(t) for t ≤ 1 and which satisfies
Z ∞
ds
|ai (x, t, u) − ai (x, t, v)| ≤ ω(|u − v|),
< +∞,
(3.6)
ω 2 (s)
1
∀i, ∀ u, v ∈ R, a.e. (x, t) ∈ Ω × (0, T ).
Suppose that
Z
ds
= +∞.
ω 2 (s)
0+
Set
Rx
ds
ε ω 2 (s)
Fε (x) =
Z
∞
0
(3.7)
x ≥ ε,
x ≤ ε,
(3.8)
ds
Fε (x)
, Hε (x) =
.
2
ω (s)
Iε
Iε =
ε
(3.9)
Since for x ≥ ε one has
Hε (x) =
Rx
ds
ε
ω 2 (s)
Iε
=
Iε −
R∞
x
Iε
ds
ω 2 (s)
.
(3.10)
It is clear that Iε → +∞ as ε → 0 and thus
Hε (x) → H(x) the Heaviside function.
(3.11)
Theorem 3.1. Let u, v be two solutions of (3.1) corresponding to initial values u0 , v0 . Under
the conditions (2.2),(3.4),(3.6),(3.7), we have
Z
+
Z
(u − v) dx ≤
Ω
(u0 − v0 )+ dx, (u)+ = max(u, 0),
Ω
which implies in particular uniqueness of the solution to (3.1).
14
(3.12)
Proof. Let u, v be two solutions of (3.1) corresponding to initial values u0 , v0 . Set
w = u − v. By difference of the equations (3.2) for u and v we have
n
X
(wt , ϕ)Qt +
ai (x, t, u) |∂xi u|pi −2 ∂xi u − |∂xi v|pi −2 ∂xi v , ∂xi ϕ Qt
(3.13)
i=1
=
n
X
{ai (x, t, v) − ai (x, t, u)} |∂xi v|pi −2 ∂xi v, ∂xi ϕ
Qt
.
i=1
We take as test function in (3.13)
Hε (w) =
Fε (w)
.
Iε
Then we obtain
Z tZ
Z Z
1 t
∂t w Hε (w) = −
ai (x, s, u) |∂xi u|pi −2 ∂xi u − |∂xi v|pi −2 ∂xi v ∂xi Fε (w)
Iε 0 Ωε
0
Ωε
(3.14)
+
1
Iε
Z tZ
0
{ai (x, s, v) − ai (x, s, u)} |∂xi v|pi −2 ∂xi v ∂xi Fε (w)
Ωε
where Ωε is the set defined by
Ωε = {x ∈ Ω| (u − v)(x, s) > ε} .
(3.15)
(outside of Ωε , Fε (w) vanishes, see (3.8)). Using (3.4) and the definition of Fε we have
for some µ > 0
∂xi w 2
pi −2
µ
ω (|∂xi u| + |∂xi v|)
Ωε
Z
∂xi w
≤
ai (x, s, u) |∂xi u|pi −2 ∂xi u − |∂xi v|pi −2 ∂xi v
ω2
Ω
Z ε
=
ai (x, s, u) |∂xi u|pi −2 ∂xi u − |∂xi v|pi −2 ∂xi v ∂xi Fε (w).
Z
(3.16)
Ωε
By (3.6) we have also
Z
p
−2
i
{ai (x, s, v) − ai (x, s, u)} |∂xi v|
∂xi v∂xi Fε (w)
Ωε
15
(3.17)
Z
≤
|ai (x, s, v) − ai (x, s, u)| |∂xi v|pi −1
Ωε
Z
|∂xi w|
|∂xi v|pi −1
≤
.
ω
Ωε
|∂xi w|
ω2
From (3.14) we then derive
Z tZ
Z Z
∂xi w 2
µ tX
(|∂x u| + |∂x v|)pi −2
∂t w Hε (w) ≤ −
i
i
I
ω
ε 0
0
Ωε
Ωε
i
Z
Z
1 tX
|∂xi w|
+
.
|∂xi v|pi −1
Iε 0 i Ωε
ω
(3.18)
(In the formulae before we did not write the summation in i). Then we proceed with
the same estimates than in the elliptic case.
· Suppose that pi < 2. Combining the estimates (2.12), (2.15) of the elliptic case
we have
∂xi w pi −1
≤
ω |∂xi v|
Ωε
Z Z
∂xi w pi
pi
+ C(δ, pi )
v|
≤δ
|∂
x
i
ω Ωε
Ωε
2
Z Z
∂xi w p
−2
2
i
≤δ
+ C(δ, pi )
(|∂xi u| + |∂xi v|)pi .
ω (|∂xi u| + |∂xi v|)
Ωε
Ωε
Z
(3.19)
· Suppose that pi ≥ 2. From the estimate (2.13) of the elliptic case we have
Z Z
∂xi w ∂xi w 2
p
−1
pi −2
i
|∂x v|
≤δ
+C(δ, pi )
(|∂xi u| + |∂xi v|)pi .
ω i
ω (|∂xi u| + |∂xi v|)
Ωε
Ωε
Ωε
Z
(3.20)
Choosing and fixing δ such that
δ and δ 2 < µ
in (3.19), (3.20) it is clear that from (3.18)-(3.20) we get for some constant C = C(δ, pi )
16
Z tZ
0
C
∂t wHε (w) ≤
Iε
Ωε
Z tXZ
0
i
(|∂xi u| + |∂xi v|)pi .
(3.21)
Ωε
Let us set
Zx
Gε (x) =
Hε (t)dt.
−∞
We have
∂t wHε (w) = ∂t Gε (w)
and the above inequality gives us
Z
Z
Z
Z
C tX
Gε (w)(t) −
Gε (w)(0) ≤
(|∂xi u| + |∂xi v|)pi .
I
ε 0
Ωε
Ωε
Ωε
i
Letting → 0, we get (3.12) since
Gε (w) → (w)+ , Iε → +∞.
Remark 3.1. The condition (3.7) allows the ai (x, t, u) to be Hölder continuous with
Hölder exponent greater or equal to 1/2.
3.2. Existence of solutions. In this section we discuss the existence of weak solutions
u ∈ V (QT ) for the problem (3.1). We show here only a sketch of the proof for a simple
case.
Considering the problem
n
X
(ut , ϕ)Qt +
ai (x, t, u) |∂xi u|pi −2 ∂xi u, ∂xi ϕ Qt = (f, ϕ)Qt , u(x, 0) = u0 ,
(3.22)
i=1
we assume that:
ai , b : QT × R → R are Carathéodory functions
(3.23)
such that a.e. (x, t) ∈ QT , ∀u ∈ R,
0 < λ ≤ λi ≤ ai (x, t, u) ≤ Λ < ∞, ∀i.
17
(3.24)
We suppose
f (x, t) ∈ V0−1,
p
~0
(QT ) .
(3.25)
Then we have
Theorem 3.2. Under the conditions (3.23)-(3.25) the problem (3.22) has at least one
weak solution u ∈ V0 p~ (QT ).
Proof. Let v(x, t) be a function such that v(x, t) ∈ Lpn (QT ) . We set
Ai (x, t) = ai (x, t, v(x, t)), ∀i.
Let us consider the problem
ut − Λ(u) = f, u(x, 0) = u0 ,
Λ(u) =
n
X
(3.26)
∂xi Ai (x, t) |∂xi u|pi −2 ∂xi u .
i=1
It is easy to verify that the operator Λ : V01, p~ (QT ) 7→ V0−1,
p
~0
(QT ) is continuous,
monotone and coercive. Thus according to ([23], Ch2.), the problem (3.26) has a
unique solution which defines an operator
Φ : v → u.
(3.27)
According to (3.24) for all solution u the following estimate
p1
n Z T
X
i
pi
kukV p~ (QT ) = sup kukL2 (Ω) +
k∂xi ukLpi (Ω)
0
0≤t≤T
0
i=1
≤ C ku0 kL2 (Ω) + kf kV −1,
0
p
~0
(QT )
= C0
holds. It is easy to verify that the operator Φ is compact and continuous from Lpn (QT )
into itself and transforms, for some R large enough, the ball of center 0 and radius R in
Lpn (QT ) into itself. According to the Schauder fixed point theorem the operator Φ has
18
at least one fixed point, which is a weak solution of problem (3.22). The theorem is
proved.
3.3. Generalizations. Let us consider more generally u ∈ V (QT ) solution to
n
X
(ut , ϕ)QT +
(ai (x, t, u, ∇u), ∂xi ϕ)QT = (f, ϕ)QT , u(x, 0) = u0 .
(3.28)
i=1
Assume that
0
ai (x, u, t, ∇u) ∈ Lpi (QT ), ∀u ∈ V01, p~ (QT ) ; f ∈ V0−1,
p
~0
(QT ) .
(3.29)
We address here the case
1 < pi < ∞, i = 1, ..., n.
(3.30)
We suppose that for some positive constant µ > 0, a.e. x ∈ Ω, ∀t, u, v ∈ R ∀η, ξ ∈ Rn
µ |ηi − ξi |2 (|ηi | + |ξi |)pi −2 − ω(|u − v|) (|ηi | + |ξi |)pi −1 |ηi − ξi |
(3.31)
≤ (ai (x, t, u, η) − ai (x, t, v, ξ), ηi − ξi ) ,
where the function ω(s) satisfies (3.7). Let u, v be two solutions of (3.28) and w =
u − v. Then taking as test function in (3.28) Hε (w) for u and v, we come to an analog
of (3.14)
Z tZ
Z Z X
n
1 t
∂t w Hε (w) = −
{ai (x, t, u, ∇u) − ai (x, t, v, ∇v)} ∂xi w Fε0 (w)
I
ε 0
0
Ωε
Ωε i=1
(3.32)
Using (3.31), we get
n X
|∂xi w|pi
|∂xi w|
pi −2
pi −1
µ
(|∂xi u| + |∂xi vi |)
−
(|∂xi u| + |∂xi vi |)
(3.33)
2
ω
ω
i=1
≤
n
X
{ai (x, t, u, ∇u) − ai (x, t, v, ∇v)} ∂xi w Fε0 (w).
i=1
Thus we arrive to an inequality like (3.18). Next repeating the arguments of the proof
of the Theorem 3.1 we obtain
Theorem 3.3. Let u, v be two solutions of (3.28) corresponding to the initial values
u0 , v0 . Under conditions (3.29)-(3.31) we have
Z
Z
+
(u − v) dx ≤ (u0 − v0 )+ dx, (u)+ = max(u, 0),
Ω
Ω
which implies in particular uniqueness of the solution (3.28).
19
(3.34)
Remark 3.2. Assertions of Theorems 3.2, 3.3 remain valid for mixed boundary conditions
u = 0 on ΓD , ai (x, t, u, ∇u)νi = 0 on ΓN ,
(3.35)
where ~ν = (ν1 , ..., ν1 ) is the unit normal vector to ΓN , Γ = ∂Ω = ΓD ∪ ΓN and
mes ΓD > 0.
References
[1] S. Antontsev, M. Chipot, and Y. Xie, Uniqueness results for equations of the p(x)-Laplacian
type, Adv. Math. Sci. Appl., 17 (2007), pp. 287–304.
[2] S. Antontsev and S. Shmarev, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Journal Nonlinear Analysis,
65 (2006), pp. 722–755.
[3]
, Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions, Elsevier, 2006. Handbook of Differential Equations. Stationary Partial Differential Equations, Elsevier, Vol. 3, Chapter 1, pp. 1-100.
[4]
, Parabolic equations with anisotropic nonstandard growth conditions, in Internat. Ser. Numer. Math. 154, Birkhäuser, Verlag Basel/Switzerland, 2006, pp. 33–44.
[5] J. Barrett and W. B. Liu, Finite element approximation of the parabolic p−Laplacian, SIAM
Journal on Numerical Analysis, 31 (1994), pp. 413–428.
[6] M. Bendahmane and K. H. Karlsen, Renormalized entropy solutions for quasi-linear
anisotropic degenerate parabolic equations, SIAM J. Math. Anal., 36 (2004), pp. 405–422.
[7]
, Nonlinear anisotropic elliptic and parabolic equations in RN with advection and lower order
terms and locally integrable data, Potential Anal., 22 (2005), pp. 207–227.
20
[8]
, Uniqueness of entropy solutions for doubly nonlinear anisotropic degenerate parabolic equations, Contemporary Mathematics, 371 (2005), pp. 1–27.
[9]
, Anisotropic doubly nonlinear degenerate parabolic equations, in Numerical mathematics
and advanced applications, Springer, Berlin, 2006, pp. 381–386.
[10]
, Anisotropic nonlinear elliptic systems with measure data and anisotropic harmonic maps
into spheres, Electron. J. Differential Equations, No. 46, (2006), pp. 1-30.
[11]
, Renormalized solutions of an anisotropic reaction-diffusion-advection system with L1 data,
Commun. Pure Appl. Anal., 5 (2006), pp. 733–762.
[12] M. Bendahmane, M. Langlais, and M. Saad, On some anisotropic reaction-diffusion systems with L1 -data modeling the propagation of an epidemic disease, Nonlinear Anal., 54 (2003),
pp. 617–636.
[13] M. Bendahmane and M. Saad, Entropy solution for anisotropic reaction-diffusion-advection
systems with L1 data, Rev. Mat. Complut., 18 (2005), pp. 49–67.
[14] P. Blanchard and E. Brüning, Variational methods in mathematical physics, Texts and
Monographs in Physics, Springer-Verlag, Berlin, 1992. A unified approach, Translated from the
German by Gillian M. Hayes.
[15] M. Chipot and M. Michaille, Uniqueness results and monotonicity properties for strongly
nonlinear elliptic variational inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4)16 (1989),
pp. 137–166.
[16]
, Uniqueness results and monotonicity properties for the solutions of some variational inequalities. Existence of a free boundary, in Free boundary problems: theory and applications, Vol.
I, (Irsee,1987), Pitman Research Notes in Mathematics Series,185, Longman Sci. Tech., Harlow,
1990, pp. 271–276.
[17] L. Feng-Quan, Anisotropic Elliptic Equations in Lm , Journal of Convex Analysis, 8 (2001),
pp. 417–422.
21
[18] I. Fragalà, F. Gazzola, and B. Kawohl, Existence and nonexistence results for anisotropic
quasilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), pp. 715–734.
[19] I. Fragalà, F. Gazzola, and G. Lieberman, Regularity and nonexistence results for
anisotropic quasilinear elliptic equations in convex domains, Discrete Contin. Dyn. Syst., (2005),
pp. 280–286.
[20] E. Kim, Existence results for singular anisotropic elliptic boundary-value problems, Electronic
Journal of Differential Equations, No.17 (2000), pp. 1–17.
[21] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural’tseva, Linear and quasilinear
equations of parabolic type, American Mathematical Society, Providence, R.I., 1967. Translated
from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23.
[22] O. A. Ladyzhenskaya and N. N. Ural0 tseva, Linear and quasilinear elliptic equations,
Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis,
Academic Press, New York, 1968.
[23] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod,
1969.
[24] B. Song, Anisotropic diffusions with singular advections and absorptions. part 1. existence, Applied Mathematics Letters, 14 (2001), pp. 811–816.
[25]
, Anisotropic diffusions with singular advections and absorptions. part 2. uniqueness, Applied Mathematics Letters, 14 (2001), pp. 817–823.
†
CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Por-
tugal,
‡
Institut für Mathematik, Abt. Angewandte Mathematik, Universität Zürich,
Winterthurerstrasse 190, CH–8057 Zürich, Switzerland
E-mail address:
†
[email protected], [email protected],†† [email protected]
22