Electronic Journal of Differential Equations, Vol. 2008(2008), No. 76, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ
SPACES
GE DONG
Abstract. This article shows the existence of solutions to the nonlinear elliptic problem A(u) = f in Orlicz-Sobolev spaces with a measure valued righthand side, where A(u) = − div a(x, u, ∇u) is a Leray-Lions operator defined
on a subset of W01 LM (Ω), with general M .
1. Introduction
Let M : R → R be an N -function; i.e. M is continuous, convex, with M (u) > 0
for u > 0, M (t)/t → 0 as t → 0, and
R u M (t)/t → ∞ as t → ∞. Equivalently, M
admits the representation M (u) = 0 φ(t)dt, where φ is the derivative of M , with
φ non-decreasing, right continuous, φ(0) = 0, φ(t) > 0 for t > 0, and φ(t) → ∞ as
t → ∞.
Rt
The N -function M̄ conjugate to M is defined by M̄ (v) = 0 ψ(s)ds, where ψ is
given by ψ(s) = sup{t : φ(t) ≤ s}.
The N -function M is said to satisfy the ∆2 condition, if for some k > 0 and
u0 > 0,
M (2u) ≤ kM (u), ∀u ≥ u0 .
Let P, Q be two N -functions, P Q means that P grows essentially less rapidly
than Q; i.e. for each ε > 0, P (t)/Q(εt) → 0 as t → ∞. This is the case if and only
if limt→∞ Q−1 (t)/P −1 (t) = 0.
Let Ω ⊂ RN be a bounded domain with the segment property. The class
1
W LM (Ω) (resp., W 1 EM (Ω)) consists of all functions u such that u and its distributional derivatives up to order 1 lie in LM (Ω) (resp., EM (Ω)).
Orlicz spaces LM (Ω) are endowed with the Luxemburg norm
Z
|u(x)| dx ≤ 1 .
kuk(M ) = inf λ > 0 :
M
λ
Ω
The classes W 1 LM (Ω) and W 1 EM (Ω) of such functions may be given the norm
X
kukΩ,M =
kDα uk(M ) .
|α|≤1
2000 Mathematics Subject Classification. 35J15, 35J20, 35J60.
Key words and phrases. Orlicz-Sobolev spaces; elliptic equation; nonlinear; measure data.
c 2008 Texas State University - San Marcos.
Submitted October 8, 2006. Published May 27, 2008.
1
2
G. DONG
EJDE-2008/76
These classes will be Banach spaces under this norm. I refer to spaces of the
forms W 1 LM (Ω) and W 1 EM (Ω) as Orlicz-Sobolev spaces. Thus W 1 LM (Ω) and
W 1 EM (Ω) can be identified with subspaces of the product of N +1 copies of LM (Ω).
Denoting this product by ΠLM , we will use the weak topologies σ(ΠLM , ΠEM̄ )
and σ(ΠLM , ΠLM̄ ). If M satisfies ∆2 condition, then LM (Ω) = EM (Ω) and
W 1 LM (Ω) = W 1 EM (Ω).
The space W01 EM (Ω) is defined as the (norm) closure of C0∞ (Ω) in W 1 EM (Ω)
and the space W01 LM (Ω) as the σ(ΠLM , ΠEM̄ ) closure of C0∞ (Ω) in W 1 LM (Ω).
Let W −1 LM̄ (Ω) (resp. W −1 EM̄ (Ω)) denote the space of distributions on which
can be written as sums of derivatives of order ≤ 1 of functions in LM̄ (Ω) (resp.
EM̄ (Ω)). It is a Banach space under the usual quotient norm (see [12]).
If the open set Ω has the segment property, then the space C0∞ (Ω) is dense in
1
W0 LM (Ω) for the modular convergence and thus for the topology σ(ΠLM , ΠLM̄ )
(cf. [12, 13]).
Let A(u) = − div a(x, u, ∇u) be a Leray-Lions operator defined on W 1,p (Ω),
1 < p < ∞. Boccardo-Gallouet [7] proved the existence of solutions for the Dirichlet
problem for equations of the form
A(u) = f
u=0
in Ω,
(1.1)
on ∂Ω,
(1.2)
where the right hand f is a bounded Radon measure on Ω (i.e. f ∈ Mb (Ω)). The
function a is supposed to satisfy a polynomial growth condition with respect to u
and ∇u.
Benkirane [4, 5] proved the existence of solutions to
A(u) + g(x, u, ∇u) = f,
(1.3)
in Orlicz-Sobolev spaces where
A(u) = − div(a(x, u, ∇u))
(1.4)
W01 LM (Ω),
is a Leray-Lions operator defined on D(A) ⊂
g is supposed to satisfy
a natural growth condition with f ∈ W −1 EM̄ (Ω) and f ∈ L1 (Ω), respectively, but
the result is restricted to N -functions M satisfying a ∆2 condition. Elmahi extend
the results of [4, 5] to general N -functions (i.e. without assuming a ∆2 -condition
on M ) in [8, 9], respectively.
The purpose of this paper is to solve (1.1) when f is a bounded Radon measure,
and the Leray-Lions operator A in (1.4) is defined on D(A) ⊂ W01 LM (Ω), with
general M . We show that the solutions to (1.1) belong to the Orlicz-Sobolev space
W01 LB (Ω) for any B ∈ PM , where PM is a special class of N -function (see below).
Specific examples to which our results apply include the following:
− div(|∇u|p−2 ∇u) = µ in Ω,
− div(|∇u|p−2 ∇u logβ (1 + |∇u|)) = µ in Ω
− div
M (|∇u|)∇u
=µ
|∇u|2
inΩ
where p > 1 and µ is a given Radon measure on Ω.
For some classical and recent results on elliptic and parabolic problems in Orlicz
spaces, I refer the reader to [2, 3, 6, 10, 11, 12, 14, 16, 18].
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ELLIPTIC EQUATIONS WITH MEASURE DATA
3
2. Preliminaries
We define a subset of N -functions as follows.
n
PM = B : R+ → R+ is an N -function, B 00 /B 0 ≤ M 00 /M 0
Z 1
o
and
B ◦ H −1 (1/t1−1/N )dt < ∞
0
where H(r) = M (r)/r. Assume that
PM 6= ∅
(2.1)
N
Let Ω ⊂ R be a bounded domain with the segment property, M, P be two
N -functions such that P M , M̄ , P̄ be the complementary functions of M, P ,
respectively, A : D(A) ⊂ W01 LM (Ω) → W −1 LM̄ (Ω) be a mapping given by A(u) =
− div a(x, u, ∇u) where a : Ω×R×RN → RN be a Caratheodory function satisfying
for a.e. x ∈ Ω and all s ∈ R, ξ, η ∈ RN with ξ 6= η:
|a(x, s, ξ)| ≤ βM (|ξ|)/|ξ|
(2.2)
[a(x, s, ξ) − a(x, s, η)][ξ − η] > 0
(2.3)
a(x, s, ξ)ξ ≥ αM (|ξ|)
(2.4)
where α, β, γ > 0.
Furthermore, assume that there exists D ∈ PM such that
D ◦ H −1 is an N -function.
(2.5)
Set Tk (s) = max(−k, min(k, s)), ∀s ∈ R, for all k ≥ 0. Define by Mb (Ω) as the
set of all bounded Radon measure defined on Ω and by T01,M (Ω) as the set of
measurable functions Ω → R such that Tk (u) ∈ W01 LM (Ω) ∩ D(A).
Assume that f ∈ Mb (Ω), and consider the following nonlinear elliptic problem
with Dirichlet boundary
A(u) = f in Ω.
(2.6)
The following lemmas can be found in [4].
Lemma 2.1. Let F : R → R be uniformly Lipschitzian, withF (0) = 0. Let M be
an N -function, u ∈ W 1 LM (Ω) (resp. W 1 EM (Ω)). Then F (u) ∈ W 1 LM (Ω) (resp.
W 1 EM (Ω)). Moreover, if the set D of discontinuity points of F 0 is finite, then
(
∂u
a.e. in {x ∈ Ω : u(x) 6∈ D}
F 0 (u) ∂x
∂(F ◦ u)
i
=
∂xi
0
a.e. in {x ∈ Ω : u(x) ∈ D}.
Lemma 2.2. Let F : R → R be uniformly Lipschitzian, with F (0) = 0. I suppose
that the set of discontinuity points of F 0 is finite. Let M be an N -function, then
the mapping F : W 1 LM (Ω) → W 1 LM (Ω) is sequentially continuous with respect to
the weak∗ topology σ(ΠLM , ΠEM̄ ).
3. Existence theorem
Theorem 3.1. Assume that (2.1)-(2.5) hold and f ∈ Mb (Ω). Then there exists at
least one weak solution of the problem
u ∈ T01,M (Ω) ∩ W01 LB (Ω),
∀B ∈ PM
Z
a(x, u, ∇u)∇φdx = hf, φi,
Ω
∀φ ∈ D(Ω)
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G. DONG
EJDE-2008/76
Proof. Denote V = W01 LM (Ω). (1) Consider the approximate equations
un ∈ V
− div a(x, un , ∇un ) = fn
(3.1)
where fn is a smooth function which converges to f in the distributional sense that
such that kfn kL1 (Ω) ≤ kf kMb (Ω) . By [4, Theorem 3.1] or [8], there exists at least
one solution {un } to (3.1).
For k > 0, by taking Tk (un ) as test function in (3.1), one has
Z
a(x, Tk (un ), ∇Tk (un ))∇Tk (un )dx ≤ Ck.
Ω
In view of (2.4), we get
Z
M (|∇Tk (un )|)dx ≤ Ck.
(3.2)
Ω
Hence ∇Tk (un ) is bounded in (LM (Ω))N . By [9] there exists u such that un → u
almost everywhere in Ω and
weakly in W01 LM (Ω) for σ (ΠLM , ΠEM̄ ).
Tk (un ) * Tk (u)
(3.3)
For t > 0, by taking Th (un − Tt (un )) as test function, we deduce that
Z
a(x, un , ∇un )∇un dx ≤ hkf kMb (Ω)
t<|un |≤t+h
which gives
1
h
Z
M (|∇un |)dx ≤ kf kMb (Ω)
t<|un |≤t+h
and by letting h → 0,
d
−
dt
Z
M (|∇un |)dx ≤ kf kMb (Ω) .
|un |>t
Let now B ∈ PM . Following the lines of [17], it is easy to deduce that
Z
B(|∇un |)dx ≤ C, ∀n.
(3.4)
Ω
We shall show that a(x, Tk (un ), ∇Tk (un )) is bounded in (LM̄ (Ω))N . Let ϕ ∈
(EM (Ω))N with kϕk(M ) = 1. By (2.2) and Young inequality, one has
Z
Z
Z
M (|∇T (u )|) k n
a(x, Tk (un ), ∇Tk (un ))ϕdx ≤ β
M̄
dx + β
M (|ϕ|)dx
|∇Tk (un )|
Ω
Ω
ZΩ
≤β
M (|∇Tk (un )|)dx + β
Ω
This
last inequality is deduced from M̄ (M (u)/u) ≤ M (u), for all u > 0, and
R
M
(|ϕ|)dx
≤ 1. In view of (3.2),
Ω
Z
a(x, Tk (un ), ∇Tk (un ))ϕdx ≤ Ck + β,
Ω
which implies {a(x, Tk (un ), ∇Tk (un ))}n being a bounded sequence in (LM̄ (Ω))N .
(2) For the rest of this article, χr , χs and χj,s will denoted respectively the
characteristic functions of the sets Ωr = {x ∈ Ω; |∇Tk (u(x))| ≤ r}, Ωs = {x ∈
Ω; |∇Tk (u(x))| ≤ s} and Ωj,s = {x ∈ Ω; |∇Tk (vj (x))| ≤ s}. For the sake of
EJDE-2008/76
ELLIPTIC EQUATIONS WITH MEASURE DATA
5
simplicity, I will write only ε(n, j, s) to mean all quantities (possibly different) such
that
lim lim lim ε(n, j, s) = 0.
s→∞ j→∞ n→∞
Take a sequence (vj ) ⊂ D(Ω) which converges to u for the modular convergence in
V (cf. [13]). Taking Tη (un − Tk (vj )) as test function in (3.1), we obtain
Z
a(x, un , ∇un )∇Tη (un − Tk (vj ))dx ≤ Cη
(3.5)
Ω
On the other hand,
Z
a(x, un , ∇un )∇Tη (un − Tk (vj ))dx
Ω
Z
=
a(x, Tk (un ), ∇Tk (un ))(∇Tk (un ) − ∇Tk (vj ))dx
{|un −Tk (vj )|≤η}∩{|un |≤k}
Z
a(x, un , ∇un )(∇un − ∇Tk (vj ))dx
+
{|un −Tk (vj )|≤η}∩{|un |>k}
Z
a(x, Tk (un ), ∇Tk (un ))(∇Tk (un ) − ∇Tk (vj ))dx
=
{|Tk un −Tk (vj )|≤η}
Z
a(x, un , ∇un )∇un dx
+
{|un −Tk (vj )|≤η}∩{|un |>k}
Z
−
a(x, un , ∇un )∇Tk (vj )dx
{|un −Tk (vj )|≤η}∩{|un |>k}
By (2.4) the second term of the right-hand side satisfies
Z
a(x, un , ∇un )∇un dx ≥ 0.
{|un −Tk (vj )|≤η}∩{|un |>k}
Since a(x, Tk+η (un ), ∇Tk+η (un )) is bounded in (LM̄ (Ω))N , there exists some hk+η ∈
(LM̄ (Ω))N such that
a(x, Tk+η (un ), ∇Tk+η (un )) * hk+η
weakly in (LM̄ (Ω))N for σ (ΠLM̄ , ΠEM ). Consequently the third term of the righthand side satisfies
Z
a(x, un , ∇un )∇Tk (vj )dx
{|un −Tk (vj )|≤η}∩{|un |>k}
Z
a(x, Tk+η (un ), ∇Tk+η (un ))∇Tk (vj )dx
=
{|un −Tk (vj )|≤η}∩{|un |>k}
Z
hk+η ∇Tk (vj )dx + ε(n)
=
{|u−Tk (vj )|≤η}∩{|u|>k}
since
∇Tk (vj )χ{|un −Tk (vj )|≤η}∩{|un |>k} → ∇Tk (vj )χ{|u−Tk (vj )|≤η}∩{|u|>k}
strongly in (EM (Ω))N as n → ∞. Hence
Z
a(x, Tk (un ), ∇Tk (un ))[∇Tk (un ) − ∇Tk (vj )]dx
{|Tk un −Tk (vj )|≤η}
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G. DONG
EJDE-2008/76
Z
≤ Cη + ε(n) +
hk+η ∇Tk (vj )dx
{|u−Tk (vj )|≤η}∩{|u|>k}
Let 0 < θ < 1. Define
Φn,k = [a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u))][∇Tk (un ) − ∇Tk (u)].
For r > 0, I have
Z
0≤
{[a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u))][∇Tk (un ) − ∇Tk (u)]}θ dx
Ωr
Z
Z
θ
=
Φn,k χ{|Tk (un )−Tk (vj )|>η} dx +
Φθn,k χ{|Tk (un )−Tk (vj )|≤η} dx
Ωr
Ωr
Using the Hölder Inequality (with exponents 1/θ and 1/(1 − θ)), the first term of
the right-side hand is less than
1−θ
θ Z
Z
χ{|Tk (un )−Tk (vj )|>η} dx
.
Φn,k dx
Ωr
Ωr
Noting that
Z
Φn,k dx
Ωr
Z
Z
=
a(x, Tk (un ), ∇Tk (un ))∇Tk (un )dx −
a(x, Tk (un ), ∇Tk (u))∇Tk (un )dx
Ωr
Ωr
Z
Z
−
a(x, Tk (un ), ∇Tk (un ))∇Tk (u)dx +
a(x, Tk (un ), ∇Tk (u))∇Tk (u)dx
Ωr
Ωr
Z
Z
M (|∇T (u)|) k
dx + β
M (|∇Tk (un )|)dx
M̄
≤ Ck + β
|∇Tk (u)|
Ωr
Ωr
Z
Z
M (|∇T (u )|) k n
+β
M̄
M (|∇Tk (u)|)dx
dx + β
|∇Tk (un )|
Ωr
Ωr
Z
+β
M (|∇Tk (u)|)dx
Ωr
Z
Z
≤ Ck + β
M (|∇Tk (u)|)dx + β
M (|∇Tk (un )|)dx
Ωr
Ω
Z
Z
Z
+β
M (|∇Tk (un )|)dx + β
M (|∇Tk (u)|)dx + β
M (|∇Tk (u)|)dx
Ω
Ωr
Ωr
≤ (2β + 1)Ck + 3M (r) meas Ω
it follows that
Z
Φθn,k χ{|Tk (un )−Tk (vj )|>η} dx ≤ C̃(meas{|Tk (un ) − Tk (vj )| > η})1−θ ,
Ωr
where C̃ = [(2β + 1)Ck + 3M (r) meas Ω]θ .
Using the Hölder Inequality (with exponents 1/θ and 1/(1 − θ)),
Z
Φθn,k χ{|Tk (un )−Tk (vj )|≤η} dx
Ωr
Z
θ Z
1−θ
≤
Φn,k χ{|Tk (un )−Tk (vj )|≤η} dx
dx
Ωr
Ωr
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ELLIPTIC EQUATIONS WITH MEASURE DATA
≤
Z
Φn,k χ{|Tk (un )−Tk (vj )|≤η} dx
θ
meas Ω
1−θ
Ωr
Hence
Z
0≤
{[a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u))][∇Tk (un ) − ∇Tk (u)]}θ dx
Ωr
1−θ
≤ C̃ meas{|Tk (un ) − Tk (vj )| > η}
Z
θ
1−θ
+
Φn,k χ{|Tk (un )−Tk (vj )|≤η} dx
meas Ω
Ωr
1−θ
= C̃ meas{|Tk (un ) − Tk (vj )| > η}
Z
+
a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u))
Ωr ∩{|Tk (un )−Tk (vj )|≤η}
1−θ
θ
meas Ω
× ∇Tk (un ) − ∇Tk (u) dx
For each s ≥ r one has
Z
0≤
a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u))
Ωr ∩{|Tk (un )−Tk (vj )|≤η}
× ∇Tk (un ) − ∇Tk (u) dx
Z
≤
a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u))
Ωs ∩{|Tk (un )−Tk (vj )|≤η}
× ∇Tk (un ) − ∇Tk (u)]dx
Z
=
a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u)χs )
Ωs ∩{|Tk (un )−Tk (vj )|≤η}
× ∇Tk (un ) − ∇Tk (u)χs dx
Z
≤
a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u)χs )
Ω∩{|Tk (un )−Tk (vj )|≤η}
× ∇Tk (un ) − ∇Tk (u)χs ]dx
Z
=
a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (vj )χj,s )
{|Tk (un )−Tk (vj )|≤η}
× ∇Tk (un ) − ∇Tk (vj )χj,s dx
Z
+
a(x, Tk (un ), ∇Tk (un )) ∇Tk (vj )χj,s − ∇Tk (u)χs dx
{|Tk (un )−Tk (vj )|≤η}
Z
+
a(x, Tk (un ), ∇Tk (vj )χj,s )
{|Tk (un )−Tk (vj )|≤η}
− a(x, Tk (un ), ∇Tk (u)χs ) ∇Tk (un )dx
Z
−
a(x, Tk (un ), ∇Tk (vj )χj,s )∇Tk (vj )χj,s dx
{|Tk (un )−Tk (vj )|≤η}
Z
a(x, Tk (un ), ∇Tk (u)χs )∇Tk (u)χs dx
+
{|Tk (un )−Tk (vj )|≤η}
= I1 (n, j, s) + I2 (n, j, s) + I3 (n, j, s) + I4 (n, j, s) + I5 (n, j, s)
7
8
G. DONG
EJDE-2008/76
On the other hand,
Z
a(x, Tk (un ), ∇Tk (un ))[∇Tk (un ) − ∇Tk (vj )]dx
{|Tk (un )−Tk (vj )|≤η}
Z
a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (vj )χj,s )
=
{|Tk (un )−Tk (vj )|≤η}
× ∇Tk (un ) − ∇Tk (vj )χj,s dx
Z
+
a(x, Tk (un ), ∇Tk (vj )χj,s ) ∇Tk (un ) − ∇Tk (vj )χj,s dx
{|Tk (un )−Tk (vj )|≤η}
Z
−
a(x, Tk (un ), ∇Tk (un ))∇Tk (vj )χ{|∇Tk (vj )|>s} dx
{|Tk (un )−Tk (vj )|≤η}
The second term of the right-hand side tends to
Z
a(x, Tk (u), ∇Tk (u)χs )[∇Tk (u) − ∇Tk (vj )χs ]dx
{|Tk (u)−Tk (vj )|≤η}
since a(x, Tk (un ), ∇Tk (u)χs )χ{|Tk (un )−Tk (vj )|≤η} tends to
a(x, Tk (u), ∇Tk (u)χs )χ{|Tk (u)−Tk (vj )|≤η}
N
in (EM̄ (Ω)) while ∇Tk (un ) − ∇Tk (vj )χs tends weakly to ∇Tk (u) − ∇Tk (vj )χs in
(LM (Ω))N for σ(ΠLM , ΠEM̄ ).
Since a(x, Tk (un ), ∇Tk (un )) is bounded in (LM̄ (Ω))N there exists some hk ∈
(LM̄ (Ω))N such that (for a subsequence still denoted by un )
weakly in (LM̄ (Ω))N for σ(ΠLM̄ , ΠEM ).
a(x, Tk (un ), ∇Tk (un )) * hk
In view of the fact that ∇Tk (vj )χ{|Tk (un )−Tk (vj )|≤η} → ∇Tk (vj )χ{|Tk (u)−Tk (vj )|≤η}
strongly in (EM (Ω))N as n → ∞ the third term of the right-hand side tends to
Z
−
hk ∇Tk (vj )χ{|∇Tk (vj )|>s} dx.
{|Tk (u)−Tk (vj )|≤η}
Hence in view of the modular convergence of (vj ) in V , one has
Z
I1 (n, j, s) ≤ Cη + ε(n) +
hk+η ∇Tk (vj )dx
{|u−Tk (vj )|≤η}∩{|u|>k}
Z
hk ∇Tk (vj )χ{|∇Tk (vj )|>s} dx
+
{|Tk (u)−Tk (vj )|≤η}
Z
−
a(x, Tk (u), ∇Tk (u)χs )[∇Tk (u) − ∇Tk (vj )χs ]dx
{|Tk (u)−Tk (vj )|≤η}
Z
= Cη + ε(n) + ε(j) +
hk ∇Tk (u)χ{|∇Tk (u)|>s} dx
Ω
Z
−
a(x, Tk (u), 0)χ{|∇Tk (u)|>s} dx
Ω
Therefore,
I1 (n, j, s) = Cη + ε(n, j, s)
(3.6)
For what concerns I2 , by letting n → ∞, one has
Z
I2 (n, j, s) =
hk [∇Tk (vj )χj,s − ∇Tk (u)χs ]dx + ε(n)
{|Tk (u)−Tk (vj )|≤η}
EJDE-2008/76
ELLIPTIC EQUATIONS WITH MEASURE DATA
9
since
a(x, Tk (un ), ∇Tk (un )) * hk
weakly in (LM̄ )N for σ(ΠLM̄ , ΠEM )
while χ{|Tk (un )−Tk (vj )|≤η} [∇Tk (vj )χj,s − ∇Tk (u)χs ] approaches
χ{|Tk (u)−Tk (vj )|≤η} [∇Tk (vj )χj,s − ∇Tk (u)χs ]
strongly in (EM )N . By letting j → ∞, and using Lebesgue theorem, then
I2 (n, j, s) = ε(n, j).
Similar tools as above, give
Z
I3 (n, j, s) = −
a(x, Tk (u), ∇Tk (u)χs )∇Tk (u)χs dx + ε(n, j)
(3.7)
(3.8)
Ω
Combining (3.6), (3.7), and (3.8), we have
Z
a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u))
Ωr ∩{|Tk (un )−Tk (vj )|≤η}
× ∇Tk (un ) − ∇Tk (u) dx
≤ ε(n, j, s).
Therefore,
Z
0≤
{[a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u))][∇Tk (un ) − ∇Tk (u)]}θ dx
Ωr
≤ C̃(meas{|Tk (un ) − Tk (vj )| > η})1−θ + (meas Ω)1−θ (ε(n, j, s))θ
Which yields, by passing to the limit superior over n, j, s and η,
Z
a(x, Tk (un ), ∇Tk (un )) − a(x, Tk (un ), ∇Tk (u))
lim
n→∞
Ωr
× ∇Tk (un ) − ∇Tk (u)
θ
dx = 0 .
Thus, passing to a subsequence if necessary, ∇un → ∇u a.e. in Ωr , and since r is
arbitrary,
∇un → ∇u
a.e. in Ω.
By (2.2) and (2.5),
Z
Z
−1 |a(x, un , ∇un )|
D◦H
dx ≤
D(|∇un |)dx ≤ C
β
Ω
Ω
Hence
a(x, un , ∇un ) * a(x, u, ∇u)
weakly for σ(ΠLD◦H −1 ΠED◦H −1 ).
Going back to approximate equations (3.1), and using φ ∈ D(Ω) as the test function,
one has
Z
a(x, un , ∇un )∇φdx = hfn , φi
Ω
in which I can pass to the limit. This completes the proof.
10
G. DONG
EJDE-2008/76
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Ge Dong
1. Department of Mathematics, Shanghai University, No. 99, Shangda Rd., Shanghai,
China
2. Department of Basic, Jianqiao College, No. 1500, Kangqiao Rd., Shanghai, China
E-mail address: dongge97@sina.com