43
Acta Math. Univ. Comenianae
Vol. LXXVII, 1(2008), pp. 43–54
REGULARITY OF WEAK SOLUTIONS
OF DEGENERATE ELIPTIC EQUATIONS
A. C. CAVALHEIRO
Abstract. In this article we establish the existence of higher order weak derivatives
of weak solutions of the Dirichlet problem for a class of degenerate elliptic equations.
1. Introduction
In this paper we shall study the existence of higher order weak derivatives (see
Theorem 3.9) of weak solutions of degenerate elliptic equations Lu = g, where L
is an elliptic operator
(1.1)
Lu = −
n
X
Dj (aij (x)Di u)(x))
i,j=1
whose coefficients aij are measurable, real-valued functions, and whose coefficient
matrix A = (aij ) is symmetric and satisfies the degenerate ellipticity condition
(1.2)
2
ω(x)|ξ| ≤
n
X
2
aij (x)ξi ξj ≤ v(x)|ξ|
i,j=1
for all ξ ∈ Rn and almost every x ∈ Ω ⊂ Rn , where Ω is a bounded open set, ω and
v are weight functions (that is, ω and v are locally integrable and nonnegative
functions on Rn ).
In general, the Sobolev spaces W k,p (Ω) without weights occurs as spaces of
solutions for elliptic and parabolic partial differential equations. For degenerate
partial differential equations, i.e., equations with various types of singularities in
the coefficients it is natural to look for solutions in weighted Sobolev spaces (see
[1], [2], [3], [4], [5] and [8]).
Received September 25, 2006; revised February 7, 2007.
2000 Mathematics Subject Classification. Primary 35J70, 35J25.
Key words and phrases. degenerate elliptic equations; weighted Sobolev spaces.
The author was partially supported by CNPq Grant 476040/2004-03.
44
A. C. CAVALHEIRO
2. Definitions and basic results
By a weight, we shall mean a locally integrable function ω on Rn such that 0 <
ω(x) < ∞ for a.e. x∈Rn . Every weight ω gives rise to a measure on the measurable
n
subsets of
will also be denoted by ω. Thus
R R through integration. This measure
ω(E) = E ω dx for measurable sets E ⊂ Rn .
Definition 2.1. Let Ω ⊂ Rn be open and let ω be a weight. For 1 < p < ∞,
we define Lp (Ω, ω), the Banach space of all measurable functions f defined on Ω
for which
Z
1/p
p
< ∞.
kf kLp (Ω,ω) =
|f (x)| ω(x) dx
Ω
Definition 2.2. Let 1 ≤ p < ∞.
(a) The weight ω belongs to the Muckenhoupt class Ap (ω∈Ap ) if there is a
constant C = Cp,ω (called Ap -constant) such that
p−1
Z
Z
1
1
−1/(p−1)
ω dx
ω
dx
≤ C,
when 1 < p < ∞
|B| B
|B| B
Z
1
1
ω dx
ess sup
≤ C,
when p = 1,
|B| B
ω
B
for every ball B ⊂ Rn , where |B| is the n-dimensional Lebesgue measure
of B.
(b) Let ω and v be weights. We shall say that the pair of weights (v, ω) satisfies
the condition Ap , 1 ≤ p < ∞, if there is a constant C such that
p−1
Z
Z
1
1
v(x) dx
ω −1/(p−1) (x) dx
≤ C,
when 1 < p < ∞,
|B| B
|B| B
Z
1
v(x) dx ≤ C ess inf ω,
when p = 1,
B
|B| B
for every ball B in Rn . The smallest constant C will be called the Ap constant for the pair (ω, v).
Remark 2.3. If (v, ω)∈Ap and ω ≤ v then ω∈Ap and v∈Ap .
α
Example 2.4. The function ω(x) = |x| , x∈Rn , is a weight Ap if and only if
−n < α < n(p − 1) (see [7, Chapter 15]).
Remark 2.5. If ω∈Ap , 1 ≤ p < ∞, then since ω −1/(p−1) is locally integrable,
when p > 1, and 1/ω is locally bounded, when p = 1, we have Lp (Ω, ω) ⊂ L1loc (Ω)
for every open set Ω and such that convergence in Lp (Ω, ω) implies local convergence in L1 (Ω). If Ω is bounded, in the same way one obtains Lp (Ω, ω) ⊂ L1 (Ω).
It thus makes sense to talk about weak derivatives of functions in Lp (Ω, ω).
Definition 2.6. We shall say that the pair of weights (v, ω) satisfies the condition Sp (1 < p < ∞) if there is a constant C (called the Sp -constant) such
45
REGULARITY OF WEAK SOLUTIONS
that
p
Z M (µχB )(x) v(x) dx ≤ Cµ(B) < ∞,
B
Z
1
for every ball B, where [M f ](x) = sup
|f (y)| dy is the Hardy-Littlewood
B3x |B| B
R
maximal function, µ = ω −1/(p−1) and µ(B) = B µ(x) dx.
Remark 2.7. If (v, ω)∈Sp , 1 < p < ∞, then (v, ω)∈Ap .
Theorem 2.8 (Muckenhoupt generalized theorem). Let 1 < p < ∞ and let
(v, ω) be a pair of weights in Rn . Then M : Lp (Rn , ω) → Lp (Rn , v) is bounded
kM f kLp (Rn ,v) ≤ CM kf kLp (Rn ,ω) ,
if and only if (v, ω)∈Sp . The constant CM is called Muckenhoupt constant and
CM depends only on n, p and the Sp -constant of (v, ω).
Proof. See Theorem 4.9, Chapter IV in [6].
Definition 2.9. Let Ω ⊂ Rn be a bounded domain and let v, ω be weights. We
define the space
W k,2 (Ω, ω, v)
Z
= u∈L2 (Ω, v) :
hA∇u, ∇ui dx < ∞ and Dα u∈L2 (Ω, ω), 2 ≤ |α| ≤ k
Ω
with the norm
1/2

Z
kukW k,2 (Ω,ω,v) = 
u2 v dx +
Ω
Z
X
hA∇u, ∇ui dx +
Ω
2 ≤ |α| ≤ k
Z
|Dα u|2 ω dx
Ω
where A = (aij )i,j=1,...,n is the coefficient matrix of the operator L.
Remark 2.10. If (v, ω)∈S2 and ω ≤ v then C ∞ (Ω) is dense in W k,2 (Ω, ω, v)
(see [1, Theorem 4.7]). In this case, we define W0k,2 (Ω, ω, v) as the closure of
C0∞ (Ω) with respect to the norm

1/2
Z
X Z
kukW k,2 (Ω,ω,v) =  hA∇u, ∇ui dx +
|Dα u|2 ω dx .
0
Ω
2≤|α|≤k
Ω
Note that, by (1.2), we have
Z
Z
Z
2
2
|∇u| ω dx ≤
hA∇u, ∇ui dx ≤
|∇u| v dx.
Ω
Ω
Ω
Definition 2.11. We say that an element u∈W 1,2 (Ω, ω, v) is a weak solution
of the equation Lu = g if
Z X
Z
n
aij Di uDj ϕ dx =
g ϕ dx
Ω i,j=1
for every
ϕ∈W01,2 (Ω, ω, v).
Ω
46
A. C. CAVALHEIRO
Remark 2.12. The existence and uniqueness result for the Dirichlet problem
(
Lu = g, in Ω
(P )
u − ψ∈W01,2 (Ω, ω, v)
where ψ ∈ W 1,2 (Ω, ω, v), can be found in [1, Theorem 4.9].
3. Differentiability of Weak Solutions
In this section we prove that weak solutions u∈W 1,2 (Ω, ω, v) of the equation
Lu = g, with some hypotheses, are twice weakly differentiable and Dij u∈L2 (Ω0 , ω)
(that is, u∈W 2,2 (Ω0 , ω, v), ∀Ω0 ⊂⊂ Ω).
Definition 3.1. Let u be a function on a bounded open set Ω ⊂ Rn and denote
by ei the unit coordinate vector in the xi direction. We define the difference
quotient of u at x in the direction ei by
(3.1)
∆hk u(x) =
u(x + hek ) − u(x)
, (0 < |h| < dist
h
(x, ∂Ω)).
Lemma 3.2. Let Ω0 ⊂⊂ Ω and 0 < |h| < dist(Ω0 , ∂Ω). If u, ϕ∈L2loc (Ω, ω),
supp(ϕ) ⊂ Ω0 and g is a measurable function with |g(x)| ≤ Cω(x), then
(a) ∆hk (uϕ)(x) = u(x + hek )∆hk ϕ(x) + ϕ(x)∆hk u(x), with 1 ≤ k ≤ n.
Z
Z
−h
(b)
g(x)u(x)∆k ϕ(x) dx = −
ϕ(x)∆hk (gu)(x) dx.
Ω
Ω
(c) If ϕ ∈ C 1 (Ω), then ∆hk (Dj ϕ)(x) = Dj (∆hk ϕ)(x).
Proof. The proof of this lemma follows trivially from the Definition 3.1.
Definition 3.3. Let ω be a weight in Rn . We say that ω is uniformly Ap in
each coordinate if
(a) ω∈Ap (Rn );
(b) ωi (t) = ω(x1 , . . . , xi−1 , t, xi+1 , . . . , xn ) is in Ap (R), for x1 , . . . , xi−1 , xi+1 , . . . ,
. . . , xn a.e., 1 ≤ i ≤ n, with Ap constant of ωi bounded independently of
x1 , . . . , xi−1 , xi+1 , . . . , xn .
1/2
Example 3.4. Let ω(x, y) = ω1 (x)ω2 (y), with ω1 (x) = |x|
|y| . We have ω is uniformly A2 in each coordinate.
and ω2 (y) =
1/2
Definition 3.5. Let v, ω be weights in Rn .
(a) We say that (v, ω) is uniformly Ap in each coordinate if (v, ω)∈Ap (Rn )
and (vi , ωi )∈Ap (R) (1 ≤ i ≤ n) with constant Ap of (vi , ωi ) bounded
independently of x1 , . . . , xi−1 , xi+1 , . . . , xn .
(b) We say that (v, ω) is uniformly Sp in each coordinate if (v, ω)∈Sp (Rn )
and (vi , ωi )∈Sp (R) (1 ≤ i ≤ n) with Sp -constant of (vi , ωi ) bounded
independently of x1 , . . . , xi−1 , xi+1 , . . . , xn .
REGULARITY OF WEAK SOLUTIONS
47
Lemma 3.6. Let u∈W 1,2 (Ω, ω, v) and let (ω, v) be uniformly S2 in each coordinate. Then for any Ω0 ⊂⊂ Ω and 0 < |h| < dist(Ω0 , ∂Ω), we have
k∆hk ukL2 (Ω0 ,v) ≤ CkDk ukL2 (Ω,ω)
(3.2)
where C = 2CM , and CM is the Muckenhoupt constant.
Proof. Case 1: Let us suppose initially that u∈C ∞ (Ω). We have,
Z
u(x + hek ) − u(x)
1 h
h
∆k u(x) =
Dk (x + ζek ) dζ
=
h
h 0
Z h
1
=
Dk u(x1 , . . . , xk−1 , xk + ζ, xk+1 , . . . , xn ) dζ.
h 0
For 1 ≤ k ≤ n, we define the functions
(
Dk u(x), if x∈Ω
Gk (x) =
0,
if x6∈Ω.
We have for x∈Ω0 ⊂⊂ Ω and h satisfying 0 < |h| < dist(Ω0 , ∂Ω),
Z
1 h
h
|Dk u(x1 , ..., xk−1 , xk + ζ, xk+1 , . . . , xn )| dζ |∆k u(x)| ≤
|h| 0
Z
1 xk +h
=
|Gk (x1 , . . . , xk−1 , t, xk+1 , . . . , xn )| dt
|h| xk
Z
1 xk +h
≤
|G
(x
,
.
.
.
,
x
,
t,
x
,
.
.
.
,
x
)|
dt
k 1
k−1
k+1
n
|h|
xk −h
x ,...,xk−1 ,xk+1 ,...,xn
≤ 2M (Gk1
)(xk ),
x ,...,xk−1 ,xk+1 ,...,xn
(xk )
Gk 1
= Gk (x1 , . . . , xk , . . . , xn ). Consequently, using the
where
d
notation dxk = dx1 . . . dxk−1 dxk+1 . . . dxn (where the hat indicates the term that
must be omitted in the product) and by Theorem 2.8, we obtain
Z
|∆hk u(x)|2 v(x) dx
Ω0
Z
x ,...,xk−1 xk+1 ,...,xn 2
≤ 22
[M (Gk1
)] (xk )v(x1 , . . . , xk , . . . , xn ) dx
Ω0
Z
x ,...,xk−1 ,xk+1 ,...,xn 2
≤4
[M (Gk1
)] (xk )v(x1 , . . . , xk , . . . , xn ) dx1 . . . dxk . . . dxn
Rn
Z
Z
x1 ,...,xk−1 ,xk+1 ,...,xn
2
2
dk
≤4
CM
|Gk
(xk )| ω(x1 , . . . , xk , . . . , xn ) dxk dx
Rn−1
R
Z
Z
2
2
= 4CM
|Gk (x)|2 ω(x) dx = 4CM
|Dk u(x)|2 ω(x) dx,
Rn
Ω
where CM is independent of x1 , . . . , xk−1 , xk+1 , . . . , xn because (v, ω) is uniformly
S2 in each coordinate. Therefore
k∆hk ukL2 (Ω0 ,v) ≤ CkDk ukL2 (Ω,ω) , where C = 2CM .
48
A. C. CAVALHEIRO
Case 2: If u ∈ W 1,2 (Ω, ω, v) then there exists a sequence {um }, um ∈ C ∞ (Ω),
Cauchy sequence in the norm k · kW 1,p (Ω,ω,v) . By Remark 2.10, we have
um → u in L2 (Ω, v),
and
Dk um → Dk u in L2 (Ω, ω).
Since (v, ω)∈S2 and ω ≤ v, we have ω∈A2 and v∈A2 . Consequently, by Remark 2.5, there exists a subsequence {umj} such that umj → u a.e. and Dk umj → Dk u
a.e.. This implies, for 0 < |h| < dist(Ω0 , ∂Ω), that
∆hk umj →∆hk u a.e..
We have {∆hk umj } is a Cauchy sequence in L2 (Ω0 , v), for any Ω0 ⊂⊂ Ω. In fact,
using the first case, we have
k∆hk umr − ∆hk ums kL2 (Ω0 ,v) = k∆hk (umr − ums )kL2 (Ω0 ,v)
≤ CkDk (umr − ums )kL2 (Ω,ω)
= CkDk umr − Dk ums kL2 (Ω,ω)
→ 0,
as mr , ms →∞.
Therefore, there exists g ∈ L2 (Ω0 , v) such that ∆hk umj → g in L2 (Ω0 , v). Consequently, there exists a subsequence ∆hk umjr −→g a.e.. We can conclude that
∆hk u = g a.e.. Hence
∆hk umj →∆hk u in L2 (Ω0 , v).
This implies that
k∆hk ukL2 (Ω0 ,v) = lim k∆hk umj kL2 (Ω0 ,v)
mj →∞
≤ C lim kDk umj kL2 (Ω,ω)
mj →∞
= CkDk ukL2 (Ω,ω) ,
that is, k∆hk ukL2 (Ω0 ,v) ≤ CkDk ukL2 (Ω,ω) .
Lemma 3.7. Let u∈Lp (Ω, ω), 1 < p < ∞, ω∈Ap and suppose there exists a
constant C such that
k∆hk ukLp (Ω0 ,ω) ≤ C,
(3.3)
k = 1, 2, . . . , n
for any Ω0 ⊂⊂ Ω and 0 < |h| < dist(Ω0 , ∂Ω) (with C independent of h). Then there
exists ϑk ∈Lp (Ω, ω) such that Dk u = ϑk in the weak sense and kDk ukLp (Ω,ω) ≤ C.
Proof. Since k∆hk ukLp (Ω0 ,ω) ≤ C, using Lp (Ω, ω) is reflexive (1 < p < ∞), there
exists a sequence {hm }, hm → 0, and a function ϑk ∈ Lp (Ω, ω), with kϑk kLp (Ω,ω) ≤
C, such that
Z
Z
(3.4)
∆hk m u(x)ϕ(x)ω(x) dx →
ϑk (x)ϕ(x)ω(x) dx
Ω
Ω
REGULARITY OF WEAK SOLUTIONS
0
49
0
for all ϕ ∈ Lp (Ω, ω). Since ω∈Ap , we have ϕ = ψ/ω∈Lp (Ω, ω) for any ψ∈C0∞ (Ω).
In fact,
Z
Z
0
0
0
|ϕ|p ω dx =
|ψ|p ω −p ω dx
Ω
Ω
Z
0
≤ Cψ
ω 1−p dx < ∞
(because ω∈Ap ).
Ω
Setting ϕ = ψ/ω in (3.4), we obtain
Z
Z
hm
∆k u(x)ψ(x) dx →
ϑk (x)ψ(x) dx, ∀ ψ ∈ C0∞ (Ω).
Ω
Ω
Now for hm < dist(supp ψ, ∂Ω), we have
Z
Z
u(x)∆k−hm ψ(x) dx
∆hk m u(x)ψ(x) dx = −
Ω
Ω
Z
→ −
u(x)Dk ψ(x) dx,
with hm → 0.
Ω
Hence
Z
Z
ϑk (x)ψ(x) dx = −
Ω
u(x)Dk ψ(x) dx,
∀ ψ ∈ C0∞ (Ω).
Ω
Therefore Dk u = ϑk in the weak sense.
Remark 3.8. If the assumptions of Lemma 3.6 are satisfied and ω ≤ v, then
we have
k∆hk ukL2 (Ω0 ,ω) ≤ k∆hk ukL2 (Ω0 ,v) ≤ CkDk ukL2 (Ω0 ,ω) .
We are able now to prove the main result of this paper.
Theorem 3.9. Let u∈W 1,2 (Ω, ω, v) be a weak solution of the equation Lu = g
in Ω, and assume that
(a) g/v ∈ L2 (Ω, v);
(b) The pair of weights (v, ω) is uniformly S2 in each coordinate;
(c) |∆hk aij (x)| ≤ C1 v(x), x∈Ω0 ⊂⊂ Ω a.e., 0 < |h| < dist(Ω0 , ∂Ω), with constant C1 is independent of Ω0 and h.
Then for any subdomain Ω0 ⊂⊂ Ω, we have u∈W 2,2 (Ω0 , ω, v) and
(3.5)
kukW 2,2 (Ω0 ,ω,v) ≤ C kukW 1,2 (Ω,ω,v) + kg/vkL2 (Ω,v)
for C = C(n, CM , C1 , d0 ), and d0 = dist(Ω0 , ∂Ω).
Proof. Since u∈W 1,2 (Ω, ω, v) is a weak solution of the equation Lu = g, then
by Definition 2.11 we have,
Z
Z
(3.6)
aij (x)Di u(x)Dj ϕ(x) dx =
g(x)ϕ(x) dx,
Ω
for all
ϕ∈W01,2 (Ω, ω, v)
Ω
(in particular for all ϕ∈C0∞ (Ω)).
50
A. C. CAVALHEIRO
∞
In (3.6) let us replace ϕ by ∆−h
k ϕ (1 ≤ k ≤ n), with ϕ∈C0 (Ω), supp(ϕ) ⊂⊂ Ω
and let |2h| < dist(supp (ϕ), ∂Ω). Then, by Lemma 3.2, we obtain
Z
Z
−
g(x)∆−h
ϕ(x)
dx
=
−
aij (x)Di u(x)Dj (∆−h
k
k ϕ(x)) dx
Ω
Ω
Z
=−
aij (x)Di u(x)∆−h
k (Dj ϕ)(x) dx
Ω
Z
=
∆hk (aij Di u)(x)Dj ϕ(x) dx
Ω
Z
=
a(x + hek )∆hk Di u(x) + Di u(x)∆hk aij (x) Dj ϕ(x) dx
ZΩ
=
(3.7)
[h∆hk aij (x) + aij (x)]∆hk Di u(x) + Di u(x)∆hk aij (x)
Ω
· Dj ϕ(x) dx.
By Lemma 3.6, if u∈W 1,2 (Ω, ω, v) we have
(3.8)
k∆hk ukL2 (Ω0 ,v) ≤ CkDk ukL2 (Ω,ω) = C̃,
∀ Ω0 ⊂⊂ Ω.
Since u ∈ L2 (Ω, v) and v ∈ A2 (see Remark 2.3 and Remark 2.7), by Lemma 3.7
we have that
(3.9) kDk ukL2 (Ω0 ,v) ≤ kDk ukL2 (Ω,v) ≤ C̃ = CkDk ukL2 (Ω,ω) , ∀ Ω0 ⊂⊂ Ω.
Consequently, in (3.7), we obtain
Z
aij (x)Di (∆hk u(x))Dj ϕ(x) dx
Ω
Z
Z
= −
g(x)∆−h
ϕ(x)
dx
−
∆hk aij (x)Di u(x)Dj ϕ(x) dx
k
Ω
Ω
Z
−
h∆hk aij (x)∆hk Di u(x)Dj ϕ(x) dx
Ω
Z
Z
h
dx +
∆k aij (x) |Di u(x)| |Dj ϕ(x)| dx
(3.10)
≤
|g(x)| ∆−h
ϕ(x)
k
Ω
Ω
Z
h
h
+ |h|
∆k aij (x) ∆k Di u(x) |Dj ϕ(x)| dx
Ω
We have, by Lemma 3.6,
Z
Z 1/2
−h
|g(x)|
v 1/2 (x) ∆−h
(x) dx
|g(x)| ∆k ϕ(x) dx =
k ϕ(x) v
v(x)
Ω
Ω
g
≤ k∆−h
k ϕkL2 (suppϕ,v)
v L2 (Ω,v)
g
kDk ϕkL2 (Ω,ω) .
≤C v L2 (Ω,v)
51
REGULARITY OF WEAK SOLUTIONS
And, using (3.9), we obtain
Z
h
∆k aij (x) |Di u(x)| |Dj ϕ(x)| dx
Ω
Z
≤ C1
v(x) |Di u(x)| |Dj ϕ(x)| dx
supp ϕ
Z
≤ C1
|Di u(x)|v 1/2 (x)|Dj ϕ(x)|v 1/2 (x) dx
suppϕ
≤ C1 C
2
Z
1/2 Z
1/2
2
|Di u(x)| ω(x) dx
|Di ϕ(x)| ω(x) dx
2
Ω
Ω
Hence, in (3.10), we obtain
Z
aij (x)Di (∆hk u(x))Dj ϕ(x) dx
Ω
g
1,2
≤ C kukW (Ω,ω,v) + (3.11)
kDϕkL2 (Ω,ω)
v L2 (Ω,ω)
Z
+ C1 |h|
v(x)|∆hk Di u(x)||Dj ϕ(x)| dx
Ω
Let Ω0 ⊂⊂ Ω. To proceed further let us take a function η ∈ C0∞ (Ω) satisfying
0 ≤ η ≤ 1, η ≡ 1 in Ω0 and with kDηk∞ ≤ 2/d0 , where d0 = dist(Ω0 , ∂Ω) and set
ϕ = η 2 ∆hk u (with |2h| < dist(supp(η), ∂Ω)). We have
Dj ϕ = 2ηDj η∆hk u + η 2 Dj (∆hk u).
g
We denote by a = kukW 1,2 (Ω,ω,v) + . In (3.11) we obtain
v L2 (Ω,v)
Z
hAηD(∆hk u), ηD(∆hk u)i dx
Ω
Z
aij (x)[η(x)Di (∆hk u(x))][η(x)Dj (∆hk u(x))] dx
Ω
g
≤ C kukW 1,2 (Ω,ω,v) + k2ηDj η∆hk u
v L2 (Ω,v)
Z
2 h
+ η ∆k (Dj u)kL2 (Ω,ω) + 2
|aij (x)||ηDi ∆hk u||Dj η∆hk u| dx
=
Ω
Z
+ C1 |h|
v|∆hk Di u||2ηDj η∆hk u + η 2 Dj ∆hk u| dx
Ω
≤ Ca 2kηDj η∆hk ukL2 (suppη,ω) + kηDj (∆hk u)kL2 (suppη,ω)
+ 2I2 + C1 |h|I3
≤ Ca kDj ηk∞ k∆hk ukL2 (suppη,ω) + kηDj ∆hk ukL2 (suppη,ω)
+ 2I2 + C1 |h|I3
≤ CakDj ηk∞ kDk ukL2 (Ω,ω) + CakηDj ∆hk ukL2 (Ω,ω) + 2I2 + C1 |h|I3 ,
52
A. C. CAVALHEIRO
i. e.
Z
(3.12)
hAηD(∆hk u),ηD(∆hk u)i dx
Ω
≤ C a2 kDj ηk∞ + CakηDj ∆hk ukL2 (Ω,ω) + 2I2 + C1 |h|I3 .
Let us estimate the integrals I2 and I3 . By (1.2), we have that |aij (x)| ≤ Cv(x)
a.e. in Ω. Using (3.8), (3.9) and Remark 3.8, we obtain
Z
I2 =
|aij ||ηDi ∆hk u||Dj η∆hk u| dx
Ω
Z
≤ C
v|ηDi ∆hk u||Dj η∆hk u| dx
suppη
≤ CkηDi ∆hk ukL2 (suppη,v) kDj η∆hk ukL2 (suppη,v)
≤ CkDi (η∆hk u) − Di η∆hk ukL2 (suppη,v) kDj ηk∞ k∆hk ukL2 (suppη,v)
≤ C CM kDi η∆hk u + ηDi ∆hk ukL2 (suppη,ω) + CkukW 1,2 (Ω,ω,v)
· kukW 1,2 (Ω,ω,v)
≤ C kDk ukL2 (Ω,ω) + kηDi ∆hk ukL2 (Ω,ω) + kukW 1,2 (Ω,ω,v)
· kukW 1,2 (Ω,ω,v)
≤ C kukW 1,2 (Ω,ω,v) + kηDi ∆hk ukL2 (Ω,ω) kukW 1,2 (Ω,ω,v)
≤ Ckuk2W 1,2 (Ω,ω,v) + CkηDi ∆hk ukL2 (Ω,ω) kukW 1,2 (Ω,ω,v)
≤ Ca2 + CakηDj ∆hk ukL2 (Ω,ω) .
(3.13)
We also have,
Z
I3 =
v|∆hk Di u||2ηDj η∆hk u + η 2 Dj ∆hk u| dx
Ω
Z
Z
≤ 2
v|∆hk Di u||ηDj η∆hk u| dx +
v|∆hk Di u||η 2 Dj ∆hk u| dx
suppη
suppη
Z
Z
= 2
v|η∆hk Di u||Dj η∆hk u| dx +
v|ηDi ∆hk u||ηDj ∆hk u|dx
suppη
≤
(3.14)
suppη
h
h
2kηDi ∆k ukL2 (suppη,v) kDj η∆k ukL2 (suppη,v)
+ kηDi ∆hk ukL2 (suppη,v) kηDj ∆hk ukL2 (suppη,v) .
Using (3.8) and (3.9), we obtain
kDj η∆hk ukL2 (suppη,v) ≤ kDj ηk∞ k∆hk ukL2 (suppη,v)
≤ CkDj ηk∞ kDk ukL2 (Ω,ω)
≤ CkDj ηk∞ kukW 1,2 (Ω,ω,v)
(3.15)
And we also have,
≤ CakDj ηk∞ .
REGULARITY OF WEAK SOLUTIONS
53
kηDi ∆hk ukL2 (suppη,v) = kDi (η∆hk u) − ∆hk uDi ηkL2 (suppη,v)
≤ kDi (η∆hk u)kL2 (suppη,v) + k∆hk uDi ηkL2 (suppη,v)
≤ CM kDi (η∆hk u)kL2 (suppη,ω) + kDi ηk∞ k∆hk ukL2 (suppη,v)
≤ CM kDi η∆hk u + ηDi ∆hk ukL2 (suppη,ω)
+ CM kDi ηk∞ kDk ukL2 (Ω,ω)
≤ CakDj ηk∞ + CM kηDi ∆hk ukL2 (suppη,ω) .
(3.16)
By condition (1.2) we have,
Z
Z
h
h
aij (x)[η(x)Dj (∆k u(x))][η(x)Di (∆k u(x))] dx ≥
|η(x)D(∆hk u(x))|2 ω(x) dx
Ω
Ω
Z
(3.17)
=
|η(x)∆hk (Du(x))|2 ω(x) dx.
Ω
We denote by b = kηD(∆hk u)kL2 (Ω,ω) . By (3.12), (3.13), (3.14), (3.15), (3.16) and
(3.17), and using Young’s inequality, we obtain
b2 ≤ Ca2 + Cab + C|h|a2 + Cab|h| + C|h|b2
ε2
ε−2 2
ε2
ε−2 2
2
a + C|h|a2 + C b2 + C
a + C b2 |h| + C|h|b2
2
2
2
2
2
2
ε
ε2 2
−2
= C + Cε + C|h| a + C + C |h| + C|h| b2 .
2
2
≤ Ca2 + C
Choose ε > 0 and h such that
Cε2
Cε2 2
1
+
|h| + C|h| ≤ ,
2
2
2
we obtain,
2
Z
g
|η∆hk Du|2 ω dx ≤ C kukW 1,2 (Ω,ω,v) + .
v L2 (Ω,v)
Ω
Using η ≡ 1 in Ω0 , we have
2
Z
g
|∆hk Du|2 ω dx ≤ C kukW 1,2 (Ω,ω,v) + .
v L2 (Ω,v)
Ω0
Then we conclude that
kDj (∆hk u)k2L2 (Ω0 ,ω) = k∆hk Dj uk2L2 (Ω0 ,ω) ≤ k∆hk Duk2L2 (Ω0 ,ω)
2
g
≤ C kukW 1,2 (Ω,ω,v) + .
v L2 (Ω,v)
By Lemma 3.7, we have there exists Djk u and
g
kDjk ukL2 (Ω0 ,ω) ≤ C kukW 1,2 (Ω,ω,v) + .
v L2 (Ω,v)
Therefore u∈W 2,2 (Ω0 , ω, v), ∀ Ω0 ⊂⊂ Ω.
54
A. C. CAVALHEIRO
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A. C. Cavalheiro, State University of Londrina, Department of Mathematics, 86051-990 Londrina
– PR, Brazil, e-mail: accava@gmail.com