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REGULARITY OF WEAK SOLUTIONS
OF DEGENERATE ELIPTIC EQUATIONS
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A. C. CAVALHEIRO
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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
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whose coefficients aij are measurable, real-valued functions, and whose coefficient matrix A = (aij )
is symmetric and satisfies the degenerate ellipticity condition
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(1.2)
2
ω(x)|ξ| ≤
n
X
2
aij (x)ξi ξj ≤ v(x)|ξ|
i,j=1
Close
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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.
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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]).
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
subsets of Rn through integration.
R
This measure will also be denoted by ω. Thus 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
< ∞.
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Definition 2.2. Let 1 ≤ p < ∞.
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(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
B ω
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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
−1/(p−1)
v(x) dx
ω
(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 .
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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]).
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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).
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α
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 that
p
Z M (µχB )(x) v(x) dx ≤ Cµ(B) < ∞,
B
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Z
|f (y)| dy is the Hardy-Littlewood maximal function,
B
Remark 2.7. If (v, ω)∈Sp , 1 < p < ∞, then (v, ω)∈Ap .
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for every ball B, where [M f ](x) = sup
|B|
B3x
R
µ = ω −1/(p−1) and µ(B) = B µ(x) dx.
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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) ,
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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
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W k,2 (Ω, ω, v)
Z
2
α
2
= u∈L (Ω, v) :
hA∇u, ∇ui dx < ∞ and D u∈L (Ω, ω), 2 ≤ |α| ≤ k
Ω
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with the norm
1/2

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Z
kukW k,2 (Ω,ω,v) = 
Ω
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u2 v dx +
Z
hA∇u, ∇ui dx +
Ω
X
2 ≤ |α| ≤ k
where A = (aij )i,j=1,...,n is the coefficient matrix of the operator L.
Z
Ω
|Dα u|2 ω dx
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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
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Ω
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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
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2≤|α|≤k
for every
Ω
ϕ∈W01,2 (Ω, ω, v).
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
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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).
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∆hk u(x) =
(3.1)
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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
I
(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
(b)
g(x)u(x)∆−h
ϕ(x)
dx
=
−
ϕ(x)∆hk (gu)(x) dx.
k
Ω
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u(x + hek ) − u(x)
, (0 < |h| < dist
h
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.
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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
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Example 3.4. Let ω(x, y) = ω1 (x)ω2 (y), with ω1 (x) = |x|
uniformly A2 in each coordinate.
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Definition 3.5. Let v, ω be weights in Rn .
1/2
and ω2 (y) = |y|
. We have ω is
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(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 .
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)
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where C = 2CM , and CM is the Muckenhoupt constant.
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Proof. Case 1: Let us suppose initially that u∈C ∞ (Ω). We have,
∆hk u(x)
=
=
Z
u(x + hek ) − u(x)
1 h
=
Dk (x + ζek ) dζ
h
h 0
Z
1 h
Dk u(x1 , . . . , xk−1 , xk + ζ, xk+1 , . . . , xn ) dζ.
h 0
For 1 ≤ k ≤ n, we define the functions
(
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Gk (x) =
Dk u(x), if x∈Ω
0,
if x6∈Ω.
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We have for x∈Ω0 ⊂⊂ Ω and h satisfying 0 < |h| < dist(Ω0 , ∂Ω),
|∆hk u(x)|
≤
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=
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≤
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x ,...,xk−1 ,xk+1 ,...,xn
≤ 2M (Gk1
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)(xk ),
x ,...,xk−1 ,xk+1 ,...,xn
dk =
where Gk1
(xk ) = Gk (x1 , . . . , xk , . . . , xn ). Consequently, using the notation dx
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
2
≤2
[M (Gk1
)] (xk )v(x1 , . . . , xk , . . . , xn ) dx
0
Z Ω
x ,...,xk−1 ,xk+1 ,...,xn 2
≤4
[M (Gk1
)] (xk )v(x1 , . . . , xk , . . . , xn ) dx1 . . . dxk . . . dxn
n
R
Z
Z
x ,...,xk−1 ,xk+1 ,...,xn
dk
≤4
C2
|G 1
(xk )|2 ω(x1 , . . . , xk , . . . , xn ) dxk dx
Rn−1
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Z
1 h
|Dk u(x1 , ..., xk−1 , xk + ζ, xk+1 , . . . , xn )| dζ |h| 0
Z
1 xk +h
|G
(x
,
.
.
.
,
x
,
t,
x
,
.
.
.
,
x
)|
dt
k 1
k−1
k+1
n
|h| xk
Z xk +h
1 |Gk (x1 , . . . , xk−1 , t, xk+1 , . . . , xn )| dt
|h| xk −h
2
= 4CM
Z
Rn
M
k
R
2
|Gk (x)|2 ω(x) dx = 4CM
Z
Ω
|Dk u(x)|2 ω(x) dx,
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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 .
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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
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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
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k∆hk umr − ∆hk ums kL2 (Ω0 ,v) = k∆hk (umr − ums )kL2 (Ω0 ,v)
≤ CkDk (umr − ums )kL2 (Ω,ω)
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= CkDk umr − Dk ums kL2 (Ω,ω)
→ 0,
as mr , ms →∞.
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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
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∆hk umj →∆hk u in L2 (Ω0 , v).
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This implies that
k∆hk ukL2 (Ω0 ,v) = lim k∆hk umj kL2 (Ω0 ,v)
mj →∞
≤ C lim kDk umj kL2 (Ω,ω)
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mj →∞
= CkDk ukL2 (Ω,ω) ,
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that is, k∆hk ukL2 (Ω0 ,v) ≤ CkDk ukL2 (Ω,ω) .
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Lemma 3.7. Let u∈Lp (Ω, ω), 1 < p < ∞, ω∈Ap and suppose there exists a constant C such
that
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k∆hk ukLp (Ω0 ,ω) ≤ C,
(3.3)
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k = 1, 2, . . . , n
for any Ω0 ⊂⊂ Ω and 0 < |h| < dist(Ω0 , ∂Ω) (with C independent of h).
ϑk ∈Lp (Ω, ω) such that Dk u = ϑk in the weak sense and kDk ukLp (Ω,ω) ≤ C.
Then there exists
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
0
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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
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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
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Now for hm < dist(supp ψ, ∂Ω), we have
Z
Z
hm
∆k u(x)ψ(x) dx = −
u(x)∆k−hm ψ(x) dx
Ω
Ω
Z
→ −
u(x)Dk ψ(x) dx,
Hence
Z
Z
ϑk (x)ψ(x) dx = −
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with hm → 0.
Ω
Ω
u(x)Dk ψ(x) dx,
∀ ψ ∈ C0∞ (Ω).
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Therefore Dk u = ϑk in the weak sense.
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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) .
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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);
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(b) The pair of weights (v, ω) is uniformly S2 in each coordinate;
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(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).
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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
aij (x)Di u(x)Dj ϕ(x) dx =
g(x)ϕ(x) dx,
(3.6)
Ω
Ω
ϕ∈W01,2 (Ω, ω, v)
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(in particular for all
for all
∞
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
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−
Ω
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ϕ∈C0∞ (Ω)).
(3.7)
g(x)∆−h
k ϕ(x) dx = −
Z
ZΩ
aij (x)Di u(x)Dj (∆−h
k ϕ(x)) dx
=−
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
=
[h∆hk aij (x) + aij (x)]∆hk Di u(x) + Di u(x)∆hk aij (x) · Dj ϕ(x) dx.
Ω
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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
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(3.9)
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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
≤
|g(x)| ∆−h
ϕ(x)
(3.10)
k
Ω
Ω
Z
h
h
∆k aij (x) ∆k Di u(x) |Dj ϕ(x)| dx
+ |h|
Ω
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We have, by Lemma 3.6,
Z Z
−h
1/2
|g(x)|
|g(x)| ∆k ϕ(x) dx =
v 1/2 (x) ∆−h
(x) dx
k ϕ(x) v
v(x)
Ω
Ω
g
≤ k∆−h
k ϕkL2 (suppϕ,v)
v L2 (Ω,v)
g
≤C kDk ϕkL2 (Ω,ω) .
v L2 (Ω,v)
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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
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≤ C1 C 2
Z
1/2 Z
1/2
|Di u(x)|2 ω(x) dx
|Di ϕ(x)|2 ω(x) dx
Ω
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Ω
Hence, in (3.10), we obtain
Z
aij (x)Di (∆hk u(x))Dj ϕ(x) dx
Ω
g
1,2
(3.11)
≤ C kukW (Ω,ω,v) + kDϕkL2 (Ω,ω)
v L2 (Ω,ω)
Z
+ C1 |h|
v(x)|∆hk Di u(x)||Dj ϕ(x)| dx
Ω
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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).
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g
. In (3.11) we obtain
We denote by a = kukW 1,2 (Ω,ω,v) + v L2 (Ω,v)
Z
hAηD(∆hk u), ηD(∆hk u)i dx
Ω
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Z
aij (x)[η(x)Di (∆hk u(x))][η(x)Dj (∆hk u(x))] dx
g
≤ C kukW 1,2 (Ω,ω,v) + k2ηDj η∆hk u
v L2 (Ω,v)
Z
+ η 2 ∆hk (Dj u)kL2 (Ω,ω) + 2
|aij (x)||ηDi ∆hk u||Dj η∆hk u| dx
=
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J
I
Ω
Ω
Z
+ C1 |h|
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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
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≤ Ca kDj ηk∞ k∆hk ukL2 (suppη,ω) + kηDj ∆hk ukL2 (suppη,ω)
+ 2I2 + C1 |h|I3
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≤ CakDj ηk∞ kDk ukL2 (Ω,ω) + CakηDj ∆hk ukL2 (Ω,ω) + 2I2 + C1 |h|I3 ,
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i. e.
Z
(3.12)
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hAηD(∆hk u),ηD(∆hk u)i dx
Ω
≤ C a2 kDj ηk∞ + CakηDj ∆hk ukL2 (Ω,ω) + 2I2 + C1 |h|I3 .
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I
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)
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≤ 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)
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≤ Ckuk2W 1,2 (Ω,ω,v) + CkηDi ∆hk ukL2 (Ω,ω) kukW 1,2 (Ω,ω,v)
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≤ Ca2 + CakηDj ∆hk ukL2 (Ω,ω) .
(3.13)
We also have,
Close
Z
v|∆hk Di u||2ηDj η∆hk u + η 2 Dj ∆hk u| dx
Z
Z
h
h
≤ 2
v|∆k Di u||ηDj η∆k u| dx +
v|∆hk Di u||η 2 Dj ∆hk u| dx
I3 =
Ω
Quit
suppη
suppη
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Z
= 2
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v|η∆hk Di u||Dj η∆hk u| dx
suppη
Z
+
v|ηDi ∆hk u||ηDj ∆hk u|dx
suppη
≤ 2kηDi ∆hk ukL2 (suppη,v) kDj η∆hk ukL2 (suppη,v)
+ kηDi ∆hk ukL2 (suppη,v) kηDj ∆hk ukL2 (suppη,v) .
(3.14)
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Using (3.8) and (3.9), we obtain
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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) ≤ CakDj ηk∞ .
(3.15)
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I
And we also have,
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)
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≤ 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)
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+ CM kDi ηk∞ kDk ukL2 (Ω,ω)
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(3.16)
≤ CakDj ηk∞ + CM kηDi ∆hk ukL2 (suppη,ω) .
By condition (1.2) we have,
Z
Z
aij (x)[η(x)Dj (∆hk u(x))][η(x)Di (∆hk u(x))] dx ≥
|η(x)D(∆hk u(x))|2 ω(x) dx
Ω
ZΩ
(3.17)
=
|η(x)∆hk (Du(x))|2 ω(x) dx.
Ω
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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
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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
ε2 2
ε
−2
= C + Cε + C|h| a + C + C |h| + C|h| b2 .
2
2
≤ Ca2 + C
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Choose ε > 0 and h such that
Cε2 2
1
Cε2
+
|h| + C|h| ≤ ,
2
2
2
we obtain,
Z
Ω
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|η∆hk Du|2 ω dx
2
g
≤ C kukW 1,2 (Ω,ω,v) + .
v L2 (Ω,v)
0
Using η ≡ 1 in Ω , we have
2
Z
g
h
2
1,2
.
|∆k Du| ω dx ≤ C kukW (Ω,ω,v) + v L2 (Ω,v)
Ω0
Then we conclude that
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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)
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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.
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1. Cavalheiro A. C., An approximation theorem for solutions of degenerate elliptic equations, Proc. of the Edinburgh
Math. Soc. 45 (2002), 363-389.
2. Chanillo S. and Wheeden R. L., Weighted Poincaré and Sobolev inequalities and estimates for the Peano maximal
functions, Am. J. Math. 107 (1985), 1119–1226.
3. Fabes E., Jerison D. and Kenig C., The Wiener Test for Degenerate Elliptic Equations, Ann. Inst. Fourier
(Grenoble), 32(3) (1982), 151–182.
4. Fabes E., Kenig C. and Serapioni R., The Local Regularity of Solutions of Degenerate Elliptic Equations, Comm.
in P.D.E., 7(1) (1982), 77–116.
5. Franchi B. and Serapioni R., Pointwise Estimates for a Class of Strongly Degenerate Elliptic Operators: a
Geometrical Approch, Ann. Scuola Norm. Sup. Pisa, 14 (1987), 527–568.
6. Garcia-Cuerva J. and Rubio de Francia J., Weighted Norm Inequalities and Related Topics, Nort-Holland
Matematics Studies 116, Elsevier Science Pub. Co., 1985.
7. Heinonen J., Kilpeläinen T. and Martio O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford
Mathematical Monographs, The Cleredon Press, Oxford University Press, New York, 1993.
8. Turesson B.O., Nonlinear Potential Theory and Weighted Sobolev Spaces, Lec. Notes in Math. 1736, SpringerVerlag 2000.
A. C. Cavalheiro, State University of Londrina, Department of Mathematics, 86051-990 Londrina – PR, Brazil,
e-mail: accava@gmail.com