TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 359, Number 12, December 2007, Pages 5899–5913
S 0002-9947(07)04238-9
Article electronically published on June 26, 2007
QUASILINEAR ELLIPTIC EQUATIONS
WITH BMO COEFFICIENTS IN LIPSCHITZ DOMAINS
SUN-SIG BYUN AND LIHE WANG
Abstract. We obtain a global W 1,q estimate for the weak solution to an
elliptic partial differential equation of p-Laplacian type with BMO coefficients
in a Lipschitz domain with small Lipschitz constant.
1. Introduction
Suppose that 1 < p < ∞. We are concerned with the following equation:
p−2
(1.1)
div (A∇u · ∇u) 2 A∇u = div |f |p−2 f in Ω,
where Ω is an open, bounded subset of Rn . The coefficients matrix A is assumed
to be essentially bounded and uniformly elliptic; that is,
A ∈ L∞ (Ω)
and
Λ−1 |ξ|2 ≤ A(x)ξ · ξ ≤ Λ|ξ|2
for some Λ > 0, a.e. x ∈ Rn and all ξ ∈ Rn . We assume as well
f ∈ Lq (Ω)
for some q ≥ p.
We are interested in the question: What is a minimal requirement on the coefficients matrix A and a more general geometric condition on the boundary of Ω on
which W 1,q estimates hold? In particular we are interested in estimates like
q
(1.2)
|∇u| dx ≤ C
|f |q dx
Ω
Ω
for some constant C independent of u and f .
This is a classical question, and there have been many works in this direction
(see e.g. [4, 6, 10, 11]). In [10, 11] the authors considered the Dirichlet problem
for (1.1) to prove the well posedness in W 1,q (Ω) under the assumptions that A is
Received by the editors August 5, 2005.
2000 Mathematics Subject Classification. Primary 35R05, 35R35; Secondary 35J15, 35J25.
Key words and phrases. W 1,p estimates, quasilinear elliptic equations, BMO space, Lipschitz
domain, maximal function, Vitali covering lemma.
The first author was supported in part by KRF-2005-003-C00016.
The second author was supported in part by NSF Grant #0401261.
c
2007
American Mathematical Society
5899
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5900
SUN-SIG BYUN AND LIHE WANG
of the space VMO and that ∂Ω is locally C 1,α , 0 < α ≤ 1. There the authors used
the sharp maximal functions and found a local version for the the sharp maximal
functions while here we will use maximum functions and simplify their proof.
Our work is very much influenced by [3, 14] and the works in [10, 11]. We
are working under the assumption that the boundary ∂Ω of the domain is locally,
the graph of a function which is required to be Lipschitz continuous (see papers
[7, 8, 9]).
Recently in [1] the author dealt with PDE (1.1) when p = 2 with zero boundary
condition to show that W 1,q estimates hold under the assumptions that A has small
BMO seminorms and that Ω has locally, small Lipschitz constants. The author
used a scaling based on the standard Lp estimates, maximal functions and a Vitali
covering lemma. The key idea is to find the decay estimates of the Hardy-Littlewood
maximal function of the gradient of solutions. This approach used in [1, 14] enables
the authors to avoid the classical one which uses integral representations.
In this work zero boundary condition is studied. More precisely, we consider the
following Dirichlet problem:
p−2
div (A∇u · ∇u) 2 A∇u
= div |f |p−2 f
in
Ω,
(1.3)
u = 0
on ∂Ω.
Definition 1.1. Weak solution of (1.3) is a function u ∈ W01,q (Ω) such that
p−2
2
(A∇u · ∇u)
A∇u · ∇ϕdx =
|f |p−2 f · ∇ϕdx
Ω
Ω
for all ϕ ∈ C0∞ (Ω).
We refer to [10, 11] for a general discussion on equation (1.1).
The main theorem is stated as follows.
Theorem 1.2. Let q > p. Then there is a small δ = δ(Λ, p, n, R) > 0 so that
for all A with A (δ, R)-vanishing, for all Ω with Ω (δ, R)-vanishing, and for all f
with f ∈ Lq (Ω; Rn ), the Dirichlet problem ( 1.3) has a unique weak solution with the
estimate
q
|∇u| dx ≤ C
|f |q dx,
Ω
Ω
where the constant C is independent of u and f .
We wish to conclude this Introduction by mentioning that a Lipschitz domain
with small Lipschitz constant exhibits the minimal geometric condition necessary
for the W 1,q regularity theory in this direction, from the point of view that the
boundary of the domain is locally the graph of a function.
2. Preliminaries
In this section we describe precisely the assumptions considered in this work on
the coefficients matrix A and the boundary ∂Ω of the domain Ω. Here we also
introduce the main tools we will use.
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QUASILINEAR ELLIPTIC EQUATIONS IN LIPSCHITZ DOMAINS
5901
In view of [12], A is assumed to be defined on Rn . For x ∈ Rn and r > 0,
Br (x) denotes an n-dimensional ball of radius r and center x. We use the following
definition.
Definition 2.1. We say that the coefficients matrix A is (δ, R)-vanishing if
p
p
1
sup sup
|A(y) − ABr (x) | p−1 dy ≤ δ p−1 .
0<r≤R x∈Rn |Br | Br (x)
∂Ω is assumed to be written locally as the graph of Lipschitz functions with
small Lipschitz norms.
Definition 2.2. We say that Ω is (δ, R)-Lipschitz if for every x0 ∈ ∂Ω and every
r ∈ (0, R], there exists a Lipschitz continuous function γ : Rn−1 → R with Lip(γ) ≤
δ such that
Ω ∩ Br (x0 ) = {x = (x1 , ..., xn−1 , xn ) = (x , xn ) ∈ Br (x0 ) : xn > γ(x )}
in some coordinate system.
We remark that one might assume that R in the definitions above to be 1 by
scaling the given equations, while δ is scaling invariant. Through this paper we
mean δ to be a small positive constant.
We will combine the compactness method, the classical Hardy-Littlewood maximal function, the Vitali covering lemma and standard arguments of measure theory.
Our compactness method is based on the following lemma:
Lemma 2.3 ([12]). If Ω is a Lipschitz domain, then W 1,p (Ω) is compactly embedded
in Lp (Ω) for all 1 < p < ∞.
We use a maximal function argument.
Definition 2.4. The Hardy-Littlewood maximal function Mf of a locally integrable function f is a function such that
1
(Mf )(x) = sup
|f (y)|dy,
r>0 |Br (x)| Br (x)
where Br (x) is the open ball of radius r centered at x.
MΩ f = M (f χΩ ) ,
if f is not defined outside Ω.
The basic properties for the Hardy-Littlewood maximal function are the following.
Lemma 2.5 ([13]).
(1) (strong p-p estimate). If f ∈ Lp (Rn ) with 1 < p ≤ ∞,
p
then Mf ∈ L (Rn ) and
(2.1)
1
f Lp ≤ Mf Lp ≤ Cf Lp .
C
(2) (weak 1-1 estimate). If f ∈ L1 (Rn ), then
(2.2)
|{x ∈ Rn : (Mf )) (x) > t}| ≤
C
t
|f (x)|dx.
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5902
SUN-SIG BYUN AND LIHE WANG
We will use the following version of the Vitali covering lemma.
Lemma 2.6 ([1]). Assume that C and D are measurable sets with C ⊂ D ⊂ B1+ .
We suppose further that there exists an > 0 such that
|C| < |B1+ |
and
for every x ∈ B1+ with |C ∩ Br (x)| ≥ |Br |, Br (x) ∩ B1+ ⊂ D.
Then we have
|C| ≤ 2(10)n |D|.
We use the following standard arguments of measure theory.
Lemma 2.7 ([2]). Suppose that f is a nonnegative and measurable function in Rn .
Suppose further that f has a compact support in a bounded subset E of Rn . Let
θ > 0 and m > 1 be constants. Then for 0 < p < ∞ we have
f ∈ Lp (E) ⇐⇒ S =
mkp x ∈ E : f (x) > θmk < ∞
k≥1
and
1
S ≤ f pLp (E) ≤ C(|E| + S),
C
where C > 0 is a constant depending only on θ, m, and p.
3. Interior regularity
With the different types of equations from those in [1] we should start out with
the definition of weak solutions concerning (1.1). Based on a scaling we consider
the following PDE:
p−2
(3.1)
div (A∇u · ∇u) 2 A∇u = div |f |p−2 f in B6 .
Definition 3.1. A weak solution of (3.1) is a function u ∈ W 1,p (B6 ) which satisfies
p−2
(A∇u · ∇u) 2 A∇u · ∇ϕdx =
|f |p−2 f · ∇ϕdx
B6
B6
for all ϕ ∈ C0∞ (B6 ).
One can prove the following interior W 1,q estimates via the same lines of ideas
considered in W 1,q boundary estimates (see Section 4).
Theorem 3.2. Let q be a real number with q > p. Then there is a small δ =
δ(Λ, p, n) > 0 so that for all A with A (δ, 6)-vanishing, if u is a weak solution of
( 3.1), then u belongs to W 1,q (B1 ) with the estimate
∇uLq (B1 ) ≤ C uLq (B6 ) + f Lq (B6 ) ,
where the constant C is independent of u and f .
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QUASILINEAR ELLIPTIC EQUATIONS IN LIPSCHITZ DOMAINS
5903
The main thing to do for the proof of the theorem above is to derive the following
lemma. One can find its proof in the same way as we will treat Lemma 4.5.
Lemma 3.3. There is a constant N1 > 0 so that for any 0 < , r ≤ 1, there
exists a small δ = δ() > 0 such that if u is a weak solution of ( 3.1), with A
(δ, 6)-vanishing and
|{x ∈ B1 (0) : M(|∇u|p )(x) > N1p } ∩ Br | ≥ |Br |,
then we have
Br ∩ B1 (0) ⊂ {x ∈ B1 (0) : M(|∇u|p ) > 1} ∪ {x ∈ B1 (0) : M(|f |p ) > δ p },
where Br denotes the ball with radius r and center in B1 (0).
4. Boundary regularity
Now we are intended to find the boundary W 1,q regularity with p < q < ∞
regarding the Dirichlet problem (1.3) under the assumptions that the coefficients
matrix is (δ, R)-vanishing and the domain is (δ, R)-Lipschitz. The important analytical tools are the maximal function and a modified Vitali covering lemma (see
Section 2 of paper [1]). As the boundary of the domain is, locally, the graph of
a function which is Lipschitz continuous, we are first concerned with boundary
estimates on flat boundaries.
Denote
TR = BR ∩ {xn = 0}, TR (x0 ) = TR + x0 for x0 ∈ Rn−1 ,
+
+
BR
= BR ∩ {xn > 0}, BR
(x0 ) = BR (x0 ) ∩ {xn > 0} for x0 ∈ Rn .
+
) is a weak solution of
Definition 4.1. We say that u ∈ W 1,q (BR
p−2
+
div (A∇u · ∇u) 2 A∇u
= div |f |p−2 f
in BR
,
(4.1)
u = 0
on TR ,
if we have
+
BR
(A∇u · ∇u)
p−2
2
A∇u · ∇ϕdx =
+
BR
|f |p−2 f · ∇ϕdx
+
for all ϕ ∈ C0∞ (BR
) and the zero extension of u is of W 1,p (BR ).
The compactness argument is based on the following observation: Since A is
(δ, R)-vanishing, one can freeze the constant coefficients AB + and use known inteR
gral estimates of the reference equation of (4.1)
p−2
+
div (AB + ∇v · ∇v) 2 AB + ∇v
= 0 in BR
,
R
R
(4.2)
v = 0 on
TR
+
+
to observe that u − v is small in LP (BR
) provided that f is small in LP (BR
),
which is possible since it is about data, and that A is small in BM O, which is the
assumption imposed on A.
Lemma 4.2. For any > 0, there exists a small δ = δ() > 0 such that if u is a
weak solution of ( 4.1) in B4+ , with
p
p
1
|A − AB + | p−1 dx ≤ δ p−1
(4.3)
4
|B4 | B4+
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5904
SUN-SIG BYUN AND LIHE WANG
and
1
|B4 |
(4.4)
B4+
|∇u|p dx ≤ 1,
1
|B4 |
B4+
|f |p dx ≤ δ p ,
then there exist a weak solution v of ( 4.2) in B4+ such that
|u − v|p dx ≤ p .
(4.5)
B4+
Proof. We argue by contradiction. If not,
∞
{uk }∞
k=1 and {fk }k=1 such that uk is a weak
p−2
div (Ak ∇uk · ∇uk ) 2 Ak ∇uk
(4.6)
uk
with
1
|B4 |
(4.7)
and
1
|B4 |
(4.8)
But we have
there would exist 0 > 0, {Ak }∞
k=1 ,
solution of
= div |fk |p−2 fk
in B4+ ,
= 0
on
T4 ,
1
p
|Ak − Ak B + | p−1 dx ≤
4
B4+
1
|∇uk | dx ≤ 1,
+
|B4 |
B4
p
k p−1
,
p
B4+
|fk |p dx ≤
1
.
kp
(4.9)
B4+
|uk − vk |p dx > p0
for any weak solution vk of
p−2
div (Ak B + ∇vk · ∇vk ) 2 Ak B + ∇vk
4
4
(4.10)
vk
= 0
= 0
in
on
B4+ ,
T4 .
Noting that uk = 0 on T4 and using (4.8), we observe that {uk }∞
k=1 is bounded
+
1,p
in W (B4 ). Consequently there exists a subsequence, which we still denote by
{uk }, and u0 ∈ W 1,p (B4+ ) such that
uk u0 in W 1,p (B4+ ),
(4.11)
Lp (B4+ ).
uk → u0 in
∞
As Ak B +
is bounded in l∞ , there exists a subsequence, which we denote by
4
k=1
{Ak }, such that
Ak → A0 in l∞
for some constant coefficients matrix A0 . Consequently (4.7) implies
p
Ak → A0 in L p−1 (B4+ ).
(4.12)
Now we want verify that u0 is a weak solution of
p−2
div (A0 ∇u0 · ∇u0 ) 2 A0 ∇u0
= 0
(4.13)
u0 = 0
in
on
B4+ ,
T4 .
Fix any ϕ ∈ C0∞ (B4+ ). Then we recall Definition 4.1 to find from (4.6) that
p−2
2
(Ak ∇uk · ∇uk )
Ak ∇uk · ∇ϕdx =
|fk |p−2 fk · ∇ϕdx.
(4.14)
B4+
B4+
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QUASILINEAR ELLIPTIC EQUATIONS IN LIPSCHITZ DOMAINS
5905
Now using (4.12), (4.11), the method of Minty (see Theorem 3 of Chapter 9 in [5]),
and (4.8) and letting k → ∞ in (4.14), we have
p−2
(A0 ∇u0 · ∇u0 ) 2 A0 ∇u0 · ∇ϕdx = 0.
(4.15)
B4+
This establishes (4.13) since u0 = 0 in T4 in the trace sense from (4.11). Taking
v = u0 and sending k → ∞, we reach a contradiction to (4.10).
Lemma 4.3. There is a constant N1 > 0 so that for any > 0, there exists a small
δ = δ() > 0 such that if u is a weak solution of ( 4.1) in B6+ , with
p
p
1
|A − AB + | p−1 dx ≤ δ p−1
4
|B4 | B4+
and
(4.16)
B1+ ∩ x ∈ B6+ : M(|∇u|p )(x) ≤ 1 ∩ x ∈ B6+ : M(|f |p )(x) ≤ δ p = ∅,
then we have
x ∈ B + : M(|∇u|p )(x) > N p ∩ B + < |B1 |.
1
6
1
(4.17)
Proof. In view of (4.16), there exists an x0 ∈ B1+ such that
1
1
(4.18)
|∇u|p dx ≤ 1 ,
|f |p dx ≤ δ p
|Br | Br (x0 )
|Br | Br (x0 )
for all r > 0. Now that B4+ (0) ⊂ B5+ (x0 ) we see from (4.18) that
n
n
5
5
1
1
|f |p dx ≤
|f |p dx ≤
δp
(4.19)
|B4 | B4+ (0)
4
|B5 | B5+ (x0 )
4
and
1
|B4 |
(4.20)
n
5
|∇u| dx ≤
.
+
4
B4 (0)
p
n
n
Applying the lemma above to the PDE (4.1), with 45 u replacing u and 45 f replacing f , we deduce that for any η > 0, there exist a small δ(η) and a corresponding
weak solution v of (4.2) in B4+ such that
(4.21)
|u − v|p dx ≤ η p
B4+
provided
(4.22)
1
|B4 |
B4+
|f |p dx ≤ δ p ,
1
|B4 |
B4+
p
4
Now choose any standard cut-off function φ ∈ C ∞ satisfying
(4.23)
p
|A − AB + | p−1 dx ≤ δ p−1 .
0 ≤ φ ≤ 1, sptφ ⊂ B3 , and φ = 1 on B2 .
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5906
SUN-SIG BYUN AND LIHE WANG
Then without loss of generality we assume φp (u − v) ∈ C0∞ (B4+ ) by approximation.
Now according to Definition 4.1 we have
p−2
(A∇u · ∇u) 2 A∇u·∇ (φp (u − v)) dx =
|f |p−2 f ·∇ (φp (u − v)) dx,
(4.24)
B4+
B4+
and
(4.25)
B4+
p−2
2
AB + ∇v · ∇v
AB + ∇v · ∇ (φp (u − v)) dx = 0.
4
4
Subtracting the identity (4.25) from the identity (4.24) and operating basic computations we write the resulting expression as
I1 = I2 + I3 + I4 + I5 + I6
for
I1
I2
p−2
p−2
φp (A∇u · ∇u) 2 A∇u − (A∇v · ∇v) 2 A∇v · (∇u − ∇v) dx,
B4+
p−2
= −p
φp−1 (u − v) (A∇u · ∇u) 2 A∇u · ∇φdx,
=
I3
= p
I4
=
I5
B4+
B4+
=
B4+
I6
= p
B4+
φp−1 (u − v) (A∇v · ∇v)
p−2
2
A∇v · ∇φdx,
p−1
φ (u − v)|f |p−2 f · ∇φ + φp |f |p−2 f · ∇(u − v) dx,
p−2
p−2
2
AB + ∇v · φp ∇(u − v)dx,
(A∇v · ∇v) 2 A∇v − AB + ∇v · ∇v
4
4
p−2
p−2
2
2
A∇v− AB + ∇v·∇v
AB + ∇v ·φp−1 (u − v)∇φ.
(A∇v·∇v)
4
B4+
4
Estimate of I1 . We divide it into two cases.
Case 1. p ≥ 2. Using the elementary inequality
p−2
p−2
(Aξ · ξ) 2 Aξ − (Aη · η) 2 Aη · (ξ − η) ≥ C|ξ − η|p
for every ξ, η ∈ Rn , we have
I1 ≥ C
B4+
|φ∇(u − v)|p dx.
Case 2. 1 < p < 2. Using the elementary inequality
p−2 |ξ − η|p ≤ C(p)τ p |ξ|p−2 ξ − |η|p−2 η · (ξ − η) + τ |η|p
for every ξ, η ∈ Rn and for every τ ∈ (0, 1], we have
p
p
φ |∇v| dx ≥ C(τ )
|φ∇(u − v)|p dx.
I1 + τ
B4+
B4+
∞
Estimate of I2 . Since A ∈ L , we readily check from the uniform ellipticity
condition and Young’s inequality with τ that
I2 ≤ C
(|φ||∇u|)p−1 (|u − v||∇φ|)dx
B4+
≤ τ
|φ| |∇u| dx + C(τ )
p
Ω4
p
B4+
|u − v|p |∇φ|p dx.
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QUASILINEAR ELLIPTIC EQUATIONS IN LIPSCHITZ DOMAINS
5907
Estimate of I3 . Similarly to the estimate of I2 , we have
p
p
I3 ≤ τ
|φ| |∇v| dx + C(τ )
|u − v|p |∇φ|p dx.
B4+
B4+
Estimate of I4 . From Young’s inequality with τ we observe
I4 ≤ C
(|φ||f |)p−1 (|u − v||∇φ|) + (|φ||f |)p−1 (|φ||∇(u − v)|)dx
≤ τ
B4+
B4+
|∇φ|p |u − v|p dx
+C(τ )
B4+
|φ|p |f |p dx + τ
B4+
|φ∇(u − v)|p dx.
Estimate of I5 . Using the elementary inequality
|(Aξ · ξ)
p−2
2
· ξ)
Aξ − (Aξ
p−2
2
≤ C(p, Λ)|A − A||ξ|
p−1
Aξ|
for every ξ, η ∈ Rn and from Young’s inequality with τ , we have
p−1
I5 ≤ C
|A − A|(|φ||∇v|)
|φ∇(u − v)|dx
B4+
p
|φ∇(u − v)|p dx + C(τ )
|A − AB4 | p−1 |φ∇v|p dx
≤ τ
B4+
B4+
≤ τ
|φ∇(u − v)|p dx + C(τ )
B4+
p
B4+
|A − AB4 | p−1 dx,
the last inequality following from the interior W 1,∞ regularity for v and (4.23).
Estimate of I6 . Similarly to the estimate of I5 , we have
I6 ≤ C
|A − AB4 |(|φ||∇v|)p−1 |(u − v)∇φ|dx
B4+
≤ τ
|u − v| |∇φ| dx + C(τ )
p
Ω4
≤ τ
p
p
B4+
|A − AB4 | p−1 |φ∇v|p dx
B4+
|∇φ|p |(u − v)|p dx + C(τ )
B4+
p
|A − AB4 | p−1 dx.
Using (4.23) and combining all the estimates I1 to I6 , we have
C(Λ, τ )
|φ∇(u − v)|p dx
≤C ·τ
B4+
B4+
|φ∇(u − v)|p dx
(|∇u| + |∇v| ) dx + C(τ )
|u − v|p dx
B4+
p
p
|f | dx + C(τ )
|A − AB4 | p−1 dx.
+C(τ )
p
+τ
p
B4+
B4+
B4+
Using (4.23) and selecting a sufficiently small constant τ > 0, we find
p
p
p
p
|∇(u−v)| dx ≤ C
|∇(u − v)| dx +
|f | dx +
|A − AB4 | p−1 dx .
B2+
B4+
B4+
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B4+
5908
SUN-SIG BYUN AND LIHE WANG
Then (4.21)-(4.22) imply
p
|∇(u − v)|p dx ≤ C η p + δ p + δ p−1 ,
(4.26)
B2+
where η = η(δ) is to be selected later.
According to W 1,∞ interior regularity for v there exists a constant N0 such that
(4.27)
∇vpL∞ (B + ) ≤ N0p .
Denote by N1 by the constant
N1p
3
(4.28)
= max{2p+1 N0p , 2n }. We claim
{x ∈ B1+ : M(|∇u|p > N1p )} ⊂ {x ∈ B1+ : MB + (|∇(u − v|p ) > N0p }.
4
To see this, now suppose that
(4.29)
x1 ∈ x ∈ B1+ : MB + (|∇(u − v)|p )(x) ≤ N0p .
4
If r ≤ 2, Br+ (x1 ) ⊂ B3+ . Thus we observe from (4.29), (4.27) that
1
2p
(4.30)
|∇u|p dx ≤
(|∇(u − v)|p + |∇v|p ) dx ≤ 2p+1 N0p .
|Br | Br+ (x1 )
|Br | Br+ (x1 )
+
If r > 2, Br+ (x1 ) ⊂ B2r
(x0 ), and so (4.18) implies
1
2n
p
(4.31)
|∇u| dx ≤
|∇u|p dx ≤ 2n .
+
|Br | Br+ (x1 )
|B2r | B2r
(x0 )
Using (4.30) and (4.31) we conclude that
(4.32)
x1 ∈ x ∈ B1+ : M(|∇u|p )(x) ≤ N1p .
Assertion (4.28) comes from (4.30) and (4.31). We consequently can calculate from
(4.28), a weak (1,1) estimate (see Lemma 2.5) and (4.26)
x ∈ B + : M(|∇u|p ) > N p ≤ x ∈ B + : M + (|∇(u − v)|p ) > N p 1
0
1
1
B4
C
|∇(u − v)|p dx
≤
N0p B2+
p
1
p
p
p−1
.
≤
pC η +δ +δ
N0
Consequently we have
p
x ∈ B + : M(|∇u|p ) > N p ≤ C η p + δ p + δ p−1 = |B1 |,
1
1
provided that we select η = η(δ), δ satisfying the last identity above. This completes the proof.
We now come to state the scaling invariant form of the lemma above. We have
Corollary 4.4. There exists a constant N1 > 0 so that for any 0 < , r < 1, there
exists a small δ = δ() > 0 such that if u is a weak solution of ( 4.1) in B6+ , with A
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QUASILINEAR ELLIPTIC EQUATIONS IN LIPSCHITZ DOMAINS
5909
(δ, 6r)-vanishing and
Br+ ∩ {x : M(|∇u|p )(x) ≤ 1} ∩ {x : M(|f |p )(x) ≤ δ p } = ∅,
then we have
x ∈ B + : M(|∇u|p )(x) > N p ∩ Br < |Br |.
1
6
Lemma 4.5. There is a constant N1 > 0 so that for any 0 < , r ≤ 1, there exists
+
(0), with
a small δ = δ() > 0 such that if u is a weak solution of ( 4.1) in B42
(δ, 42)-vanishing and
|{x ∈ B1+ (0) : M(|∇u|p )(x) > N1p } ∩ Br | ≥ |Br |,
(4.33)
then we have
(4.34) Br ∩ B1+ (0) ⊂ {x ∈ B1+ (0) : M(|∇u|p ) > 1} ∪ {x ∈ B1+ (0) : M(|f |p ) > δ p },
where Br denotes the ball with radius r and center in B1+ (0).
Proof. We argue by contradiction. If Br satisfies (4.33) and the conclusion (4.34)
is false, then there exists x0 ∈ Br ∩ B1+ (0) such that
1
1
p
|∇u| dx ≤ 1,
|f |p dx ≤ δ p
|Bρ | Bρ (x0 )
|Bρ | Bρ (x0 )
for all ρ > 0. If B6r ∩ {xn = 0} = ∅, this is an interior estimate (see Lemma 3.3).
So suppose that (x , 0) ∈ B6r ∩ {xn = 0}. Now observe
+
+
B6r
⊂ B7r
(x , 0).
+
(x , 0), with
Applying Corollary 4.4 to the ball B7r
7n
replacing we obtain
|{x ∈ B1+ : M(|∇u|p )(x) > N1p } ∩ Br |
+
≤ |{x ∈ B1+ : M(|∇u|p )(x) > N1p } ∩ B7r
(x , 0)|
+
< n |B7r
|
7
= |Br+ |.
Then we reach a contradiction to (4.33).
Now take N1 , , and the corresponding δ > 0 given by the lemma above and set
1 = 2(10)n .
+
Corollary 4.6. Let u be a weak solution of ( 4.1) in B42
and k be a positive integer.
Assume that A is (δ, 42)-vanishing. Assume further that
x ∈ B + : M(|∇u|p ) > N p < |B1 |.
1
1
Then we have
p k x ∈ B1+ : M(|∇u|p ) > (N1 ) ≤
k
k−i i1 {x ∈ B1+ : M(|f |p ) > δ p (N1p ) } + k1 x ∈ B1+ : M(|∇u|p ) > 1 .
i=1
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5910
SUN-SIG BYUN AND LIHE WANG
Proof. We want to prove this lemma by induction on k. The case k = 1 follows
from Lemma 4.5 and Lemma 2.6 on
C =
x ∈ B1+ : M(|∇u|p ) > N1p ,
D =
x ∈ B1+ : M(|f |p ) > δ p ∪ x ∈ B1+ : M(|∇u|p ) > 1 .
Suppose then that the conclusion is valid for some positive integer k ≥ 2. Set
u1 = Nu1 and f1 = Nf1 . Thus u1 is the weak solution of
p−2
+
in B42
,
= div |f1 |p−2 f1
div (A∇u1 · ∇u1 ) 2 A∇u1
on
T42
u1 = 0
and
x ∈ B + : M(|∇u1 |p )(x) > N p < |B1 |.
1
1
Then by the induction assumption
(k+1)p
|{x ∈ B1+ : M(|∇u|p ) > N1
= |{x ∈
≤
k
B1+
: M(|∇u1 | ) >
p
}|
N1kp }|
(k−i)p
i1 |{x ∈ B1+ : M(|f1 |p ) > δ p N1
i=1
+k1 |{x
k+1
}|
∈ B1+ : M(|∇u1 |p ) > 1}|
(k+1−i)p
i1 |{x ∈ B1+ : M(|f |p ) > δ p N1
=
i=1
+k+1
|{x
1
}|
∈ B1+ : M(|∇u|p ) > 1}|.
These estimates in turn complete the induction on k.
Finally, in view of Corollary 4.6 we have the following boundary estimates.
Theorem 4.7. Let q be a real number with q > p. There is a small δ = δ(p, n, Λ) >
+
, with A uniformly elliptic and (δ, 42)0 so that if u is a weak solution of ( 4.1) in B42
+
q
n
vanishing and f ∈ L (B42 , R ), then u belongs to W 1,q (B1+ ) with the estimate
q
|∇u| dx ≤ C
(|u|q + |f |q )dx,
B1+
B6+
where the constant C is independent of u and f .
Proof. According to standard arguments of measure theory (See Lemma 2.7), there
exists a constant C = C(δ, N1p , q) such that
(4.35)
∞
q
kq
k
(N1p ) p |{x ∈ B6+ : M(|f |p ) > δ p (N1p ) }| ≤ CM(|f |p ) p q + ≤ Cf qLq (B + ) .
L p (B6 )
k=1
6
The last estimate follows from the Lq -estimate of the Hardy-Littlewood maximal
function. Now we may with no loss suppose
(4.36)
|{x ∈ Ω : M(|∇u|p ) > N1p } ∩ B1+ | < |B1 |.
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QUASILINEAR ELLIPTIC EQUATIONS IN LIPSCHITZ DOMAINS
5911
Then from Corollary 4.6 and (4.36) we have
∞
q
kp
(N1p )
k=1
≤
∞
k
|{x ∈ B1+ : M(|∇u|p ) > (N1p ) }|
k
N1qk
i=1
k=1
+k1 |{x
=
+
∞
p(k−i)
i1 |{x ∈ B1+ : M(|f |p ) > δ p N1
∈
B1+
}|
: M(|∇u| ) > 1}|
p
i
(N1q 1 )
i=1
∞
∞
q
(k−i) p
(N1p )
|{x ∈
B1+
: M(|f | ) > δ
p
p
p(k−i)
N1
}|
k=i
(N1q 1 )k |{x ∈ B6+ : M(|∇u|p ) > 1}|
k=1
≤ Cf qLq (B + )
6
∞
k
(N1q 1 )
k=1
≤ Cf qLq (B + ) ,
6
provided > 0 is selected small enough to have
(N1q 1 ) = (N1q ) 2(10)n < 1.
Then
∞
q
kp
(N1p )
k
|{x ∈ B1+ : M(|∇u|p ) > (N1p ) }| ≤ Cf qLq (B + ) .
6
k=1
Thus we have from Lemma 2.5 that
q
M(|∇u|p ) ∈ L p (B1+ )
and
∇u ∈ Lq (B1+ ; Rn )
with the estimate
∇uLq (B + ) ≤ uLq (B + ) + f Lq (B + ) ,
1
6
6
and we are done.
5. Flattening argument
In the general case we choose any point x0 ∈ ∂Ω. We may assume that
Ω ∩ Br (x0 ) = {x ∈ Br (x0 ) : xn > γ(x )}
for some constant r > 0 and some Lipschitz continuous function
γ : Rn−1 → R
with Lip(γ) small, where Lip(γ) denotes the Lipschitz constant of γ.
Now define
yi = xi = Φi (x)(1 ≤ i ≤ n − 1) and yn = xn − γ(x ) = Φn (x)
and write
y = Φ(x) and x = Φ−1 (y) = Ψ(y).
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5912
SUN-SIG BYUN AND LIHE WANG
Choose s > 0 so small that Bs+ lies in Φ(Ω ∩ Br (x0 )) and define
u1 (y) = u(Ψ(y))
for all y ∈
If u is a weak solution of
p−2
div (A∇u · ∇u) 2 A∇u
= div |f |p−2 f
(5.1)
u = 0
Bs+ .
in
on
Ω ∩ Br (x0 ),
∂Ω ∩ Br (x0 ),
then u1 is a weak solution of
(5.2)
p−2
= div |f1 |p−2 f1
div (A1 ∇u1 · ∇u1 ) 2 A1 ∇u1
u1 = 0
in
on
Bs+ (Φ(x0 )) ,
Ts (Φ(x0 )) .
Here
(5.3)
A1 (y) = [∇Φ]T (Ψ(y) · A(Ψ(y)) · [∇Φ] (Ψ(y))
and
(5.4)
T
f1 (y) = [∇Φ] (Ψ(y)) · f (Ψ(y)).
Then it is straightforward to check that
(5.5)
[A1 ]BM O ≤ C [A]BM O + ∇γL∞ (Rn−1 ) ≤ C ([A]BM O + Lip(γ)) ,
where [A1 ]BM O denotes the BMO seminorm of A1 and [A]BM O denotes the BMO
seminorm of A.
Now we remark that γ in the above is Lipschitz continuous with small Lipschitz
constant if and only if it is in W 1,∞ with small ∇γL∞ (see Theorem 4 of chapter
5 in [5]). Recalling the assumptions announced in the Introduction that Ω is (δ, R)Lipschitz and that A is (δ, R)-vanishing, it follows easily from (5.5) and (5.3) that A1
is (δ, R)-vanishing and that A1 is uniformly elliptic. Hence we can find a boundary
estimate for the case that Ω is (δ, R)-Lipschitz and A is (δ, R)-vanishing via the
approached used in Section 4.
We are finally set to give our proof of Theorem 1.2.
Proof. Once we established the boundary Lq (q > p) estimates for the gradient of
u in B1+ in Theorem 4.7 we can get the proof by standard scaling, covering and
flattening arguments along with the interior estimate and a duality argument. Acknowledgments
The authors wish to thank Professor Shulin Zhou for some very helpful conversations on this work.
References
1. S. Byun, Elliptic equations with BMO coefficients in Lipschitz domains, Trans. Amer. Math.
Soc., 357 (2005), 1025–1046. MR2110431 (2005i:35054)
2. L.A. Caffarelli and X. Cabré, Fully nonlinear elliptic equations, vol. 43, Amer. Math. Soc.,
Providence, RI, 1995. MR1351007 (96h:35046)
3. L.A. Caffarelli and I. Peral, On W 1,p estimates for elliptic equations in divergence form,
Comm. Pure Appl. Math., 51, (1998), 1–21. MR1486629 (99c:35053)
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
QUASILINEAR ELLIPTIC EQUATIONS IN LIPSCHITZ DOMAINS
5913
4. E. DiBenedetto and J. Manfredi, On the higher integrability of the gradient of weak solutions
of certain degenerate elliptic systems, Amer. J. Math., 115 (5), 1993, 1107–1134. MR1246185
(94i:35077)
5. L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, (1998), xviii+662 pp. ISBN: 0-8218-0772-2.
MR1625845 (99e:35001)
6. T. Iwaniec, Projections onto gradient fields and Lp -estimates for degenerate elliptic operators,
Studia Math., 75, 1983, 293–312. MR0722254 (85i:46037)
7. D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J.
Funct. Anal., 130, (1995), 161–219. MR1331981 (96b:35042)
8. D. Jerison and C. Kenig, The logarithm of the Poisson kernel of a C 1 domain has vanishing
mean oscillation, Trans. Amer. Math. Soc. 273, (1982), 781–794. MR0667174 (83k:31004)
9. D. Jerison and C. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math.
Soc. (N.S.), 4, (1981), 203–207. MR0598688 (84a:35064)
10. J. Kinnunen and S. Zhou, A Local estimate for nonlinear equations with discontinuous
coefficients, Comm. partial differential equations, 24, (1999), 2043–2068. MR1720770
(2000k:35084)
11. J. Kinnunen and S. Zhou, A boundary estimate for nonlinear equations with discontinuous coefficients, Differential and integral equations, 14 (2001), 475–492. MR1799417 (2002a:35070)
12. P.W. Jones, Extension theorems for BMO, Indiana Univ. Math. J., 29, (1980), 41–66.
MR0554817 (81b:42047)
13. E. M. Stein, Harmonic Analysis, Princeton University Press, Princeton, NJ, 1993. MR1232192
(95c:42002)
14. L. Wang, A geometric approach to the Calderón-Zygmund estimates, Acta Mathematica
Sinica, 19, (2003), 381–396. MR1987802 (2004e:42033)
Department of Mathematical Sciences, Seoul National University, Seoul 151-747,
Korea
E-mail address: [email protected]
Department of Mathematics, University of Iowa, Iowa City, Iowa 52242 – and – College of Sciences, Xian Jiaotong University, Xian 710049, People’s Republic of China
E-mail address: [email protected]
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