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Weighted gradient estimates for the parabolic p -Laplacian equations
Nonlinear Analysis 112 (2015) 58–68
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Weighted gradient estimates for the parabolic p-Laplacian equations Fengping Yao Department of Mathematics, Shanghai University, Shanghai 200444, China
article
info
abstract
Article history: Received 25 July 2014 Accepted 9 September 2014 Communicated by Enzo Mitidieri
In this paper we obtain the weighted Lq estimates of the gradients of weak solutions for the quasilinear parabolic equations of p-Laplacian type with small BMO coefficients. For every ε ∈ (0, q − 1) if |∇ u|p ∈ L1w,+εloc (ΩT ) and |f|p ∈ Lqw,loc (ΩT ), then we prove that
|∇ u|p ∈ Lqw,loc (ΩT ).
MSC: 35K55 35K65
Our results improve the known results for such equations using a harmonic analysis free technique. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Lq estimates Weighted Regularity Gradient Divergence Parabolic p-Laplacian
1. Introduction Many authors [6,8,7,10–12,16,19,20] have studied the regularity estimates for weak solutions of the following quasilinear elliptic equations of p-Laplacian type div (A∇ u · ∇ u)
p−2 2
A∇ u = div(|f|p−2 f)
in Ω
and the general case. Recently, Byun and Ryu [5] obtained the global weighted Lq estimates with q ∈ (p, +∞) for the gradient of weak solutions of the general nonlinear elliptic equations of p-Laplacian type. In this paper we are concerned with the following quasilinear parabolic equation of p-Laplacian type ut − div (A∇ u · ∇ u)(p−2)/2 A∇ u = div(|f|p−2 f) in ΩT = Ω × (0, T ]
(1.1)
for 1 ≤ 2n/(n + 2) < p < ∞ and n ≥ 2, where Ω is an open bounded domain in R and ∂p ΩT = ∂ Ω × [0, T ] ∪ Ω × {t = 0} is the parabolic boundary of ΩT . Moreover, f = (f 1 , . . . , f n ) is a given vector field and A = {aij (x)}n×n is a symmetric matrix with measurable coefficients satisfying the uniformly parabolic condition n
Λ−1 |ξ |2 ≤ A(x, t )ξ · ξ ≤ Λ|ξ |2
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.na.2014.09.010 0362-546X/© 2014 Elsevier Ltd. All rights reserved.
(1.2)
F. Yao / Nonlinear Analysis 112 (2015) 58–68
59
for all ξ ∈ Rn and almost every (x, t ) ∈ Rn × R, where Λ is a positive constant. We remark that the lower bound 2n/(n + 2) on the exponent p is standard and unavoidable in the theory of the parabolic p-Laplacian operator. Differently from the elliptic case, the quasilinear parabolic equation of p-Laplacian type is not homogeneous even if f ≡ 0, which is the main difficulty. Kinnunen and Lewis [15] obtained a reverse Hölder inequality of the gradient for the weak solutions of (1.1), which implies the local higher integrability of the gradient. In [23] Misawa proved that |Du|p ∈ Lq , for every q ≥ 1 if f ∈ L∞ for the weak solutions of (1.1). Furthermore, Acerbi and Mingione [1] obtained the following estimates in Sobolev spaces
|F |p ∈ Lqloc (ΩT ) ⇒ |∇ u|p ∈ Lqloc (ΩT ) for any q ≥ 1 for the weak solutions of the following parabolic p-Laplacian systems ut − div a(x, t )|∇ u|p−2 ∇ u = div |F |p−2 F
in ΩT
(1.3)
with small BMO coefficients. Recently, some authors [2,3] studied the corresponding estimates for weak solutions of the evolutionary p(x, t )-Laplacian system. Let Q (θ , ρ) = Bρ × (−θ , θ] be a centered parabolic cylinder. Especially when θ = ρ 2 = r 2 , we shall simply write Qr ≡ Q (r 2 , r ). Throughout this paper we assume that the coefficients of A = {aij } are in parabolic BMO spaces and their parabolic semi-norm are small enough. More precisely, we have the following definition. Definition 1.1 (Small BMO Semi-Norm Condition). We say that the matrix A of coefficients is (δ, R)-vanishing if
sup −
Qz (θ,ρ) Qz (θ,ρ)
|A(ζ ) − AQz (θ,ρ) |dζ ≤ δ,
and the supremum is taken among all cylinders of the type Qz (θ , ρ) with ρ ≤ R and θ ≤ R2 , where z = (x, t ), ζ = (y, s) ∈ Rn × R. As usual, the solutions of (1.1) are taken in a weak sense. We now state the definition of weak solutions. p
p
p
2 Definition 1.2. Assume that f ∈ Lloc (ΩT ). A function u ∈ Lloc (ΩT ) ∩ L∞ loc (0, T ; Lloc (Ω )) with |∇ u| ∈ Lloc (ΩT ) is a local weak solution of (1.1) in ΩT if for any compact subset K of Ω and for any subinterval [t1 , t2 ] of (0, T ] we have
t2
K
uϕ dx +
t2
t1
t1
−uϕt + (A∇ u · ∇ u)
p−2 2
t2 t1
K
|f|p−2 f · ∇ϕ dxdt
A∇ u · ∇ϕ dxdt = − K
for any ϕ ∈ C0∞ (K × [t1 , t2 ]). Here for convenience of the readers, we shall give some definitions and properties on the weighted Lebesgue spaces (see [4,14,17,18,21,22,24–27]). Definition 1.3. Aq for some q > 1 is the class of the reverse Hölder weights: w ∈ Aq if w ∈ L1loc (Rn+1 ), w > 0 almost everywhere and there exists a constant C such that for all balls Qr = Br × (−r 2 , r 2 ] of Rn+1 ,
q −1 −1 − w(z ) dz − w(z ) q−1 dz ≤ C. Qr
Qr
Moreover, we denote A∞ =
Aq
1
and
w(Qr ) =
w(z ) dz .
Qr
Furthermore, the corresponding weighted Lebesgue space Lpw (Qr ) for any p ≥ 1 consists of all functions h which satisfy
∥h∥Lpw (Qr ) =:
|h| w(z ) dz p
1/p
< ∞.
Qr
Remark 1.4. We remark that Aq1 ⊂ Aq2 for any 1 < q1 ≤ q2 < ∞ (see [26, p. 195]). Lemma 1.5. If w ∈ Aq with q > q1 > 1, then we have Lqw (Qr ) ⊂ Lq1 (Qr ).
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F. Yao / Nonlinear Analysis 112 (2015) 58–68
Proof. Using Hölder’s inequality, we have
|f |q1 dz =
q1
q1
|f |q1 w(z ) q w(z )− q dz Qr
Qr
|f | w(z ) dz q
≤
qq1
Qr
w(z )
1− qq1
q
− q−1q
1
dz
.
Qr
Moreover, we find that w ∈ Aq/q1 in view of Remark 1.4 and the fact that w ∈ Aq with q > q1 > 1. Furthermore, from Definition 1.3 we conclude that
q1 q |Qr | q−q1 − q/q1 −1 − q−1q 1 1 − w(z ) dz ≤ C dz = − w(z ) . w(Qr ) Qr Qr Thus, if f ∈ Lqw (Qr ), then we have
q1
|f | w(z ) dz q
|f | dz ≤ C
qq1
q 1− q1
|Qr |
Qr
Qr
|Qr | w(Qr )
qq1
≤ C,
since w ∈ L1loc (Rn ) and w > 0 almost everywhere. This finishes our proof.
Lemma 1.6. Assume that w ∈ Aq for some q > 1 and g ∈ Lqw (Rn+1 ). Then for any q > β > 1 we have (1)
w
z ∈ Rn+1 : M g (z ) > µ
≤
C
µq
Rn+1
|g (z )|q w(z ) dz for any µ > 0.
(2)
Rn+1
|g (z )|q w(z ) dz = q
∞
λq−1 w z ∈ Rn+1 : |g (z )| > µ dµ 0 ∞ = (q − β) λq−β−1 |g |β w(z ) dz dµ. |g |>µ
0
Next, we shall give some lemmas on the properties of Aq weight. Lemma 1.7 ([4,27]). If w ∈ Aq for some q > 1 and Qr ⊂ QR ⊂ Rn+1 , then there exist α > 0 and a positive constant C1 > 0 such that
1 C1
|Qr | |QR |
q ≤
w(Qr ) ≤ C1 w(QR )
|Qr | |QR |
α
.
Now let us state the main result of this work. Theorem 1.8. Assume that w(z ) ∈ Aq for some q > 1. Let u be a local weak solution of (1.1). Then for every ε ∈ (0, q − 1) there exists a small δ = δ(n, p, q, ε, Λ, R) > 0 so that for each uniformly parabolic and (δ, R)-vanishing A, and for all f with |f|p ∈ Lqw,loc (ΩT ), if |∇ u|p ∈ L1w,+εloc (ΩT ), then we have |∇ u|p ∈ Lqw,loc (ΩT ) with the estimate
|∇ u| w(z ) dz pq
1q
|∇ u|
≤C
Q1
p(1+ε)
w(z ) dz
d 1+ε
|f| w(z ) dz pq
+
Q2
dq
+1 ,
Q2
where Q2 ⊂ ΩT , the constant C is independent of u and f, and
1 ≤ d :=
p p − (p − 2)q
for p ≥ 2,
1
2p 2p − (2 − p)nq1
for
(1.4)
2n n+2
< p < 2.
Here
p − 1 p − 2 1 < q1 = 2n + (2 − n)p 2n(2 − p) q,
for p > 2 and q > for
2n
n+2 elsewhere.
p p−2
,
< p < 2 and q >
2p
(2 − p)n
,
(1.5)
F. Yao / Nonlinear Analysis 112 (2015) 58–68
61
Remark 1.9. Assume that 2n/(n + 2) < p < 2. We can check that 2p
(2 − p)n
>1
and 2n + (2 − n)p 2n(2 − p)
2p
=
1 + (2−p)n > 1. 2
2. Proof of the main result To start with, let u be a local weak solution of problem (1.1) in ΩT with Q2 ⊂ ΩT and A be uniformly parabolic. We write 1
λ0d =
1
w (Q2 )
|∇ u|p1 w(z ) dz
p1
1
1
+
δ
Q2
1
w (Q2 )
|f|p1 w(z ) dz
p1
1
+ 1,
(2.1)
.
(2.2)
Q2
where p1 = p(1 + ε) ∈ (p, pq) with ε ∈ (0, q − 1), d is defined in (1.4) and E (λ) = {z ∈ Q1 : |∇ u| > λ} for λ > 0. Moreover, for any domain Q in Rn+1 we write
J [Q ] =:
1
w (Q )
|∇ u| w(z ) dz p1
p1
1
+
1
Q
δ
1
w (Q )
|f| w(z ) dz p1
p1
1
Q
(n+2)dq1
Lemma 2.1. Given λ ≥ λ∗ =: C1d 20 p λ0 , where d is defined in (1.4), there exists a family of disjoint cylinders {Qi0 }i∈N with zi = (xi , ti ) ∈ E (λ), 0 < ρi = ρ(zi ) ≤ 1/10 and
Qi0
Qzi (λ2−p ρi2 , ρi )
=
for p ≥ 2,
Qzi ρi2 , λ(p−2)/2 ρi
for 2n/(n + 2) < p < 2,
such that J Qi0 = λ,
J Qzi (θ , ρ) < λ for any Qzi (θ , ρ) ⊃ Qi0 ,
(2.3)
and E (λ) ⊂
Qi1 ∪ negligible set ,
i∈N
where
j Qi
=:
5jQi0
Qzi (5j)2 λ2−p ρi2 , 5jρi
=
(p−2)/2
Qzi (5j) ρ , 5jλ
2
2 i
for p ≥ 2,
ρi
for 2n/(n + 2) < p < 2,
(2.4)
for j = 1, 2. Proof. Case 1: p ≥ 2. We first claim that sup
J Qw λ2−p ρ 2 , ρ
sup
w∈Q1 1/10≤ρ≤1
≤ λ.
(2.5)
To prove this, fix any w ∈ Q1 and 1/10 ≤ ρ ≤ 1. Then from Lemma 1.7 and the fact that p1 ∈ (p, pq) we find that
1
w Qw λ2−p ρ 2 , ρ
≤
p1
|∇ u| w(z ) dz
1
p1
(
Qw λ2−p ρ 2 ,ρ
)
w (Q2 ) 1 · w (Q2 ) w Qw λ2−p ρ 2 , ρ
p1
|∇ u| w(z ) dz p1
Q2
1
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F. Yao / Nonlinear Analysis 112 (2015) 58–68
≤ C1
≤ C1 λ
q1
2n+2
1 p1
λ2−p ρ n+2
1− 2p q1
20
1
w (Q2 )
(n+2)q1
p
1
w (Q2 )
|∇ u| w(z ) dz p1
p1
1
Q2
|∇ u|p1 w(z ) dz
p1
1
,
Q2
where q1 ∈ (1, q] is defined in (1.5). Here we have used the fact that w ∈ Aq1 since w ∈ Aq and Remark 1.4. Similarly, we have
p1
1
w Qw λ2−p ρ 2 , ρ
1
|f| w(z ) dz p1
(
Qw λ2−p ρ 2 ,ρ
)
≤ C1 λ
1− 2p q1
20
(n+2)q1 p
1
w (Q2 )
|f|p1 w(z ) dz
p1
1
.
Q2
Consequently it follows from the two inequalities above and the definition of λ0 that J Qw λ2−p ρ 2 , ρ
≤ C1 λ = C1 λ
(p−2)q1 p
(p−2)q1 p
20 20
(n+2)q1 p
(n+2)q1 p
1
λ0d
1−
λ0
(p−2)q1 p
for any w ∈ Q1 , 1/10 ≤ ρ ≤ 1 and λ ≥ λ∗ =: C1d 20 2. We first conclude that
1−
= λ∗
(n+2)dq1 p
(p−2)q1 p
λ
(p−2)q1 p
≤λ
λ0 , where d is defined in (1.4), which implies that (2.5) holds.
lim J Qw (λ2−p ρ 2 , ρ) > λ
ρ→0
for a.e. w ∈ E (λ), which implies that there exists some ρ ∈ (0, 1] satisfying J Qw (λ2−p ρ 2 , ρ) > λ.
Furthermore, from (2.5) we can select a radius ρw ∈ (0, 1/10] such that
ρw =: max ρ | J [Qw (λ2−p ρ 2 , ρ)] = λ, 0 < ρ ≤ 1/10 . Thus, we observe that 2 J Qw (λ2−p ρw , ρw ) = λ and
J Qw (λ2−p ρ 2 , ρ) < λ for ρw < ρ ≤ 1.
Therefore, applying Vitali’s covering lemma, we can find a family of disjoint cylinders {Qi0 }i∈N = {Qzi (λ2−p ρi2 , ρi )}i∈N , zi = (xi , ti ) ∈ E (λ) so that the results of the lemma is true for p ≥ 2. Case 2: 2n/(n + 2) < p < 2. Let λ ≥ λ∗ = C1d 20 claim that sup
J Qw ρ 2 , λ(p−2)/2 ρ
sup
w∈Q1 1/10≤ρ≤1
(n+2)dq1 p
λ0 , where d is defined in (1.4). Then similarly to (2.5), we can
≤ λ.
(2.6)
To prove this, fix any w ∈ Q1 and 1/10 ≤ ρ ≤ 1. Then
p1
1
1
|∇ u| w(z ) dz Qw ρ 2 , λ(p−2)/2 ρ Qw ρ 2 ,λ(p−2)/2 ρ p1 1 1 w (Q2 ) p1 · |∇ u| w(z ) dz ≤ 2 ( p − 2 )/ 2 w (Q2 ) Q2 w Qw ρ , λ ρ 1 q p 1 2n+2
≤ C1 ≤ C1 λ
1
1
λ(p−2)/2 ρ n+2 (2−p)nq1 2p
We remark that
(2 − p)n 2p
p1
< 1,
20
(n+2)q1 p
1
w (Q2 )
1
|∇ u|p1 w(z ) dz
w (Q2 ) Q2 p1 1 p1 . |∇ u| w(z ) dz Q2
p1
F. Yao / Nonlinear Analysis 112 (2015) 58–68
63
since p > 2n/(n + 2). Similarly, we have
p1
1
w Qw ρ 2 , λ(p−2)/2 ρ
1
|f| w(z ) dz
≤ C1 λ
p1
Qw
(
ρ 2 ,λ(p−2)/2 ρ
)
(2−p)nq1
20
2p
(n+2)q1
p
1
w (Q2 )
|f| w(z ) dz p1
p1
1
.
Q2
Consequently, it follows from the two inequalities above and the definition of λ0 that J Qw ρ 2 , λ(p−2)/2 ρ
≤ C1 λ = C1 λ
(2−p)nq1 2p
(2−p)nq1 2p
20 20
(n+2)q1 p
(n+2)q1 p
1
λ0d
2p−(2−p)nq1 2p
λ0
for any w ∈ Q1 , 1/10 ≤ ρ ≤ 1 and λ ≥ λ∗ =: C1d 20 similarly to that of Case 1.
(n+2)dq1 p
1−
= λ∗
(2−p)nq1 2p
λ
(2−p)nq1
≤λ
2p
λ0 , which implies that (2.6) holds. So we can complete the proof
Corollary 2.2. Under the same hypothesis and results as those in Lemma 2.1, we have
w
Qi0
≤C
1
λp1
|∇ u| w(z ) dz + p1
z ∈Qi0 :|∇ u|> λ 4
1
λp1 δ p1
|f| w(z ) dz , p1
z ∈Qi0 :|f|> δλ 4
pq
where C = 1−221−p . Proof. From (2.3) we find that
J
Qi0
=
p1
1
w Qi
0
1
|∇ u| w(z ) dz p1
Qi0
+
1
δ
w Qi
p1
1
0
|f| w(z ) dz
1
p1
Qi0
= λ,
which implies that
w
Qi0
2p1
≤
λp1
|∇ u| w(z ) dz + p1
Qi0
2p1
λp1 δ p1
|f|p1 w(z ) dz ,
Qi0
(2.7)
since either of the following inequalities must be true
λ 2
≤
w
p1
1 Qi0
Qi0
|∇ u|p1 w(z ) dz
1
or
λ 2
≤
1
δ
w
p1
1
Qi0
|f|p1 w(z ) dz
Qi0
1
.
Therefore, by splitting the right-side two integrals in (2.7) as follows we have
2p1 w Qi0 ≤ p λ1
z ∈Qi0 :|∇ u|> λ 4
p1 |∇ u| w(z ) dz +
w Qi0 2p1
+
2p1
λp1 δ p1
Thus we have concluded with the desired estimate since p1 ∈ (p, pq).
z ∈Qi0 :|f|> δλ 4
p1 |f| w(z ) dz +
w Qi0 2p1
.
In view of Lemma 2.1, given λ ≥ λ∗ , we can construct the family of cylinders {Qi0 }i∈N , where zi ∈ E (λ). Fix any i ∈ N. Then it follows from Lemma 2.1 that
1
w Qij
p1
1
|∇ u| w(z ) dz p1
j
≤ λ and
Qi
1
p1
1
w Qij
|f| w(z ) dz p1
j
≤ δλ
(2.8)
Qi
j
for j = 1, 2, where Qi is defined in Lemma 2.1. Corollary 2.3. Assume that 1 < p < p2 < p1 < pq. Under the same hypothesis and results as those in Lemma 2.1, we have
p1 2 p2 − |∇ u| dz ≤ C λ and j
Qi
for j = 1, 2.
p1 2 p2 − |f| dz ≤ C δλ, j
Qi
(2.9)
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F. Yao / Nonlinear Analysis 112 (2015) 58–68
Proof. Using Hölder’s inequality and (2.8), we have
p1 p1 2 2 p2 p2 − − |∇ u|p2 dz = − |∇ u|p2 w(z ) p1 w(z ) p1 dz j
j
Qi
Qi
p11 pp1 −pp2 w Qij 1 2 p2 1 − |∇ u|p1 w(z ) dz − w(z ) p1 −p2 dz ≤ j j j j Qi Qi Qi w Qi
pp1 −1 p11 2 − p11 −1 , ≤ λ − w(z ) dz · − w(z ) p2 dz j
j
Qi
Qi
which implies that
p1 2 − |∇ u|p2 dz ≤ C λ, j
Qi
in view of Definition 1.3 and w ∈ Ap1 /p2 since w ∈ Aq and 1 < p1 /p2 < q. Similarly, we have
p1 2 p2 − |f| dz ≤ C δλ. j
Qi
Thus we finish the proof.
Furthermore, we can obtain the following result. Lemma 2.4. For any ϵ > 0, there exists a small δ = δ(ϵ) > 0 such that if u is a local weak solution of (1.1) in ΩT with Q2 ⊂ ΩT ,
− |A − AQ1 |dz ≤ δ,
(2.10)
Q1
p1 2 p2 − |∇ u| dz ≤ C and
p1 2 p2 − |f| dz ≤ C δ,
(2.11)
Q1
Q1
then there exists N0 > 1 such that sup |∇v| ≤ N0
(2.12)
Q1/2
and
− |∇(u − v)|p dz ≤ ϵ,
(2.13)
Q1
where v is the weak solution of
vt − div (AQ1 ∇v · ∇v)(p−2)/2 AQ1 ∇v = 0 v=u
in Q1 , on ∂p Q1 .
(2.14)
Proof. If u and v are the weak solutions of (1.1) and (2.14) respectively, then by choosing the test function ϕ = u − v , after a direct calculation we show the resulting expression as I1 + I2 = I3 + I4 , where I1 =
I2 =
d dt
|v − u|2
Q1
Q1
2
|v(x, 1) − u(x, 1)|2
dz = B1
2
dz ,
(AQ1 ∇v · ∇v)(p−2)/2 AQ1 ∇v − (AQ1 ∇ u · ∇ u)(p−2)/2 AQ1 ∇ u · ∇(v − u)dz ,
F. Yao / Nonlinear Analysis 112 (2015) 58–68
65
(A∇ u · ∇ u)(p−2)/2 A∇ u − (AQ1 ∇ u · ∇ u)(p−2)/2 AQ1 ∇ u · ∇(v − u)dz , I3 = Q1
|f|p−2 f · ∇(v − u)dz .
I4 = Q1
Actually, (2.12) can follow from (2.11), (2.13), Hölder’s inequality and the interior regularity of weak solutions for degenerate parabolic equations (see [9, Chapter 8, Theorems 5.1 and 5.2]). Estimate of I2 . We divide into two cases. Case 1. p ≥ 2. Using the elementary inequality
(AQ1 ξ · ξ )(p−2)/2 AQ1 ξ − (AQ1 η · η)(p−2)/2 AQ1 η · (ξ − η) ≥ C |ξ − η|p
for every ξ , η ∈ Rn with C = C (p, Λ), we have
|∇(u − v)|p dz .
I2 ≥ C Q1
Case 2. 2n/(n + 2) < p < 2. Using the elementary inequality
|ξ − η|p ≤ C τ (p−2)/p (AQ1 ξ · ξ )(p−2)/2 AQ1 ξ − (AQ1 η · η)(p−2)/2 AQ1 η · (ξ − η) + τ |η|p for every ξ , η ∈ Rn and every τ ∈ (0, 1) with C = C (p, Λ), we have I2 + τ
|∇ u|p dz ≥ C (τ ) Q1
|∇(u − v)|p dz . Q1
Estimate of I3 . Using the elementary inequality
(Aξ · ξ )(p−2)/2 Aξ − (AQ ξ · ξ )(p−2)/2 AQ ξ ≤ C |A − AQ | |ξ |p−1 1 1 1 for every ξ , η ∈ Rn with C = C (p, Λ), and then using Young’s inequality with τ and Hölder’s inequality, from (2.11) we have
|A − AQ1 | |∇ u|p−1 |∇(u − v)|dz p ≤ C (τ ) |A − AQ1 | p−1 |∇ u|p dz + τ |∇(u − v)|p dz
I3 ≤ C
Q1
Q1
≤ C (τ )
Q1
p2p−p
pp2 (p−1)(p2 −p)
2
|A − AQ1 | dz Q1 p2 −p |∇(u − v)|p dz , ≤ C (τ )δ p2 + τ
|∇ u|p2 dz
pp
2
+τ
Q1
|∇(u − v)|p dz Q1
Q1
since
pp2
|A − AQ1 |
(p−1)(p2 −p) dz
p2p−p 2
≤ (2Λ)
p2 +p2 −p p2 (p−1)
p2p−p
Q1
|A − AQ1 |dz
2
≤ Cδ
p2 −p p2
Q1
as a consequence of (1.2) and (2.10). Estimate of I4 . Using Young’s inequality with τ , Hölder’s inequality and (2.11), we have I4 ≤ τ
|∇(u − v)|p dz + C (τ ) Q1
≤τ
|∇(u − v)|p dz + C (τ )
|f|p dz Q1
Q1
≤τ
|f|p2 dz
pp
2
Q1
|∇(u − v)|p dz + C (τ )δ p . Q1
Combining all the estimates of Ii (1 ≤ i ≤ 4), we obtain C (τ )
|∇(u − v)| dz ≤ 2τ p
Q1
|∇(u − v)| dz + τ p
Q1
|∇ u| dz + C δ p
Q1
p2 −p p2
+δ
p
.
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F. Yao / Nonlinear Analysis 112 (2015) 58–68
Selecting a small constant τ > 0 such that 0 < τ ≪ δ < 1, and then using Hölder’s inequality and (2.11), we conclude that
p2 −p p p p2 − |∇(u − v)| dz ≤ C δ + δ +δ =ϵ Q1
by selecting δ satisfying the last inequality above. This completes our proof.
Let δ in (2.1) and Definition 1.1 be the same as that in Lemma 2.4. We may as well assume that R = 2 by a scaling in Definition 1.1. Therefore
− A − AQ j dz ≤ δ j
(2.15)
i
Qi
for j = 0, 1, 2, since ρi = ρ(zi ) ≤ 1/10 and zi ∈ Q1 . Corollary 2.5. Assume that λ ≥ λ∗ . For any ϵ > 0, there exists a small δ = δ(ϵ) > 0 such that if u is a local weak solution of (1.1) in ΩT with Qi2 ⊂ ΩT , then there exists N0 > 1 such that sup |∇vλi | ≤ N0 Qi1
and − |∇(u/λ − vλi )|p dz ≤ ϵ,
(2.16)
Qi2
where vλi is the weak solution of (2.14) in Qi2 . j
Proof. Case 1: p ≥ 2. From the definitions of Qi for j = 0, 1, 2 in Lemma 2.1 we define
i 2 −p 2 uλ (x, t ) = u 10ρi x, λ (10ρi ) t /(λ10ρi ) fiλ (x, t ) = f 10ρi x, λ2−p (10ρi )2 t /λ i Aλ (x, t ) = A 10ρi x, λ2−p (10ρi )2 t z = (x, t ) ∈ Q1 , which implies that uiλ is a local weak solution of
(uiλ )t − div (Aiλ ∇ uiλ · ∇ uiλ )(p−2)/2 Aiλ ∇ uiλ = div(|fiλ |p−2 fiλ ) in Q1 . Furthermore, from (2.9) and (2.15) we observe that
p1 2 − |∇ uiλ (x)|p2 dz ≤ 1, Q1
p1 2 − |fiλ |p2 dz ≤ δ and − |Aiλ − Aiλ Q1 |dz ≤ δ. Q1
Q1
Then from Lemma 2.4 we have sup |∇v| ≤ N0
and
Q1/2
− |∇(uiλ − v)|p dz ≤ ϵ, Q1
where v satisfies
vt − div (Aiλ Q1 ∇v · ∇v)(p−2)/2 Aiλ Q1 ∇v = 0 v = uiλ
in Q1 , on ∂p Q1 .
Now we define vλi in Qi2 by
v(x, t ) = vλi 10ρi x, λ2−p (10ρi )2 t /(λ10ρi ) z = (x, t ) ∈ Q1 . Then changing variables, we recover the conclusion of Lemma 2.4 for p ≥ 2. Case 2: 2n/(n + 2) < p < 2. We define
(p−2)/2 i 10ρi x, (10ρi )2 t /(λp/2 10ρi ) uλ (x, t ) = u λ fiλ (x, t ) = f λ(p−2)/2 10ρi x, (10ρi )2 t /λ i Aλ (x, t ) = A λ(p−2)/2 10ρi x, (10ρi )2 t z = (x, t ) ∈ Q1 . Thus we complete the proof similarly to that of Case 1.
In the following it is sufficient to consider the proof of Theorem 1.8 as an a priori estimate, therefore assuming a priori q that |∇ u|p ∈ Lw,loc (ΩT ). This assumption can be removed in a standard way via an approximation argument like the one in [1,5]. Now we are ready to prove the main result, Theorem 1.8.
F. Yao / Nonlinear Analysis 112 (2015) 58–68
67
Proof. From Corollary 2.5 we have
z ∈ Q 1 : |∇ u| > 2N0 λ = z ∈ Q 1 : |∇(u/λ)| > 2N0 i i ≤ z ∈ Qi1 : |∇(u/λ − vλi )| > N0 + z ∈ Qi1 : |∇vλi | > N0 = z ∈ Qi1 : |∇(u/λ − vλi )| > N0 1 ϵ|Qi2 | 2n+2 ϵ|Qi1 | ≤ p |∇(u/λ − vλi )|p dz ≤ = p p N0
N0
Qi2
N0
for any λ ≥ λ∗ , which implies that
w
≤ C ϵ α w Qi1 ≤ C ϵ α w Qi0
z ∈ Qi1 : |∇ u| > 2N0 λ
in view of Lemma 1.7. Therefore, it follows from Corollary 2.2 that
w
z∈
Qi1
: |∇ u| > 2N0 λ ≤ C ϵ α
1
λp1
z ∈Qi0 :|∇ u|> λ 4
|∇ u| w(z ) dz +
1
p1
λp1 δ p1
|f| w(z ) dz ,
|f| w(z ) dz
p1
z ∈Qi0 :|f|> δλ 4
where C = C (n, p, q, α). Recalling the fact that the cylinders {Qi0 } are disjoint and
Qi1 ∪ negligible set ⊃ E (λ) = {z ∈ Q1 : |∇ u| > λ}
i∈N
for any λ ≥ λ∗ , and then summing up on i ∈ N in the inequality above, we have
w ({z ∈ Q1 : |∇ u| > 2N0 λ}) ≤
w
1
z ∈ Qi1 : |∇ u| > 2N0 λ
i
≤ Cϵ
α
λp1
z ∈Q2 :|∇ u|> λ 4
|∇ u| w(z ) dz +
1
p1
λp1 δ p1
p1
z ∈Q2 :|f|> δλ 4
for any λ ≥ λ∗ . Moreover, recalling Lemma 1.6, we have
|∇ u|pq w(z ) dz = q(2N0 )pq
= q(2N0 )
µq − 1 w
µ>0
Q1 pq
µq − 1 w
z ∈ Q1 : |∇ u|p > (2N0 )p µ
dµ
p
λ∗
z ∈ Q1 : |∇ u|p > (2N0 )p µ
dµ
0
+ q(2N0 )pq
∞ p
λ∗
µq−1 w
z ∈ Q1 : |∇ u|p > (2N0 )p µ
dµ
=: J1 + J2 . Estimate of J1 . From the definitions of λ∗ and λ0 we deduce that pq 0
J1 ≤ C λpq ∗ w (Q1 ) ≤ C λ
|∇ u|p(1+ε) w(z ) dz
≤ C2
1dq +ε
|f|pq w(z ) dz
+
d
+1 ,
Q2
Q2
where C2 = C2 (n, p, q, δ, w), since
λ
pq 0
1
1dq +ε
p(1+ε)
1
1
p(1+ε)
|∇ u| w(z ) dz + |f| w (Q2 ) Q2 δ w (Q2 ) Q2 1dq d +ε |f|pq w(z ) dz + 1 ≤ C2 + |∇ u|p(1+ε) w(z ) dz
≤C
Q2
w(z ) dz
1dq +ε
+1
Q2
by using Hölder’s inequality and in view of the fact that p1 = p(1 + ε) < pq. p Estimate of J2 . Set λ = µ1/p in (2.17). Then for any µ ≥ λ∗ and ε ∈ (0, q − 1) we see that
w
z ∈ Q1 : |∇ u|p > (2N0 )p µ
≤ Cϵ
α
µ
q−ε−2
p(1+ε)
µ z ∈Q2 :|∇ u|p > p 4
|∇ u|
µq−ε−2 w(z ) dz + δ p1
p(1+ε)
z ∈Q2 :|f|p >
δp µ 4p
|f|
w(z ) dz .
(2.17)
68
F. Yao / Nonlinear Analysis 112 (2015) 58–68
Furthermore, from Lemma 1.6 we observe that J2 ≤ C ϵ
α
∞
µq−ε−2
0
1
∞
µ z ∈Q2 :|∇ u|p > p 4
|∇ u|p(1+ε) w(z ) dzdµ
p(1+ε)
µ w(z ) dzdµ |f| δp µ δ p1 0 z ∈Q2 :|f|p > p 4 |f|pq w(z ) dz , |∇ u|pq w(z ) dz + C4 ≤ C3 ϵ α +
q−ε−2
Q2
Q2
where C3 = C3 (n, p, q) and C4 = C4 (n, p, q, ϵ, δ). Therefore, combining the estimates of J1 and J2 , we obtain
|∇ u| w(z ) dz ≤ C3 ϵ pq
Q1
α
|∇ u| w(z ) dz + C5 pq
Q2
|∇ u| Q2
p(1+ε)
w(z ) dz
1dq +ε
+
|f| w(z ) dz pq
d
+1 ,
Q2
where C5 = C5 (n, p, q, ϵ, δ, w). Selecting suitable ϵ such that C3 ϵ α = 1/2, and reabsorbing at the right-side first integral in the inequality above by a covering and iteration argument (see Lemma 2.1, Chapter 3 in [13]), we can finish the proof of Theorem 1.8. Acknowledgment The author wishes to thank the anonymous reviewer for the valuable comments and suggestions to improve the paper. This work is supported in part by the NSFC (11471207) and the Innovation Program of Shanghai Municipal Education Commission (14YZ027). References [1] E. Acerbi, G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2007) 285–320. [2] P. Baroni, V. Bögelein, Calderón–Zygmund estimates for parabolic p(x, t )-Laplacian systems, Rev. Mat. Iberoam., to appear. [3] V. Bögelein, F. Duzaar, Higher integrability for parabolic systems with non-standard growth and degenerate diffusions, Publ. Mat. 55 (1) (2011) 201–250. [4] S. Byun, D. Palagachev, S. Ryu, Weighted W 1,p estimates for solutions of nonlinear parabolic equations over non-smooth domains, Bull. Lond. Math. Soc. 45 (4) (2013) 765–778. [5] S. Byun, S. Ryu, Global weighted estimates for the gradient of solutions to nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2) (2013) 291–313. [6] S. Byun, L. Wang, Quasilinear elliptic equations with BMO coefficients in Lipschitz domains, Trans. Amer. Math. Soc. 359 (12) (2007) 5899–5913. [7] S. Byun, L. Wang, Nonlinear gradient estimates for elliptic equations of general type, Calc. Var. Partial Differential Equations 45 (3–4) (2012) 403–419. [8] S. Byun, L. Wang, S. Zhou, Nonlinear elliptic equations with BMO coefficients in Reifenberg domains, J. Funct. Anal. 250 (1) (2007) 167–196. [9] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, Berlin, 1993. [10] E. DiBenedetto, J. Manfredi, On the higer integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115 (1993) 1107–1134. [11] F. Duzaar, G. Mingione, Gradient estimates via non-linear potentials, Amer. J. Math. 133 (4) (2011) 1093–1149. [12] F. Duzaar, G. Mingione, Gradient estimates via linear and nonlinear potentials, J. Funct. Anal. 259 (11) (2010) 2961–2998. [13] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, NJ, 1983. [14] J. Jiménez Urrea, The Cauchy problem associated to the Benjamin equation in weighted Sobolev spaces, J. Differential Equations 254 (4) (2013) 1863–1892. [15] J. Kinnunen, J.L. Lewis, Higher integrability for parabolic systems of p-Laplacian type, Duke Math. J. 102 (2000) 253–271. [16] J. Kinnunen, S. Zhou, A local estimate for nonlinear equations with discontinuous coefficients, Comm. Partial Differential Equations 24 (1999) 2043–2068. [17] A. Kufner, Weighted Sobolev Spaces, John Wiley & Sons, Inc., New York, 1985, A Wiley-Interscience Publication, Translated from the Czech. [18] A. Kufner, A.M. Sändig, Some Applications of Weighted Sobolev Spaces, in: Teubner-Texte zur Mathematik, Band 100, Teubner Verlagsgesellschaft, 1987. [19] T. Kuusi, G. Mingione, Universal potential estimates, J. Funct. Anal. 262 (10) (2012) 4205–4269. [20] T. Kuusi, G. Mingione, Linear potentials in nonlinear potential theory, Arch. Ration. Mech. Anal. 207 (1) (2013) 215–246. [21] T. Mengesha, N. Phuc, Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains, J. Differential Equations 250 (5) (2011) 2485–2507. [22] T. Mengesha, N. Phuc, Global estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Ration. Mech. Anal. 203 (1) (2012) 189–216. [23] M. Misawa, Lq estimates of gradients for evolutional p-Laplacian systems, J. Differential Equations 219 (2) (2005) 390–420. [24] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972) 207–226. [25] N. Phuc, Weighted estimates for nonhomogeneous quasilinear equations with discontinuous coefficients, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (1) (2011) 1–17. [26] E. Stein, Harmonic Analysis, Princeton University Press, Princeton, NJ, 1993. [27] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, in: Pure Appl. Math., vol. 123, Academic Press, Inc., Orlando, FL, 1986.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Weighted gradient estimates for the parabolic p-Laplacian equations Fengping Yao Department of Mathematics, Shanghai University, Shanghai 200444, China
article
info
abstract
Article history: Received 25 July 2014 Accepted 9 September 2014 Communicated by Enzo Mitidieri
In this paper we obtain the weighted Lq estimates of the gradients of weak solutions for the quasilinear parabolic equations of p-Laplacian type with small BMO coefficients. For every ε ∈ (0, q − 1) if |∇ u|p ∈ L1w,+εloc (ΩT ) and |f|p ∈ Lqw,loc (ΩT ), then we prove that
|∇ u|p ∈ Lqw,loc (ΩT ).
MSC: 35K55 35K65
Our results improve the known results for such equations using a harmonic analysis free technique. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Lq estimates Weighted Regularity Gradient Divergence Parabolic p-Laplacian
1. Introduction Many authors [6,8,7,10–12,16,19,20] have studied the regularity estimates for weak solutions of the following quasilinear elliptic equations of p-Laplacian type div (A∇ u · ∇ u)
p−2 2
A∇ u = div(|f|p−2 f)
in Ω
and the general case. Recently, Byun and Ryu [5] obtained the global weighted Lq estimates with q ∈ (p, +∞) for the gradient of weak solutions of the general nonlinear elliptic equations of p-Laplacian type. In this paper we are concerned with the following quasilinear parabolic equation of p-Laplacian type ut − div (A∇ u · ∇ u)(p−2)/2 A∇ u = div(|f|p−2 f) in ΩT = Ω × (0, T ]
(1.1)
for 1 ≤ 2n/(n + 2) < p < ∞ and n ≥ 2, where Ω is an open bounded domain in R and ∂p ΩT = ∂ Ω × [0, T ] ∪ Ω × {t = 0} is the parabolic boundary of ΩT . Moreover, f = (f 1 , . . . , f n ) is a given vector field and A = {aij (x)}n×n is a symmetric matrix with measurable coefficients satisfying the uniformly parabolic condition n
Λ−1 |ξ |2 ≤ A(x, t )ξ · ξ ≤ Λ|ξ |2
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.na.2014.09.010 0362-546X/© 2014 Elsevier Ltd. All rights reserved.
(1.2)
F. Yao / Nonlinear Analysis 112 (2015) 58–68
59
for all ξ ∈ Rn and almost every (x, t ) ∈ Rn × R, where Λ is a positive constant. We remark that the lower bound 2n/(n + 2) on the exponent p is standard and unavoidable in the theory of the parabolic p-Laplacian operator. Differently from the elliptic case, the quasilinear parabolic equation of p-Laplacian type is not homogeneous even if f ≡ 0, which is the main difficulty. Kinnunen and Lewis [15] obtained a reverse Hölder inequality of the gradient for the weak solutions of (1.1), which implies the local higher integrability of the gradient. In [23] Misawa proved that |Du|p ∈ Lq , for every q ≥ 1 if f ∈ L∞ for the weak solutions of (1.1). Furthermore, Acerbi and Mingione [1] obtained the following estimates in Sobolev spaces
|F |p ∈ Lqloc (ΩT ) ⇒ |∇ u|p ∈ Lqloc (ΩT ) for any q ≥ 1 for the weak solutions of the following parabolic p-Laplacian systems ut − div a(x, t )|∇ u|p−2 ∇ u = div |F |p−2 F
in ΩT
(1.3)
with small BMO coefficients. Recently, some authors [2,3] studied the corresponding estimates for weak solutions of the evolutionary p(x, t )-Laplacian system. Let Q (θ , ρ) = Bρ × (−θ , θ] be a centered parabolic cylinder. Especially when θ = ρ 2 = r 2 , we shall simply write Qr ≡ Q (r 2 , r ). Throughout this paper we assume that the coefficients of A = {aij } are in parabolic BMO spaces and their parabolic semi-norm are small enough. More precisely, we have the following definition. Definition 1.1 (Small BMO Semi-Norm Condition). We say that the matrix A of coefficients is (δ, R)-vanishing if
sup −
Qz (θ,ρ) Qz (θ,ρ)
|A(ζ ) − AQz (θ,ρ) |dζ ≤ δ,
and the supremum is taken among all cylinders of the type Qz (θ , ρ) with ρ ≤ R and θ ≤ R2 , where z = (x, t ), ζ = (y, s) ∈ Rn × R. As usual, the solutions of (1.1) are taken in a weak sense. We now state the definition of weak solutions. p
p
p
2 Definition 1.2. Assume that f ∈ Lloc (ΩT ). A function u ∈ Lloc (ΩT ) ∩ L∞ loc (0, T ; Lloc (Ω )) with |∇ u| ∈ Lloc (ΩT ) is a local weak solution of (1.1) in ΩT if for any compact subset K of Ω and for any subinterval [t1 , t2 ] of (0, T ] we have
t2
K
uϕ dx +
t2
t1
t1
−uϕt + (A∇ u · ∇ u)
p−2 2
t2 t1
K
|f|p−2 f · ∇ϕ dxdt
A∇ u · ∇ϕ dxdt = − K
for any ϕ ∈ C0∞ (K × [t1 , t2 ]). Here for convenience of the readers, we shall give some definitions and properties on the weighted Lebesgue spaces (see [4,14,17,18,21,22,24–27]). Definition 1.3. Aq for some q > 1 is the class of the reverse Hölder weights: w ∈ Aq if w ∈ L1loc (Rn+1 ), w > 0 almost everywhere and there exists a constant C such that for all balls Qr = Br × (−r 2 , r 2 ] of Rn+1 ,
q −1 −1 − w(z ) dz − w(z ) q−1 dz ≤ C. Qr
Qr
Moreover, we denote A∞ =
Aq
1
and
w(Qr ) =
w(z ) dz .
Qr
Furthermore, the corresponding weighted Lebesgue space Lpw (Qr ) for any p ≥ 1 consists of all functions h which satisfy
∥h∥Lpw (Qr ) =:
|h| w(z ) dz p
1/p
< ∞.
Qr
Remark 1.4. We remark that Aq1 ⊂ Aq2 for any 1 < q1 ≤ q2 < ∞ (see [26, p. 195]). Lemma 1.5. If w ∈ Aq with q > q1 > 1, then we have Lqw (Qr ) ⊂ Lq1 (Qr ).
60
F. Yao / Nonlinear Analysis 112 (2015) 58–68
Proof. Using Hölder’s inequality, we have
|f |q1 dz =
q1
q1
|f |q1 w(z ) q w(z )− q dz Qr
Qr
|f | w(z ) dz q
≤
qq1
Qr
w(z )
1− qq1
q
− q−1q
1
dz
.
Qr
Moreover, we find that w ∈ Aq/q1 in view of Remark 1.4 and the fact that w ∈ Aq with q > q1 > 1. Furthermore, from Definition 1.3 we conclude that
q1 q |Qr | q−q1 − q/q1 −1 − q−1q 1 1 − w(z ) dz ≤ C dz = − w(z ) . w(Qr ) Qr Qr Thus, if f ∈ Lqw (Qr ), then we have
q1
|f | w(z ) dz q
|f | dz ≤ C
qq1
q 1− q1
|Qr |
Qr
Qr
|Qr | w(Qr )
qq1
≤ C,
since w ∈ L1loc (Rn ) and w > 0 almost everywhere. This finishes our proof.
Lemma 1.6. Assume that w ∈ Aq for some q > 1 and g ∈ Lqw (Rn+1 ). Then for any q > β > 1 we have (1)
w
z ∈ Rn+1 : M g (z ) > µ
≤
C
µq
Rn+1
|g (z )|q w(z ) dz for any µ > 0.
(2)
Rn+1
|g (z )|q w(z ) dz = q
∞
λq−1 w z ∈ Rn+1 : |g (z )| > µ dµ 0 ∞ = (q − β) λq−β−1 |g |β w(z ) dz dµ. |g |>µ
0
Next, we shall give some lemmas on the properties of Aq weight. Lemma 1.7 ([4,27]). If w ∈ Aq for some q > 1 and Qr ⊂ QR ⊂ Rn+1 , then there exist α > 0 and a positive constant C1 > 0 such that
1 C1
|Qr | |QR |
q ≤
w(Qr ) ≤ C1 w(QR )
|Qr | |QR |
α
.
Now let us state the main result of this work. Theorem 1.8. Assume that w(z ) ∈ Aq for some q > 1. Let u be a local weak solution of (1.1). Then for every ε ∈ (0, q − 1) there exists a small δ = δ(n, p, q, ε, Λ, R) > 0 so that for each uniformly parabolic and (δ, R)-vanishing A, and for all f with |f|p ∈ Lqw,loc (ΩT ), if |∇ u|p ∈ L1w,+εloc (ΩT ), then we have |∇ u|p ∈ Lqw,loc (ΩT ) with the estimate
|∇ u| w(z ) dz pq
1q
|∇ u|
≤C
Q1
p(1+ε)
w(z ) dz
d 1+ε
|f| w(z ) dz pq
+
Q2
dq
+1 ,
Q2
where Q2 ⊂ ΩT , the constant C is independent of u and f, and
1 ≤ d :=
p p − (p − 2)q
for p ≥ 2,
1
2p 2p − (2 − p)nq1
for
(1.4)
2n n+2
< p < 2.
Here
p − 1 p − 2 1 < q1 = 2n + (2 − n)p 2n(2 − p) q,
for p > 2 and q > for
2n
n+2 elsewhere.
p p−2
,
< p < 2 and q >
2p
(2 − p)n
,
(1.5)
F. Yao / Nonlinear Analysis 112 (2015) 58–68
61
Remark 1.9. Assume that 2n/(n + 2) < p < 2. We can check that 2p
(2 − p)n
>1
and 2n + (2 − n)p 2n(2 − p)
2p
=
1 + (2−p)n > 1. 2
2. Proof of the main result To start with, let u be a local weak solution of problem (1.1) in ΩT with Q2 ⊂ ΩT and A be uniformly parabolic. We write 1
λ0d =
1
w (Q2 )
|∇ u|p1 w(z ) dz
p1
1
1
+
δ
Q2
1
w (Q2 )
|f|p1 w(z ) dz
p1
1
+ 1,
(2.1)
.
(2.2)
Q2
where p1 = p(1 + ε) ∈ (p, pq) with ε ∈ (0, q − 1), d is defined in (1.4) and E (λ) = {z ∈ Q1 : |∇ u| > λ} for λ > 0. Moreover, for any domain Q in Rn+1 we write
J [Q ] =:
1
w (Q )
|∇ u| w(z ) dz p1
p1
1
+
1
Q
δ
1
w (Q )
|f| w(z ) dz p1
p1
1
Q
(n+2)dq1
Lemma 2.1. Given λ ≥ λ∗ =: C1d 20 p λ0 , where d is defined in (1.4), there exists a family of disjoint cylinders {Qi0 }i∈N with zi = (xi , ti ) ∈ E (λ), 0 < ρi = ρ(zi ) ≤ 1/10 and
Qi0
Qzi (λ2−p ρi2 , ρi )
=
for p ≥ 2,
Qzi ρi2 , λ(p−2)/2 ρi
for 2n/(n + 2) < p < 2,
such that J Qi0 = λ,
J Qzi (θ , ρ) < λ for any Qzi (θ , ρ) ⊃ Qi0 ,
(2.3)
and E (λ) ⊂
Qi1 ∪ negligible set ,
i∈N
where
j Qi
=:
5jQi0
Qzi (5j)2 λ2−p ρi2 , 5jρi
=
(p−2)/2
Qzi (5j) ρ , 5jλ
2
2 i
for p ≥ 2,
ρi
for 2n/(n + 2) < p < 2,
(2.4)
for j = 1, 2. Proof. Case 1: p ≥ 2. We first claim that sup
J Qw λ2−p ρ 2 , ρ
sup
w∈Q1 1/10≤ρ≤1
≤ λ.
(2.5)
To prove this, fix any w ∈ Q1 and 1/10 ≤ ρ ≤ 1. Then from Lemma 1.7 and the fact that p1 ∈ (p, pq) we find that
1
w Qw λ2−p ρ 2 , ρ
≤
p1
|∇ u| w(z ) dz
1
p1
(
Qw λ2−p ρ 2 ,ρ
)
w (Q2 ) 1 · w (Q2 ) w Qw λ2−p ρ 2 , ρ
p1
|∇ u| w(z ) dz p1
Q2
1
62
F. Yao / Nonlinear Analysis 112 (2015) 58–68
≤ C1
≤ C1 λ
q1
2n+2
1 p1
λ2−p ρ n+2
1− 2p q1
20
1
w (Q2 )
(n+2)q1
p
1
w (Q2 )
|∇ u| w(z ) dz p1
p1
1
Q2
|∇ u|p1 w(z ) dz
p1
1
,
Q2
where q1 ∈ (1, q] is defined in (1.5). Here we have used the fact that w ∈ Aq1 since w ∈ Aq and Remark 1.4. Similarly, we have
p1
1
w Qw λ2−p ρ 2 , ρ
1
|f| w(z ) dz p1
(
Qw λ2−p ρ 2 ,ρ
)
≤ C1 λ
1− 2p q1
20
(n+2)q1 p
1
w (Q2 )
|f|p1 w(z ) dz
p1
1
.
Q2
Consequently it follows from the two inequalities above and the definition of λ0 that J Qw λ2−p ρ 2 , ρ
≤ C1 λ = C1 λ
(p−2)q1 p
(p−2)q1 p
20 20
(n+2)q1 p
(n+2)q1 p
1
λ0d
1−
λ0
(p−2)q1 p
for any w ∈ Q1 , 1/10 ≤ ρ ≤ 1 and λ ≥ λ∗ =: C1d 20 2. We first conclude that
1−
= λ∗
(n+2)dq1 p
(p−2)q1 p
λ
(p−2)q1 p
≤λ
λ0 , where d is defined in (1.4), which implies that (2.5) holds.
lim J Qw (λ2−p ρ 2 , ρ) > λ
ρ→0
for a.e. w ∈ E (λ), which implies that there exists some ρ ∈ (0, 1] satisfying J Qw (λ2−p ρ 2 , ρ) > λ.
Furthermore, from (2.5) we can select a radius ρw ∈ (0, 1/10] such that
ρw =: max ρ | J [Qw (λ2−p ρ 2 , ρ)] = λ, 0 < ρ ≤ 1/10 . Thus, we observe that 2 J Qw (λ2−p ρw , ρw ) = λ and
J Qw (λ2−p ρ 2 , ρ) < λ for ρw < ρ ≤ 1.
Therefore, applying Vitali’s covering lemma, we can find a family of disjoint cylinders {Qi0 }i∈N = {Qzi (λ2−p ρi2 , ρi )}i∈N , zi = (xi , ti ) ∈ E (λ) so that the results of the lemma is true for p ≥ 2. Case 2: 2n/(n + 2) < p < 2. Let λ ≥ λ∗ = C1d 20 claim that sup
J Qw ρ 2 , λ(p−2)/2 ρ
sup
w∈Q1 1/10≤ρ≤1
(n+2)dq1 p
λ0 , where d is defined in (1.4). Then similarly to (2.5), we can
≤ λ.
(2.6)
To prove this, fix any w ∈ Q1 and 1/10 ≤ ρ ≤ 1. Then
p1
1
1
|∇ u| w(z ) dz Qw ρ 2 , λ(p−2)/2 ρ Qw ρ 2 ,λ(p−2)/2 ρ p1 1 1 w (Q2 ) p1 · |∇ u| w(z ) dz ≤ 2 ( p − 2 )/ 2 w (Q2 ) Q2 w Qw ρ , λ ρ 1 q p 1 2n+2
≤ C1 ≤ C1 λ
1
1
λ(p−2)/2 ρ n+2 (2−p)nq1 2p
We remark that
(2 − p)n 2p
p1
< 1,
20
(n+2)q1 p
1
w (Q2 )
1
|∇ u|p1 w(z ) dz
w (Q2 ) Q2 p1 1 p1 . |∇ u| w(z ) dz Q2
p1
F. Yao / Nonlinear Analysis 112 (2015) 58–68
63
since p > 2n/(n + 2). Similarly, we have
p1
1
w Qw ρ 2 , λ(p−2)/2 ρ
1
|f| w(z ) dz
≤ C1 λ
p1
Qw
(
ρ 2 ,λ(p−2)/2 ρ
)
(2−p)nq1
20
2p
(n+2)q1
p
1
w (Q2 )
|f| w(z ) dz p1
p1
1
.
Q2
Consequently, it follows from the two inequalities above and the definition of λ0 that J Qw ρ 2 , λ(p−2)/2 ρ
≤ C1 λ = C1 λ
(2−p)nq1 2p
(2−p)nq1 2p
20 20
(n+2)q1 p
(n+2)q1 p
1
λ0d
2p−(2−p)nq1 2p
λ0
for any w ∈ Q1 , 1/10 ≤ ρ ≤ 1 and λ ≥ λ∗ =: C1d 20 similarly to that of Case 1.
(n+2)dq1 p
1−
= λ∗
(2−p)nq1 2p
λ
(2−p)nq1
≤λ
2p
λ0 , which implies that (2.6) holds. So we can complete the proof
Corollary 2.2. Under the same hypothesis and results as those in Lemma 2.1, we have
w
Qi0
≤C
1
λp1
|∇ u| w(z ) dz + p1
z ∈Qi0 :|∇ u|> λ 4
1
λp1 δ p1
|f| w(z ) dz , p1
z ∈Qi0 :|f|> δλ 4
pq
where C = 1−221−p . Proof. From (2.3) we find that
J
Qi0
=
p1
1
w Qi
0
1
|∇ u| w(z ) dz p1
Qi0
+
1
δ
w Qi
p1
1
0
|f| w(z ) dz
1
p1
Qi0
= λ,
which implies that
w
Qi0
2p1
≤
λp1
|∇ u| w(z ) dz + p1
Qi0
2p1
λp1 δ p1
|f|p1 w(z ) dz ,
Qi0
(2.7)
since either of the following inequalities must be true
λ 2
≤
w
p1
1 Qi0
Qi0
|∇ u|p1 w(z ) dz
1
or
λ 2
≤
1
δ
w
p1
1
Qi0
|f|p1 w(z ) dz
Qi0
1
.
Therefore, by splitting the right-side two integrals in (2.7) as follows we have
2p1 w Qi0 ≤ p λ1
z ∈Qi0 :|∇ u|> λ 4
p1 |∇ u| w(z ) dz +
w Qi0 2p1
+
2p1
λp1 δ p1
Thus we have concluded with the desired estimate since p1 ∈ (p, pq).
z ∈Qi0 :|f|> δλ 4
p1 |f| w(z ) dz +
w Qi0 2p1
.
In view of Lemma 2.1, given λ ≥ λ∗ , we can construct the family of cylinders {Qi0 }i∈N , where zi ∈ E (λ). Fix any i ∈ N. Then it follows from Lemma 2.1 that
1
w Qij
p1
1
|∇ u| w(z ) dz p1
j
≤ λ and
Qi
1
p1
1
w Qij
|f| w(z ) dz p1
j
≤ δλ
(2.8)
Qi
j
for j = 1, 2, where Qi is defined in Lemma 2.1. Corollary 2.3. Assume that 1 < p < p2 < p1 < pq. Under the same hypothesis and results as those in Lemma 2.1, we have
p1 2 p2 − |∇ u| dz ≤ C λ and j
Qi
for j = 1, 2.
p1 2 p2 − |f| dz ≤ C δλ, j
Qi
(2.9)
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F. Yao / Nonlinear Analysis 112 (2015) 58–68
Proof. Using Hölder’s inequality and (2.8), we have
p1 p1 2 2 p2 p2 − − |∇ u|p2 dz = − |∇ u|p2 w(z ) p1 w(z ) p1 dz j
j
Qi
Qi
p11 pp1 −pp2 w Qij 1 2 p2 1 − |∇ u|p1 w(z ) dz − w(z ) p1 −p2 dz ≤ j j j j Qi Qi Qi w Qi
pp1 −1 p11 2 − p11 −1 , ≤ λ − w(z ) dz · − w(z ) p2 dz j
j
Qi
Qi
which implies that
p1 2 − |∇ u|p2 dz ≤ C λ, j
Qi
in view of Definition 1.3 and w ∈ Ap1 /p2 since w ∈ Aq and 1 < p1 /p2 < q. Similarly, we have
p1 2 p2 − |f| dz ≤ C δλ. j
Qi
Thus we finish the proof.
Furthermore, we can obtain the following result. Lemma 2.4. For any ϵ > 0, there exists a small δ = δ(ϵ) > 0 such that if u is a local weak solution of (1.1) in ΩT with Q2 ⊂ ΩT ,
− |A − AQ1 |dz ≤ δ,
(2.10)
Q1
p1 2 p2 − |∇ u| dz ≤ C and
p1 2 p2 − |f| dz ≤ C δ,
(2.11)
Q1
Q1
then there exists N0 > 1 such that sup |∇v| ≤ N0
(2.12)
Q1/2
and
− |∇(u − v)|p dz ≤ ϵ,
(2.13)
Q1
where v is the weak solution of
vt − div (AQ1 ∇v · ∇v)(p−2)/2 AQ1 ∇v = 0 v=u
in Q1 , on ∂p Q1 .
(2.14)
Proof. If u and v are the weak solutions of (1.1) and (2.14) respectively, then by choosing the test function ϕ = u − v , after a direct calculation we show the resulting expression as I1 + I2 = I3 + I4 , where I1 =
I2 =
d dt
|v − u|2
Q1
Q1
2
|v(x, 1) − u(x, 1)|2
dz = B1
2
dz ,
(AQ1 ∇v · ∇v)(p−2)/2 AQ1 ∇v − (AQ1 ∇ u · ∇ u)(p−2)/2 AQ1 ∇ u · ∇(v − u)dz ,
F. Yao / Nonlinear Analysis 112 (2015) 58–68
65
(A∇ u · ∇ u)(p−2)/2 A∇ u − (AQ1 ∇ u · ∇ u)(p−2)/2 AQ1 ∇ u · ∇(v − u)dz , I3 = Q1
|f|p−2 f · ∇(v − u)dz .
I4 = Q1
Actually, (2.12) can follow from (2.11), (2.13), Hölder’s inequality and the interior regularity of weak solutions for degenerate parabolic equations (see [9, Chapter 8, Theorems 5.1 and 5.2]). Estimate of I2 . We divide into two cases. Case 1. p ≥ 2. Using the elementary inequality
(AQ1 ξ · ξ )(p−2)/2 AQ1 ξ − (AQ1 η · η)(p−2)/2 AQ1 η · (ξ − η) ≥ C |ξ − η|p
for every ξ , η ∈ Rn with C = C (p, Λ), we have
|∇(u − v)|p dz .
I2 ≥ C Q1
Case 2. 2n/(n + 2) < p < 2. Using the elementary inequality
|ξ − η|p ≤ C τ (p−2)/p (AQ1 ξ · ξ )(p−2)/2 AQ1 ξ − (AQ1 η · η)(p−2)/2 AQ1 η · (ξ − η) + τ |η|p for every ξ , η ∈ Rn and every τ ∈ (0, 1) with C = C (p, Λ), we have I2 + τ
|∇ u|p dz ≥ C (τ ) Q1
|∇(u − v)|p dz . Q1
Estimate of I3 . Using the elementary inequality
(Aξ · ξ )(p−2)/2 Aξ − (AQ ξ · ξ )(p−2)/2 AQ ξ ≤ C |A − AQ | |ξ |p−1 1 1 1 for every ξ , η ∈ Rn with C = C (p, Λ), and then using Young’s inequality with τ and Hölder’s inequality, from (2.11) we have
|A − AQ1 | |∇ u|p−1 |∇(u − v)|dz p ≤ C (τ ) |A − AQ1 | p−1 |∇ u|p dz + τ |∇(u − v)|p dz
I3 ≤ C
Q1
Q1
≤ C (τ )
Q1
p2p−p
pp2 (p−1)(p2 −p)
2
|A − AQ1 | dz Q1 p2 −p |∇(u − v)|p dz , ≤ C (τ )δ p2 + τ
|∇ u|p2 dz
pp
2
+τ
Q1
|∇(u − v)|p dz Q1
Q1
since
pp2
|A − AQ1 |
(p−1)(p2 −p) dz
p2p−p 2
≤ (2Λ)
p2 +p2 −p p2 (p−1)
p2p−p
Q1
|A − AQ1 |dz
2
≤ Cδ
p2 −p p2
Q1
as a consequence of (1.2) and (2.10). Estimate of I4 . Using Young’s inequality with τ , Hölder’s inequality and (2.11), we have I4 ≤ τ
|∇(u − v)|p dz + C (τ ) Q1
≤τ
|∇(u − v)|p dz + C (τ )
|f|p dz Q1
Q1
≤τ
|f|p2 dz
pp
2
Q1
|∇(u − v)|p dz + C (τ )δ p . Q1
Combining all the estimates of Ii (1 ≤ i ≤ 4), we obtain C (τ )
|∇(u − v)| dz ≤ 2τ p
Q1
|∇(u − v)| dz + τ p
Q1
|∇ u| dz + C δ p
Q1
p2 −p p2
+δ
p
.
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F. Yao / Nonlinear Analysis 112 (2015) 58–68
Selecting a small constant τ > 0 such that 0 < τ ≪ δ < 1, and then using Hölder’s inequality and (2.11), we conclude that
p2 −p p p p2 − |∇(u − v)| dz ≤ C δ + δ +δ =ϵ Q1
by selecting δ satisfying the last inequality above. This completes our proof.
Let δ in (2.1) and Definition 1.1 be the same as that in Lemma 2.4. We may as well assume that R = 2 by a scaling in Definition 1.1. Therefore
− A − AQ j dz ≤ δ j
(2.15)
i
Qi
for j = 0, 1, 2, since ρi = ρ(zi ) ≤ 1/10 and zi ∈ Q1 . Corollary 2.5. Assume that λ ≥ λ∗ . For any ϵ > 0, there exists a small δ = δ(ϵ) > 0 such that if u is a local weak solution of (1.1) in ΩT with Qi2 ⊂ ΩT , then there exists N0 > 1 such that sup |∇vλi | ≤ N0 Qi1
and − |∇(u/λ − vλi )|p dz ≤ ϵ,
(2.16)
Qi2
where vλi is the weak solution of (2.14) in Qi2 . j
Proof. Case 1: p ≥ 2. From the definitions of Qi for j = 0, 1, 2 in Lemma 2.1 we define
i 2 −p 2 uλ (x, t ) = u 10ρi x, λ (10ρi ) t /(λ10ρi ) fiλ (x, t ) = f 10ρi x, λ2−p (10ρi )2 t /λ i Aλ (x, t ) = A 10ρi x, λ2−p (10ρi )2 t z = (x, t ) ∈ Q1 , which implies that uiλ is a local weak solution of
(uiλ )t − div (Aiλ ∇ uiλ · ∇ uiλ )(p−2)/2 Aiλ ∇ uiλ = div(|fiλ |p−2 fiλ ) in Q1 . Furthermore, from (2.9) and (2.15) we observe that
p1 2 − |∇ uiλ (x)|p2 dz ≤ 1, Q1
p1 2 − |fiλ |p2 dz ≤ δ and − |Aiλ − Aiλ Q1 |dz ≤ δ. Q1
Q1
Then from Lemma 2.4 we have sup |∇v| ≤ N0
and
Q1/2
− |∇(uiλ − v)|p dz ≤ ϵ, Q1
where v satisfies
vt − div (Aiλ Q1 ∇v · ∇v)(p−2)/2 Aiλ Q1 ∇v = 0 v = uiλ
in Q1 , on ∂p Q1 .
Now we define vλi in Qi2 by
v(x, t ) = vλi 10ρi x, λ2−p (10ρi )2 t /(λ10ρi ) z = (x, t ) ∈ Q1 . Then changing variables, we recover the conclusion of Lemma 2.4 for p ≥ 2. Case 2: 2n/(n + 2) < p < 2. We define
(p−2)/2 i 10ρi x, (10ρi )2 t /(λp/2 10ρi ) uλ (x, t ) = u λ fiλ (x, t ) = f λ(p−2)/2 10ρi x, (10ρi )2 t /λ i Aλ (x, t ) = A λ(p−2)/2 10ρi x, (10ρi )2 t z = (x, t ) ∈ Q1 . Thus we complete the proof similarly to that of Case 1.
In the following it is sufficient to consider the proof of Theorem 1.8 as an a priori estimate, therefore assuming a priori q that |∇ u|p ∈ Lw,loc (ΩT ). This assumption can be removed in a standard way via an approximation argument like the one in [1,5]. Now we are ready to prove the main result, Theorem 1.8.
F. Yao / Nonlinear Analysis 112 (2015) 58–68
67
Proof. From Corollary 2.5 we have
z ∈ Q 1 : |∇ u| > 2N0 λ = z ∈ Q 1 : |∇(u/λ)| > 2N0 i i ≤ z ∈ Qi1 : |∇(u/λ − vλi )| > N0 + z ∈ Qi1 : |∇vλi | > N0 = z ∈ Qi1 : |∇(u/λ − vλi )| > N0 1 ϵ|Qi2 | 2n+2 ϵ|Qi1 | ≤ p |∇(u/λ − vλi )|p dz ≤ = p p N0
N0
Qi2
N0
for any λ ≥ λ∗ , which implies that
w
≤ C ϵ α w Qi1 ≤ C ϵ α w Qi0
z ∈ Qi1 : |∇ u| > 2N0 λ
in view of Lemma 1.7. Therefore, it follows from Corollary 2.2 that
w
z∈
Qi1
: |∇ u| > 2N0 λ ≤ C ϵ α
1
λp1
z ∈Qi0 :|∇ u|> λ 4
|∇ u| w(z ) dz +
1
p1
λp1 δ p1
|f| w(z ) dz ,
|f| w(z ) dz
p1
z ∈Qi0 :|f|> δλ 4
where C = C (n, p, q, α). Recalling the fact that the cylinders {Qi0 } are disjoint and
Qi1 ∪ negligible set ⊃ E (λ) = {z ∈ Q1 : |∇ u| > λ}
i∈N
for any λ ≥ λ∗ , and then summing up on i ∈ N in the inequality above, we have
w ({z ∈ Q1 : |∇ u| > 2N0 λ}) ≤
w
1
z ∈ Qi1 : |∇ u| > 2N0 λ
i
≤ Cϵ
α
λp1
z ∈Q2 :|∇ u|> λ 4
|∇ u| w(z ) dz +
1
p1
λp1 δ p1
p1
z ∈Q2 :|f|> δλ 4
for any λ ≥ λ∗ . Moreover, recalling Lemma 1.6, we have
|∇ u|pq w(z ) dz = q(2N0 )pq
= q(2N0 )
µq − 1 w
µ>0
Q1 pq
µq − 1 w
z ∈ Q1 : |∇ u|p > (2N0 )p µ
dµ
p
λ∗
z ∈ Q1 : |∇ u|p > (2N0 )p µ
dµ
0
+ q(2N0 )pq
∞ p
λ∗
µq−1 w
z ∈ Q1 : |∇ u|p > (2N0 )p µ
dµ
=: J1 + J2 . Estimate of J1 . From the definitions of λ∗ and λ0 we deduce that pq 0
J1 ≤ C λpq ∗ w (Q1 ) ≤ C λ
|∇ u|p(1+ε) w(z ) dz
≤ C2
1dq +ε
|f|pq w(z ) dz
+
d
+1 ,
Q2
Q2
where C2 = C2 (n, p, q, δ, w), since
λ
pq 0
1
1dq +ε
p(1+ε)
1
1
p(1+ε)
|∇ u| w(z ) dz + |f| w (Q2 ) Q2 δ w (Q2 ) Q2 1dq d +ε |f|pq w(z ) dz + 1 ≤ C2 + |∇ u|p(1+ε) w(z ) dz
≤C
Q2
w(z ) dz
1dq +ε
+1
Q2
by using Hölder’s inequality and in view of the fact that p1 = p(1 + ε) < pq. p Estimate of J2 . Set λ = µ1/p in (2.17). Then for any µ ≥ λ∗ and ε ∈ (0, q − 1) we see that
w
z ∈ Q1 : |∇ u|p > (2N0 )p µ
≤ Cϵ
α
µ
q−ε−2
p(1+ε)
µ z ∈Q2 :|∇ u|p > p 4
|∇ u|
µq−ε−2 w(z ) dz + δ p1
p(1+ε)
z ∈Q2 :|f|p >
δp µ 4p
|f|
w(z ) dz .
(2.17)
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F. Yao / Nonlinear Analysis 112 (2015) 58–68
Furthermore, from Lemma 1.6 we observe that J2 ≤ C ϵ
α
∞
µq−ε−2
0
1
∞
µ z ∈Q2 :|∇ u|p > p 4
|∇ u|p(1+ε) w(z ) dzdµ
p(1+ε)
µ w(z ) dzdµ |f| δp µ δ p1 0 z ∈Q2 :|f|p > p 4 |f|pq w(z ) dz , |∇ u|pq w(z ) dz + C4 ≤ C3 ϵ α +
q−ε−2
Q2
Q2
where C3 = C3 (n, p, q) and C4 = C4 (n, p, q, ϵ, δ). Therefore, combining the estimates of J1 and J2 , we obtain
|∇ u| w(z ) dz ≤ C3 ϵ pq
Q1
α
|∇ u| w(z ) dz + C5 pq
Q2
|∇ u| Q2
p(1+ε)
w(z ) dz
1dq +ε
+
|f| w(z ) dz pq
d
+1 ,
Q2
where C5 = C5 (n, p, q, ϵ, δ, w). Selecting suitable ϵ such that C3 ϵ α = 1/2, and reabsorbing at the right-side first integral in the inequality above by a covering and iteration argument (see Lemma 2.1, Chapter 3 in [13]), we can finish the proof of Theorem 1.8. Acknowledgment The author wishes to thank the anonymous reviewer for the valuable comments and suggestions to improve the paper. This work is supported in part by the NSFC (11471207) and the Innovation Program of Shanghai Municipal Education Commission (14YZ027). References [1] E. Acerbi, G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2007) 285–320. [2] P. Baroni, V. Bögelein, Calderón–Zygmund estimates for parabolic p(x, t )-Laplacian systems, Rev. Mat. Iberoam., to appear. [3] V. Bögelein, F. Duzaar, Higher integrability for parabolic systems with non-standard growth and degenerate diffusions, Publ. Mat. 55 (1) (2011) 201–250. [4] S. Byun, D. Palagachev, S. Ryu, Weighted W 1,p estimates for solutions of nonlinear parabolic equations over non-smooth domains, Bull. Lond. Math. Soc. 45 (4) (2013) 765–778. [5] S. Byun, S. Ryu, Global weighted estimates for the gradient of solutions to nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2) (2013) 291–313. [6] S. Byun, L. Wang, Quasilinear elliptic equations with BMO coefficients in Lipschitz domains, Trans. Amer. Math. Soc. 359 (12) (2007) 5899–5913. [7] S. Byun, L. Wang, Nonlinear gradient estimates for elliptic equations of general type, Calc. Var. Partial Differential Equations 45 (3–4) (2012) 403–419. [8] S. Byun, L. Wang, S. Zhou, Nonlinear elliptic equations with BMO coefficients in Reifenberg domains, J. Funct. Anal. 250 (1) (2007) 167–196. [9] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, Berlin, 1993. [10] E. DiBenedetto, J. Manfredi, On the higer integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115 (1993) 1107–1134. [11] F. Duzaar, G. Mingione, Gradient estimates via non-linear potentials, Amer. J. Math. 133 (4) (2011) 1093–1149. [12] F. Duzaar, G. Mingione, Gradient estimates via linear and nonlinear potentials, J. Funct. Anal. 259 (11) (2010) 2961–2998. [13] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, NJ, 1983. [14] J. Jiménez Urrea, The Cauchy problem associated to the Benjamin equation in weighted Sobolev spaces, J. Differential Equations 254 (4) (2013) 1863–1892. [15] J. Kinnunen, J.L. Lewis, Higher integrability for parabolic systems of p-Laplacian type, Duke Math. J. 102 (2000) 253–271. [16] J. Kinnunen, S. Zhou, A local estimate for nonlinear equations with discontinuous coefficients, Comm. Partial Differential Equations 24 (1999) 2043–2068. [17] A. Kufner, Weighted Sobolev Spaces, John Wiley & Sons, Inc., New York, 1985, A Wiley-Interscience Publication, Translated from the Czech. [18] A. Kufner, A.M. Sändig, Some Applications of Weighted Sobolev Spaces, in: Teubner-Texte zur Mathematik, Band 100, Teubner Verlagsgesellschaft, 1987. [19] T. Kuusi, G. Mingione, Universal potential estimates, J. Funct. Anal. 262 (10) (2012) 4205–4269. [20] T. Kuusi, G. Mingione, Linear potentials in nonlinear potential theory, Arch. Ration. Mech. Anal. 207 (1) (2013) 215–246. [21] T. Mengesha, N. Phuc, Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains, J. Differential Equations 250 (5) (2011) 2485–2507. [22] T. Mengesha, N. Phuc, Global estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Ration. Mech. Anal. 203 (1) (2012) 189–216. [23] M. Misawa, Lq estimates of gradients for evolutional p-Laplacian systems, J. Differential Equations 219 (2) (2005) 390–420. [24] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972) 207–226. [25] N. Phuc, Weighted estimates for nonhomogeneous quasilinear equations with discontinuous coefficients, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (1) (2011) 1–17. [26] E. Stein, Harmonic Analysis, Princeton University Press, Princeton, NJ, 1993. [27] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, in: Pure Appl. Math., vol. 123, Academic Press, Inc., Orlando, FL, 1986.