Gradient estimates for elliptic equations with measurable nonlinearities

Accepted Manuscript Gradient estimates for elliptic equations with measurable nonlinearities

Youchan Kim

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S0021-7824(17)30179-4 https://doi.org/10.1016/j.matpur.2017.11.003 MATPUR 2959

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Journal de Mathématiques Pures et Appliquées

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13 October 2016

Please cite this article in press as: Y. Kim, Gradient estimates for elliptic equations with measurable nonlinearities, J. Math. Pures Appl. (2017), https://doi.org/10.1016/j.matpur.2017.11.003

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Gradient estimates for elliptic equations with measurable nonlinearities Youchan Kim National Institute for Mathematical Sciences, Daejeon 34047, South Korea

Abstract We obtain Calder´ on-Zygmund type estimate for nonlinear elliptic equations of pLaplacian type, under the condition that the associated nonlinearity is allowed to be merely measurable in one spatial variable, but has locally small mean oscillation in the remaining spatial variables. This is the minimal regularity requirement on the associated nonlinearity for Calder´on-Zygmund type estimate, in the sense that if the associated nonlinearity is allowed to be merely measurable with respect to two independent spatial variables then Calder´ on-Zygmund type estimate fails in general. R´ esum´ e Nous obtenons une estimation du type Calder´on-Zygmund pour des ´equations elliptiques p-Laplaciennes avec la condition que la non-lin´earit´e associ´ee est permise d’ˆetre seulement mesurable en dimension d’un espace, mais avoir une petite oscillation moyenne localement dans les dimensions sup´erieures. Celle-ci est la r´egularit´e minimale sur la non-lin´earit´e pour des estimations du type Calder´ onZygmund dans le sens o` u si la non-lin´earit´e associ´ee est permise d’ˆetre seulement mesurable par rapport aux deux spatiales variables, alors l’estimation du type Calder´on-Zygmund ´echoue en g´en´eral. Keywords: Calder´ on-Zygmund type estimates, Nonlinear elliptic equations, Measurable nonlinearities 2010 MSC: Primary 35J60, 35B65 Secondary 35R05

1. Introduction We study nonlinear elliptic equations of p-Laplacian type div a(Du, x) = div (|F |p−2 F )

in Ω

p ∈ (1, ∞),

to find a minimal regularity requirement on the nonlinearity a, under which we establish the classical Calder´ on-Zygumnd type theory F ∈ Lqloc (Ω)

=⇒

Du ∈ Lqloc (Ω)

q ∈ [p, ∞).

(1)

Email address: [email protected] (Youchan Kim)

Preprint submitted to Elsevier

November 13, 2017

We assume that the associated nonlinearity a(ξ, x1 , x ) has no regularity assumption in one spatial variable, say x1 , but has locally small BMO (bounded mean oscillation) semi-norm in the remaining spatial variables, say x = (x2 , · · · , xn ). With this assumption, the nonlinearity a(ξ, x1 , x ) is allowed to be merely measurable in x1 variable, and might have big jumps in x1 variable. If the coefficients of linear elliptic equations are allowed to be merely measurable, then (1) fails in general, see for instance [38] and [48]. So to achieve (1) for elliptic equations, a suitable regularity condition is required on the coefficients or the nonlinearities. Indeed, Caffarelli and Peral [16] showed (1) for linear ellliptic equations when the coefficients have locally small BMO (bounded mean oscillation) semi-norm in x. Moreover, (1) holds for nonlinear elliptic equations with p ∈ (1, ∞) if the nonlinearity a(ξ, x) has locally small BMO semi-norm in x, see [12, 13, 40, 41] and the references therein. Then a natural question arises: “The coefficients or nonlinearities must be regular in all spatial variables to achieve (1) for elliptic equations?”. The example in [48] shows that if the coefficients of linear elliptic equations are allowed to be merely measurable with respect to two independent spatial variables, then (1) fails in general. So for (1) to be true, the coefficients of linear elliptic equations should be merely measurable in only one spatial variable, but regular in the remaining spatial variables. Indeed, Byun and Wang [15], and Dong and D. Kim [24] independently showed that (1) holds for linear elliptic equations when the coefficients are allowed to be merely measurable in one spatial variable, but regular in the remaining spatial variables. These works are extended by Dong and D. Kim to linear elliptic and parabolic systems in [25, 26], and higher order elliptic and parabolic systems in [27]. The estimate using weighted measure was done by Byun and Palagachev in [11], and using weighted Sobolev spaces was obtained by Dong and D. Kim in [28]. But as stated in [26, Remark 2.8], the methods used in [11, 15, 24, 25, 26, 27, 28] heavily relied on the linear structure of equations, and might not be applied to general nonlinear problems. Later, the author and Byun [9] and the author and Ryu [39] extended the results in [15, 24] to nonlinear elliptic and parabolic equations with measurable nonlinearities when p = 2, and obtained (1) by using a new approach based on Poincar´e’s inequality and Campanato type embeddings. However, the approach in [9] again relied on the linear growth condition of the equations, and could not handle the case p = 2. On the other-hand, if the associated nonlinearity has locally small BMO semi-norm in x, then (1) holds for p ∈ (1, ∞), see for instance [12]. So it is natural to ask that if (1) holds for p ∈ (1, ∞) when the associated nonlinearity is allowed to be merely measurable in one spatial variable, but regular in the remaining spatial variables. Indeed, we show in this paper that (1) even holds for nonlinear elliptic equations with measurable nonlinearities when p ∈ (1, ∞). To the best of our knowledge, this is the first regularity result in the category of elliptic equations with measurable nonlinearities and p-growth condition. To obtain (1), we use a new method based on a Moser type iteration to obtain a local higher integrability result for the limiting equations. Once the local higher integrability result for the limiting equations is obtained, we use a perturbation 2

method based on the work of Acerbi and Mingione in [2] to obtain (1). We establish the desired Calder´ on-Zygmund type estimate by using higher integrability instead of Lipschitz regularity, and we refer to the papers [4, 5, 30, 50] for this approach. There is a vast of literature for Calder´ on-Zygmund type estimate on elliptic equations, and we introduce some related results. Calder´on-Zygmund type estimate can be obtained for generalized Lebesgue spaces such as Weighted Lorentz spaces and Lebesgue spaces of variable exponent, see [1, 22, 23, 36, 46, 47] and the references therein. Also Calder´on-Zygmund type estimate has been established for obstacle problems, see [3, 6, 8, 31, 51] and the references therein. Recently, there has been a great success for the regularity theory on double phase problems as in [18, 19], and Colombo and Mingione obtained Calder´ onZygmund type estimate for double phase problems in [20]. Composite material is an actively studied topic in the regularity theory of elliptic equations, and has been considered by many authors, see for instance [7, 10, 17, 44, 45]. The problem in mind is related to thermal or electrical conduction of composite materials with laminate, core, shielding and coating structures, where big jumps of conductivity or resistance might occur between the conductors and insulators. For these structures, one can find a suitable local coordinate transformation under which the nonlinearity is allowed to be merely measurable in one of spatial variables, say x1 , but regular in the remaining spatial variables. The model in our hand appears in many applications, and insulating an electric wire or a water pipe is a good example to represent our problem. This paper is organized as follows. In the next section, we introduce the notation, the main assumptions and our main result. In Section 3, we cover the preliminaries. In Section 4, we prove the higher integrability result for limiting equations. In Section 5, we derive the comparison estimates. In the last section, the desired Calder´on-Zygmund type estimate is obtained. 2. Definitions and main result Let y = (y1 , y  ) ∈ Rn be a typical point, and r > 0 be a radius. We introduce the following notations: 1. y = (y1 , y  ) = (y1 , · · · , yn ) ∈ Rn .   2. Qr (y  ) = x = (x2 , · · · , xn ) ∈ Rn−1 : max |xi − yi | < r is the open 2≤i≤n

cube in Rn−1 with center y  and size r.   3. Qr (y) = x ∈ Rn : max |xi − yi | < r = (y1 − r, y1 + r) × Qr (y  ) is the 1≤i≤n

open cube in Rn with center y and size r.   4. Br (y) = x ∈ Rn : |x − y| < r is the open ball in Rn with center y and radius r. 5. Qr = Qr (0 ), Qr = Qr (0) and Br = Br (0).

3

Suppose that a : Rn × Rn → Rn satisfies  a(ξ, x) is measurable in x for every ξ ∈ Rn , a(ξ, x) is C 1 -regular in ξ for every x ∈ Rn ,

(2)

and the following ellipticity and growth conditions:  p−1 1 |a(ξ, x)| + |Dξ a(ξ, x)|(|ξ|2 + s2 ) 2 ≤ Λ(|ξ|2 + s2 ) 2 , p−2 Dξ a(ξ, x)ζ, ζ ≥ λ(|ξ|2 + s2 ) 2 |ζ|2 ,

(3)

for every x, ξ, ζ ∈ Rn and for some constants 0 < λ ≤ Λ, s > 0 if 1 < p < 2 and s ≥ 0 if p ≥ 2. For an open bounded domain Ω ⊂ Rn (n ≥ 2), let u be a weak solution of div a(Du, x1 , x ) = div (|F |p−2 F ) p

in Ω,

n

(4) 1,p

where p ∈ (1, ∞) and F ∈ L (Ω, R ). The function u ∈ W (Ω) is a weak solution of (4) if it satisfies   a(Du, x1 , x ), Dϕ dx = |F |p−2 F, Dϕ dx for any ϕ ∈ W01,p (Ω). Ω

Ω

To measure the oscillation of a(ξ, x1 , x ) being averaged in x variables over Qr (y), we define a function θ(a, Qr (y)) on Qr (y) as below   a(ξ, x1 , x ) − a ¯Qr (y ) (ξ, x1 ) , (5) θ(a, Qr (y))(x) = sup p−1 ξ∈Rn \{0} (|ξ|2 + s2 ) 2 where a ¯

Qr (y  )

 (ξ, x1 ) = −

Qr (y  )

a(ξ, x1 , z  ) dz  .

Then one can easily check from (3) and (5) that |θ(a, Qr (y))| ≤ 2Λ.

(6)

Now, we state the main theorem of this paper. Theorem 2.1. For any q ∈ [p, ∞), there exists a small universal constant δ = δ(n, p, q, λ, Λ) ∈ (0, 1] such that if u is a weak solution of (4) and  sup sup − θ(a, Qr (y))(x) dx ≤ δ, 0
then we have   q − |Du| dx ≤ C − QR

Q2R

Qr (y)

2

2

p 2

(|Du| + s ) dx

pq

 +C−

Q2R

|F |q dx,

for any Q2R ⊂⊂ Ω and for some constant C = C(n, p, q, λ, Λ). Remark 2.2. For the sake of convenience and simplicity, we employ the letters C > 0 throughout this paper to denote any constants which can be explicitly computed in terms of known quantities such as n, p, q, λ, Λ. Thus the exact value denoted by C may change from line to line in a given computation. 4

3. Preliminaries We introduce some elementary inequalities and an iteration lemma which will be used in this paper. Lemma 3.1. Let α > 0 be an arbitrary constant. Then we have α (b1 + b2 )α ≤ 2α (bα 1 + b2 )

(b1 , b2 ≥ 0).

Lemma 3.2. For any 1 < p < 2, there exists κ0 = κ0 (n, p, λ, Λ) ∈ (0, 1) such that p

κ ∈ (0, κ0 ) ⇒ |ξ − ζ|p ≤ Cκp (|ξ|2 + s2 ) 2 + C1 κp−2 (|ξ|2 + |ζ|2 + s2 )

p−2 2

|ξ − ζ|2 .

Proof. In view of Young’s inequality, we have |ξ − ζ|p = κ

p(2−p) 2

(|ξ|2 + |ζ|2 + s2 )

p(2−p) 4

[κ

p(p−2) 2

p

(|ξ|2 + |ζ|2 + s2 )

≤ κp (|ξ|2 + |ζ|2 + s2 ) 2 + Cκp−2 (|ξ|2 + |ζ|2 + s2 )

p−2 2

p(p−2) 4

|ξ − ζ|p ]

|ξ − ζ|2 .

Then from Lemma 3.1, we obtain p

p

p

(|ξ|2 + |ζ|2 + s2 ) 2 ≤ C(|ξ|2 + |ζ − ξ|2 + s2 ) 2 ≤ C|ζ − ξ|p + C(|ξ|2 + s2 ) 2 , and the lemma follows by choosing κ0 = κ0 (n, p, λ, Λ) sufficiently small. Lemma 3.3. [37, Lemma 4.3] Let h : [R, 2R] → [0, ∞) be a bounded function such that A +B (R ≤ r1 < r2 ≤ 2R), h(r1 ) ≤ κh(r2 ) + (r2 − r1 )α for some κ ∈ [0, 1). Then we have

 A +B h(r1 ) ≤ C(α, κ) (r2 − r1 )α

(R ≤ r1 < r2 ≤ 2R).

3.1. Regularization To regularize the nonlinearity a, define φ ∈ Cc∞ (Rn ) as a standard mollifier: ⎧  1 ⎨ if |x| < 1, C2 exp (7) φ(x) = |x|2 − 1 ⎩ 0 if |x| ≥ 1, where C2 > 0 is a constant chosen so that  φ(x) dx = 1. Rn

Under the assumptions (2) and (3), let a (ξ, x) be a regularization of a(ξ, x):   a (ξ, x) = a(ξ − y, x − z)φ(y)φ(z) dydz (0 <  < 1). (8) Rn

Rn

For the ellipticity and growth condition of a (ξ, x), we refer to [33, Lemma 2], [32, Lemma 3.1] and [35, Lemma 2.4]. 5

1

Lemma 3.4. Let s = (s2 + 2 ) 2 . Then for (8), we have  a (ξ, x) is C 1 -regular in ξ ∈ Rn for every x ∈ Rn , a (ξ, x) is C 1 -regular in x ∈ Rn for every ξ ∈ Rn , and

⎧ 2 2 1 2 2 p−1 ⎪ ⎨ |a (ξ, x)| + |Dξ a (ξ, x)|(|ξ| + s ) 2 ≤ C(n, p)Λ(|ξ| + s ) 2 , p−1 |Dx a (ξ, x)| ≤ C(n, p)Λ−1 (|ξ|2 + s2 ) 2 , ⎪ p−2 ⎩ Dξ a (ξ, x)ζ, ζ ≥ C(n, p)λ(|ξ|2 + s2 ) 2 |ζ|2 ,

for every x, ξ, ζ ∈ Rn and for some constant C(n, p) > 0. By a standard approximation argument, one can derive the next lemma. Lemma 3.5. Let U be an open bounded domain in Rn and G ∈ Lp (U, Rn ). Under the assumptions (2) and (3), let w be a weak solution of div a(Dw, x) = div (|G|p−2 G) in U, and w be the weak solution of  div a (Dw , x) = div (|G|p−2 G) in U, on ∂U. w = w Then we have Dw − Dw Lp (U ) → 0 as  0. 4. Higher integrability result for the limiting equations In this section, we derive a local higher integrability result for the limiting equations, which are homogeneous elliptic equations with the nonlinearity a being independent of x variable. Here, we briefly explain the main idea for proving a local higher integrability result in the special case p ≥ 2 and s = 0, but this approach still works for the general case p ∈ (1, ∞) and s > 0 with minor modifications. Under the assumptions of (2) and (3), let w be a weak solution of a limiting equation div a(Dw, x1 ) = 0.

(9)

Since a(ξ, x1 ) has no regularity assumption in x1 variable, |Dw| might not be differentiable in the weak sense. However, from the fact that a(ξ, x1 ) is independent of x variables, one can prove that |Dw|

p−2 2

|DDx w| ∈ L2loc .

Next, let G = |Dx w|p−1 + ν0 |a1 (Dw, x1 )|. Then we show in (29) and Lemma 4.4 that there exists a universal constants ν0 > 0 such that |Dw|p−1  G  |Dw|p−1

and 6

|DG|  |Dw|p−2 |DDx w|.

For a universal constant ν ∈ (0, ν0 ) chosen to be later, we define p−2

Gν = |Dx w| + νG− p−1 |a1 (Dw, x1 )|,

(10)

and prove in Lemma 4.3 and (37) that ν|Dw|  Gν  |Dw|

and

D(|Dx w|) − ν|DDx w|  |DGν |  |DDx w|. (11)

Then by using (11), we show that if αν << 1, then    α+p 2   2  p 2 2 ν Gα+p |Dη|2 dx.  η dx  (α + p) D Gν ν

(12)

So by taking the universal constant ν so that αν << 1, one can prove a local higher integrability result for Gν with Sobolev embedding and a Moser type iteration. Then a local higher integrability result for Dw follows from (11) and (12). To prove (12), we differentiate (9) by xk variable for 2 ≤ k ≤ n, and then 2 test by Dk w Gα ν η , where η is a smooth cut-off function and α ≥ 0, to find that    2 α−1 2 0 = aij (x)|Dw|p−2 Dkj w Dki w Gα η dx ν η + αDk w Di Gν Gν  (13) + 2 aij (x)|Dw|p−2 Dkj w Dk w Gα η D η dx, i ν where aij (x) is uniformly elliptic. In view of (10) and (11), we have η2 αaij (x)|Dw|p−2 Dkj w Dk w Di Gν Gα−1 ν 2  2  α|Dw|p−2 D(|Dx w|) |Dx w|Gα−1 η 2 − αν|Dw|p−2 |DDx w|2 Gα ν νη , and (13) implies that if αν << 1, then  2  2 p−2  |Dw|p−2 |DDx w|2 Gα η 2 dx D(|Dx w|) |Dx w|Gα−1 ν η + α|Dw| ν  2  |Dw|p Gα ν |Dη| dx, which we prove in Lemma 4.5. So we find that (12) follows from (11). 4.1. Higher integrability result for the regularized problems To derive a local higher integrability result, we use an approximation argument. In this subsection, we prove a local higher integrability result for the regularized problems. And in the next subsection, we use an approximation argument to obtain the desired local higher integrability result. For p ∈ (1, ∞), assume that a : Rn × R → Rn satisfies  a(ξ, x1 ) is measurable in x1 for every ξ ∈ Rn , (14) a(ξ, x1 ) is C 1 -regular in ξ for every x1 ∈ R, 7

and



|a(ξ, x1 )| + |Dξ a(ξ, x1 )|(|ξ|2 + s2 ) 2 ≤ Λ(|ξ|2 + s2 )

1

p−1 2

Dξ a(ξ, x1 )ζ, ζ ≥ λ(|ξ|2 + s2 )

p−2 2

,

(15)

|ζ|2 ,

for every ξ, ζ ∈ Rn , x1 ∈ R, and for some constants 0 < λ ≤ Λ and s ≥  > 0. Let w be a weak solution of the following homogeneous equation div a(Dw, x1 ) = 0 in Q3 .

(16)

Now, we define a constant ν0 ∈ (0, 1/4) which will be used in this section: ν0 :=

p − 1  −2p −1 p  λ  · 2 λΛ ∈ 0, . p 4Λ

(17)

p−1

1

)2 In Lemma 4.2, we prove that (|ξ  |2 +s2 ) 2 +ν0 λ−1 (|a1 (ξ, x1 )|2 +Λ2 s2(p−1)  2 2 p−1 1 2 is comparable to (|ξ| +s ) . To do it, we obtain a lower bound for |a (ξ, x1 )|. Lemma 4.1. Under the assumption (14) and (15), we have |a1 (ξ, x1 )| ≥ 2−p λ|ξ1 |p−1 − (2ν0 )−1 λ(|ξ  |2 + s2 )

p−1 2

− Λsp−1 . 

Proof. Let 0 = (0, · · · , 0) ∈ Rn−1 . First, we claim that |a1 (ξ1 , 0 , x1 )| ≥ 2−p+1 λ(|ξ1 |2 + s2 )

p−2 2

|ξ1 | − Λsp−1 . 

(18)

If ξ1 = 0 then (18) holds trivially. Suppose that ξ1 = 0. Then (15) implies a1 (ξ1 , 0 , x1 )ξ1 = a(ξ1 , 0 , x1 ) − a(0, 0 , x1 ), (ξ1 , 0 ) + a(0, 0 , x1 ), (ξ1 , 0 )   1 d [a(tξ1 , 0 , x1 )] dt, (ξ1 , 0 ) + a(0, 0 , x1 ), (ξ1 , 0 ) = 0 dt  1 Dξ a(tξ1 , 0 , x1 )(ξ1 , 0 ), (ξ1 , 0 ) dt + a(0, 0 , x1 ), (ξ1 , 0 ) =  ≥

0

0

1

λ(|tξ1 |2 + s2 )

p−2 2

|ξ1 |2 dt − Λsp−1 |ξ1 |. 

If 1 < p < 2, then we have  1  1 p−2 p−2 p−2 (|tξ1 |2 + s2 ) 2 dt ≥ (|ξ1 |2 + s2 ) 2 dt ≥ 2−p+1 (|ξ1 |2 + s2 ) 2 , 0

0

and if p ≥ 2, then we have  1  p−2 (|tξ1 |2 + s2 ) 2 dt ≥ 0

1 1 2

tp−2 (|ξ1 |2 + s2 )

p−2 2

dt ≥ 2−p+1 (|ξ1 |2 + s2 )

Thus we obtain |a1 (ξ1 , 0 , x1 )ξ1 | ≥ 2−p+1 λ(|ξ1 |2 + s2 ) 8

p−2 2

|ξ1 |2 − Λsp−1 |ξ1 |, 

p−2 2

.

and we find that the claim (18) holds. It follows from (15) that  1     d 1 1 1    a (tξ + (1 − t)(ξ1 , 0 ), x1 ) dt |a (ξ, x1 ) − a (ξ1 , 0 , x1 )| ≤  0 dt  1    Dξ a1 tξ + (1 − t)(ξ1 , 0 ), x1 |(0, ξ  )| dt (19) ≤ 0

≤ Λ|ξ  | If 1 < p < 2, then we have  1  2  2 2 p−2 2 (|ξ1 | + |tξ | + s ) dt ≤ 0

1 0



1 0

(|ξ1 |2 + |tξ  |2 + s2 )

tp−2 (|ξ|2 + s2 )

p−2 2

If p ≥ 2, then we have  1 p−2 p−2 (|ξ1 |2 + |tξ  |2 + s2 ) 2 dt ≤ (|ξ|2 + s2 ) 2 ≤ 0

p−2 2

dt ≤

dt.

p−2 p · (|ξ|2 + s2 ) 2 . p−1

p−2 p · (|ξ|2 + s2 ) 2 . p−1

Thus, we find from (19) that |a1 (ξ, x1 ) − a1 (ξ1 , 0 , x1 )| ≤

p−2 pΛ|ξ  | · (|ξ|2 + s2 ) 2 . p−1

(20)

In view of the triangular inequality, (18) and (20), we have |a1 (ξ, x)| ≥ |a1 (ξ1 , 0 , x1 )| − |a1 (ξ, x) − a1 (ξ1 , 0 , x1 )| ≥ 2−p+1 λ(|ξ1 |2 + s2 ) To estimate the term

p−2 2

|ξ1 | −

(21) p−2 pΛ|ξ  | · (|ξ|2 + s2 ) 2 − Λsp−1 .  p−1

p−2 pΛ|ξ  | · (|ξ|2 + s2 ) 2 , we claim that p−1

p−1 p 2−1 p  2p −1 p pΛ|ξ  | · (|ξ|2 + s2 ) 2 ≤ 2−p λ|ξ1 |p + · 2 Λλ λ(|ξ  |2 + s2 ) 2 . p−1 p−1

(22)

p−1 in Lemma 3.1, we find that 2 p−1 p−1  pΛ|ξ  | pΛ|ξ  |  p−1 p−1 · (|ξ1 |2 + |ξ  |2 + s2 ) 2 ≤ · 2 |ξ1 | + 2p−1 (|ξ  |2 + s2 ) 2 . (23) p−1 p−1

By taking α =

1

We estimate the right-hand side of (23). Let γ = 2−(p−1) λ1− p . Then by Young’s p inequality and γ p−1 ≤ 2−p λ, we have     pΛ|ξ  | p−1 p−1 p · 2 |ξ1 | · γ|ξ1 |p−1 · 2p−1 Λγ −1 |ξ  | = p−1 p−1  p p 1 (24) ≤ γ p−1 |ξ1 |p + · 2p−1 Λγ −1 |ξ  |p p−1  p p · 22p−2 Λλ−1 λ|ξ  |p . ≤ 2−p λ|ξ1 |p + p−1 9

Since p > 1 and 2p−1 ≤ 2p(2p−2) , one can easily check that  p p−1 p p pΛ|ξ  | p−1  2 · 2 (|ξ | + s2 ) 2 ≤ · 22p−2 Λλ−1 λ(|ξ  |2 + s2 ) 2 . p−1 p−1

(25)

Then we discover from (23), (24) and (25) that the claim (22) holds. Next, we 1 divide each side of (22) by (|ξ|2 + s2 ) 2 to find that p−2 p−1 2−1 p  2p −1 p pΛ|ξ  | · (|ξ|2 + s2 ) 2 ≤ 2−p λ|ξ1 |p−1 + · 2 Λλ λ(|ξ  |2 + s2 ) 2 p−1 p−1

= 2−p λ|ξ1 |p−1 + (2ν0 )−1 λ(|ξ  |2 + s2 )

p−1 2

,

and the lemma follows from (21). Next, we find that (|ξ  |2 + s2 ) p−1 parable to (|ξ|2 + s2 ) 2 .

p−1 2

1

+ ν0 λ−1 (|a1 (ξ, x1 )|2 + Λ2 s2(p−1) ) 2 is com-

Lemma 4.2. Under the assumption (14) and (15), we have 2−2p ν0 (|ξ|2 + s2 )

p−1 2

≤ (|ξ  |2 + s2 )

p−1 2

≤ 2(|ξ|2 + s2 )

p−1 2

1

+ ν0 λ−1 (|a1 (ξ, x1 )|2 + Λ2 s2(p−1) )2 

(26)

,

for any ξ ∈ Rn and x1 ∈ R. Proof. The second inequality is clear from (15) and (17). So we only prove the first inequality. Since 4Λν0 < λ and p > 1, we have from Lemma 4.1 that (|ξ  |2 + s2 )

p−1 2

1

+ ν0 λ−1 (|a1 (ξ, x1 )|2 + Λ2 s2(p−1) )2 

≥ (|ξ  |2 + s2 )

p−1 2

+ 2−p ν0 |ξ1 |p−1 − 2−1 (|ξ  |2 + s2 )

p−1 2

− 4−1 sp−1 

p−1 2

≥ 4−1 (|ξ  |2 + s2 ) + 2−p ν0 |ξ1 |p−1   p−1 ≥ 2−p ν0 (|ξ  |2 + s2 ) 2 + |ξ1 |p−1 . By taking α = (|ξ  |2 + s2 )

(27)

p−1 in Lemma 3.1, we have 2

p−1 2

+ |ξ1 |p−1 ≥ 2−

p−1 2

(|ξ  |2 + s2 + |ξ1 |2 )

p−1 2

≥ 2−p (|ξ|2 + s2 )

p−1 2

,

and the first in equality in (26) follows from (27). For a weak solution w of (16), define G : Q2 → R+ and Gν : Q2 → R+ as  p−1 1 G = (|Dx w|2 + s2 ) 2 + ν0 λ−1 (|a1 (Dw, x1 )|2 + Λ2 s2(p−1) )2 ,  (28) p−2 1 1 2. ) Gν = (|Dx w|2 + s2 ) 2 + νλ−1 G− p−1 (|a1 (Dw, x1 )|2 + Λ2 s2(p−1)  Then from Lemma 4.2, we obtain 2−2p ν0 (|Dw|2 + s2 )

p−1 2

≤ G ≤ 2(|Dw|2 + s2 )

p−1 2

Also we have the following lower and upper bound for Gν . 10

in Q2 .

(29)

  2p2 Lemma 4.3. For any ν ∈ 0, 2− p−1 ν02 , we have 1

1

ν(|Dw|2 + s2 ) 2 ≤ CGν ≤ C(|Dw|2 + s2 ) 2

in Q2 .

(30)

Proof. We first prove the second inequality in (30). (15) and (29) imply that 1

(|a1 (Dw, x1 )|2 + Λ2 2(p−1) ) 2 ≤ C(|Dw|2 + s2 )

p−1 2

≤ CG.

Then from (28), (29) and the fact that 0 < ν < ν0 < 1, we obtain p−2

1

1

1

Gν ≤ (|Dx w|2 + s2 ) 2 + Cνλ−1 G− p−1 G ≤ CG p−1 ≤ C(|Dw|2 + s2 ) 2 , which proves the second inequality in (30). Next, we prove the first inequality in (30). If 1 < p < 2, then (29) implies p−2

p−2

G p−1 ≥ 2 p−1 (|Dw|2 + s2 )

p−2 2

2p2

≥ 2− p−1 ν0 (|Dw|2 + s2 )

p−2 2

.

 1  Also if p ≥ 2, then from (29) and the fact that 2−2p ν0 p−1 ≤ 1, we have  p−2 p−1  p−2 p−2 G p−1 ≥ 2−2p ν0 (|Dw|2 + s2 ) 2 p−1 ≥ 2−2p ν0 (|Dw|2 + s2 ) 2 . Thus by considering two cases: 1 < p < 2 and p ≥ 2, we find that p−2

2p2

G p−1 ≥ 2− p−1 ν0 (|Dw|2 + s2 )

p−2 2

.

(31)

2p2

Since 0 < ν ≤ 2− p−1 ν02 , we discover from (28), (29) and (31) that p−2

p−2

1

1

G p−1 Gν = G p−1 (|Dx w|2 + s2 ) 2 + νλ−1 (|a1 (Dw, x1 )|2 + Λ2 s2(p−1) )2  2p2

p−1

1

≥ 2− p−1 ν0 (|Dx w|2 + s2 ) 2 + νλ−1 [|a1 (Dw, x1 )|2 + Λ2 s2(p−1) ]2   p−1 1 ≥ νν0−1 (|Dx w|2 + s2 ) 2 + ν0 λ−1 (|a1 (Dw, x1 )|2 + Λ2 s2(p−1) )2  ≥ νν0−1 G 1

p−2

≥ Cνν0−1 (|Dw|2 + s2 ) 2 G p−1 in Q2 . Since ν0 is a universal constant, the first inequality in (30) follows. Next, we assume that Dw ∈ L∞ (Q2 )

and

DDx w ∈ L2 (Q2 ).

(32)

Then we have the following lemma. Lemma 4.4. Under the assumption (32), if w is a weak solution of (16) then     D1 [a1 (Dw, x1 )] + Dx [a(Dw, x1 )] + |DG| ≤ C(|Dw|2 + s2 ) p−2 2 |DD  w| in Q . x 2 11

Proof. By a direct calculation and (32), for 1 < k ≤ n we have Dk [a(Dw, x1 )] = Dξj a(Dw, x1 )Dkj w

in Q2 .

Then for 1 < k ≤ n, we have from (15) that |Dk [a(Dw, x1 )]| ≤ |Dξj a(Dw, x1 )Dkj w| ≤ C(|Dw|2 + s2 )

p−2 2

|DDx w|

in Q2 . Since w is a weak solution of (16), we have the weak derivative  Di [ai (Dw, x1 )] in Q2 . D1 [a1 (Dw, x1 )] = − 1
In light of (33) and (34), we discover that   D1 [a1 (Dw, x1 )] ≤ C(|Dw|2 + s2 ) p−2 2 |DD  w| x

in Q2 .

(33)

(34)

(35)

By a direct calculation, we find that  p−1 1 )2 DG = D (|Dx w|2 + s2 ) 2 + ν0 λ−1 (|a1 (Dw, x1 )|2 + Λ2 s2(p−1)   p−3 = (p − 1)(|Dx w|2 + s2 ) 2 Dl wDDl w 1
+

ν0 a1 (Dw, x1 )D[a1 (Dw, x1 )] 2(p−1)

λ(|a1 (Dw, x1 )|2 + Λ2 s

1

)2

in Q2 . Thus (33) and (35) imply that |DG| ≤ C(|Dw|2 + s2 )

p−2 2

|DDx w|

in Q2 .

(36)

Then the lemma follows from (33), (35) and (36). By a direct calculation, we have Di Gν = Ai1 + νAi2



where Ai1 and

2

= Di [(|Dx w| +

1 s2 ) 2 ]

=

(37)

in Q2 , 1
Dli wDl w 1

(|Dx w|2 + s2 ) 2

 p−2 1 )2 . Ai2 = Di λ−1 G− p−1 (|a1 (Dw, x1 )|2 + Λ2 s2(p−1) 

Then from (29) and Lemma 4.4, we have |Ai1 | + |Ai2 | ≤ C|DDx w|

(38)   2p2 Lemma 4.5. There exists a small constant ν = ν(n, p, q, λ, Λ) ∈ 0, 2− p−1 ν02 such that under the assumption (32), if w is a weak solution of (16) then for any η ∈ Cc∞ (Q2 ) and α ∈ [0, 12q] we have   α+p 2 2 p−2 2 |DD  w|2 dx ≤ C η 2 Gα (|Dw| + s ) (|Dw|2 + s2 ) 2 |Dη|2 dx. x ν  Q2

in Q2 .

Q2

12

p−2

Proof. Recall from (32) that (|Dw|2 + s2 ) 2 |DDx w|2 ∈ L1 (Q2 ) with s > 0. Let 1 < k ≤ n. We test (16) by Dk ϕ ∈ Cc∞ (Q2 ), and then use integration by parts to find that   i 0= a (Dw, x1 )Di Dk ϕ dx = − Dξj ai (Dw, x1 )Dkj w Di ϕ dx. (39) Q2

Q2

2 In view of Lemma 4.4, we test (39) by ϕ = Dk w Gα ν η to find that    2 0= dx. Dξj ai (Dw, x1 )Dkj w Di Dk w Gα νη

(40)

Q2

One can directly check that   2 2 α−1 = Dki w Gα Di Gν η 2 + 2Dk w Gα Di Dk w Gα νη ν η + αDk w Gν ν ηDi η. (41) In light of (37) and (38), we find that Di Gν = Ai1 + νAi2 in Q2

and

|Ai1 | + |Ai2 | ≤ C|DDx w| in Q2 ,

(42)

where   p−2 1 1 )2 . Ai1 = Di (|Dx w|2 +s2 ) 2 and Ai2 = Di λ−1 G− p−1 (|a1 (Dw, x1 )|2 +Λ2 s2(p−1)  Since 1 < k ≤ n is an arbitrary constant, (40), (41) and (42) imply that I1 + I2 + I3 + I4 = 0, where

(43)

⎧   i ⎪ I1 = η 2 Gα ⎪ ν Dξj a (Dw, x1 ) Dkj w Dki w dx, ⎪ ⎪ Q ⎪ 2 1
Q2

We first estimate I1 . From (15), we have  2 2 p−2 2 |DD  w|2 dx. I1 ≥ λ η 2 Gα x ν (|Dw| + s ) Q2

(44)

To estimate I2 , we write   p−2  1 2 1 I = αλ η 2 Gα−1 (|Dw|2 + s2 ) 2 D (|Dx w|2 + s2 ) 2  (|Dx w|2 + s2 ) 2 dx. ν Q2

13

Then we use (15) to discover that  Dξj ai (Dw, x1 ) Dkj w Dk w Ai1 1
  1 1 1 = Dξj ai (Dw, x1 )Dj (|Dx w|2 + s2 ) 2 Di (|Dx w|2 + s2 ) 2 (|Dx w|2 + s2 ) 2  p−2  1 2 1 ≥ λ(|Dw|2 + s2 ) 2 D (|Dx w|2 + s2 ) 2  (|Dx w|2 + s2 ) 2 ,

which implies I2 ≥ I.

(45)

Now, we estimate I3 . We find from (42) and Young’s inequality that      Dkj wDk w|Ai2 | ν  1
 1 1  ≤ Cν(|Dx w|2 + s2 ) 2 D (|Dx w|2 + s2 ) 2 |DDx w|     1 1 2 ≤ (|Dx w|2 + s2 ) 2 λ(nΛ)−1 D (|Dx w|2 + s2 ) 2  + Cν 2 |DDx w|2 .

Since α ∈ [0, 12q], it follows from (15) and (28) that  p−2 1 2 |I3 | ≤ I + Cαν η 2 Gα−1 (|Dw|2 + s2 ) 2 (|Dx w|2 + s2 ) 2 |DDx w|2 dx ν Q2  2 2 2 p−2 2 |DD  w|2 dx, ≤ I + 12C3 qν η 2 Gα x ν (|Dw| + s ) Q2

for some constant C3 = C3 (n, p, q, λ, Λ). Then we choose a small constant ν = ν(n, p, q, λ, Λ) > 0 such that   2p2 and 48C3 qν 2 ≤ λ. ν ∈ 0, 2− p−1 ν02 So it follows that |I3 | ≤ I +

λ 4

 Q2

2 2 η 2 Gα ν (|Dw| + s )

p−2 2

|DDx w|2 dx.

(46)

It only remains to estimate I4 . From (15) and Young’s inequality, we have  1 2 2 p−2 2 (|Dw|2 + s2 ) 2 |DD  w||Dη| dx |I4 | ≤ C |η|Gα x ν (|Dw| + s )  Q  2 λ 2 2 p−2 2 |DD  w|2 dx ≤ η 2 Gα (47) x ν (|Dw| + s ) 4 Q2  2 2 p−2 2 (|Dw|2 + s2 )|Dη|2 dx. +C Gα ν (|Dw| + s )  Q2

Now, we combine (43), (44), (45), (46) and (47) to find that   2 2 p−2 2 2 p 2 2 |DD  w|2 dx ≤ C 2 η 2 Gα (|Dw| + s ) Gα x ν  ν (|Dw| + s ) |Dη| dx, Q2

Q2

and we discover that the lemma holds. 14

Let 2∗ be the Sobolev conjugate  2n ∗ 2 = n−2 4 We set χ =

n > 2, n = 2.

2∗ > 1. Then we have a reverse H¨older type inequality. 2

Lemma 4.6. Under the assumption (32), if w is a weak solution of (16) and α ∈ [0, 12q], then    1 (|Dw|2 + s2 ) 12  (α+p)χ ≤ C (|Dw|2 + s2 ) 2 Lα+p (Q2 ) . (48) L (Q1 ) Proof. Let ν = ν(n, p, q, λ, Λ) > 0 be the constant in Lemma 4.5. We choose a cut-off function η ∈ Cc∞ (Q2 ) such that 0 ≤ η ≤ 1,

η = 1 in Q1 ,

|Dη| ≤ C.

Then from Lemma 4.5, we have   2 2 p−2 2 |DD  w|2 dx ≤ C η 2 Gα (|Dw| + s ) x ν 

(|Dw|2 + s2 )

Q2

Q2

α+p 2

|Dη|2 dx. (49)

In light of (37), (38) and Lemma 4.3, we have   α+p 2  α + p 2     · DGν η + Gν Dη  Gα+p−2 D Gν 2 η  =  ν 2 2 2 α+p−2 ≤ Cη |DGν | Gν + C|Dη|2 Gνα+p 2 2 ≤ Cη 2 Gα ν (|Dw| + s )

p−2 2

|DDx w|2 + C(|Dw|2 + s2 )

α+p 2

|Dη|2

in Q2 . Thus we use (49) and the Sobolev embedding to find that 

   α+p 2∗ 22∗ 2  2    α+p ≤C Gν η  dx D Gν 2 η  dx Q2 Q  2 α+p (|Dw|2 + s2 ) 2 |Dη|2 dx. ≤C

(50)

Q2

From Lemma 4.3 and the fact that χ =

2∗ , we have 2 2∗

χ

ν χ (|Dw|2 + s2 ) 2 ≤ CGχν = CGν2 . Thus we discover from (50) that ν

α+p



2

(|Dw| + Q1

s2 )

(α+p)χ 2

χ1 dx

 ≤C

Q2

(|Dw|2 + s2 )

Since ν = ν(n, p, q, λ, Λ) ∈ (0, 1), we find that the lemma holds. 15

α+p 2

dx.

Remark 4.7. In the proof of Lemma 4.5, the small constant ν is chosen so that 48C3 qν 2 ≤ λ. Thus the constant C in (48) might blow up as q  ∞. Now, we have the following conclusion of this subsection. Lemma 4.8. Under the assumption (32), if w is a weak solution of (16), then    1 (|Dw|2 + s2 ) 12  p+q ≤ C (|Dw|2 + s2 ) 2 Lp (Q2 ) . L (Q1 ) Proof. Let ν = ν(n, p, q, λ, Λ) be the constant in Lemma 4.5. Since χ > 1, there exists a integer m0 = m0 (n, p, q) ≥ 0 such that pχm0 < (p + q)χ ≤ pχm0 +1 < (p + q)χ2 .

(51)

m

By taking α = pχ − p ≤ 12q for m = 0, · · · , m0 , we find from Lemma 4.6 that    1 (|Dw|2 + s2 ) 12  pχm+1 ≤ C (|Dw|2 + s2 ) 2 Lpχm (Q2 ) (52) L (Q1 ) for m = 0, · · · , m0 . The estimate (52) is invariant under scaling, and we obtain     2 2 12   (|Dw|2 + s2 ) 12  pχm+1 L (Q −m ) ≤ C (|Dw| + s ) Lpχm (Q −m+1 ) 2

2

for any m = 0, · · · , m0 , which implies     2 2 12   (|Dw|2 + s2 ) 12  pχm0 +1 L (Q2−m0 ) ≤ C (|Dw| + s ) Lp (Q2 ) , and so it follows from (51) that   (|Dw|2 + s2 ) 12  p+q L (Q

2−m0

  2 2 12   ) ≤ C (|Dw| + s ) Lp (Q2 ) .

(53)

The estimate (53) is invariant under scaling and translation, and from the fact that m0 = m0 (n, p, q), we find that the lemma holds. Remark 4.9. To show Theorem 2.1, we only need to prove Dw ∈ Lq+ (Q1 ) for some fixed  > 0. But we proved that Dw ∈ Lp+q (Q1 ) for the simplicity of the calculation in the proof of Theorem 2.1. 4.2. Higher integrability result for the limiting equations Assume that a : Rn × R → Rn satisfies  a(ξ, x1 ) is measurable in x1 for every ξ ∈ Rn , a(ξ, x1 ) is C 1 -regular in ξ for every x1 ∈ R, and



|a(ξ, x1 )| + |Dξ a(ξ, x1 )|(|ξ|2 + s2 ) 2 ≤ Λ(|ξ|2 + s2 )

1

p−1 2

Dξ a(ξ, x1 )ζ, ζ ≥ λ(|ξ|2 + s2 )

p−2 2

, |ζ|2 ,

for every ξ, ζ ∈ Rn , x1 ∈ R, and for some constants 0 < λ ≤ Λ, s > 0 if 1 < p < 2 and s ≥ 0 if p ≥ 2. Let w be a weak solution of the following homogeneous equation div a(Dw, x1 ) = 0 in Q3 . Then we have the following lemma. 16

(54)

Lemma 4.10. Let w be a weak solution of (54). Then we have 1

Dw Lp+q (Q1 ) ≤ C (|Dw|2 + s2 ) 2 Lp (Q3 ) , Proof. Without loss of generality, we may denote a(ξ, x) = a(ξ, x1 , x ) := a(ξ, x1 ). Then w is a weak solution of div a(Dw, x) = 0 in Q3 . By using the mollifier (7), let a (ξ, x) be a regularization of a(ξ, x):   a(ξ − y, x − z)φ(y)φ(z) dydz (0 <  < 1). a (ξ, x) = Rn

(55)

Rn

1

Then for s = (s2 + 2 ) 2 , we have from Lemma 3.4 that  a (ξ, x) is C 1 -regular in ξ ∈ Rn for every x ∈ Rn ,

(56)

a (ξ, x) is C 1 -regular in x ∈ Rn for every ξ ∈ Rn , and

⎧ p−1 1 ⎪ |a (ξ, x)| + |Dξ a (ξ, x)|(|ξ|2 + s2 ) 2 ≤ C(n, p)Λ(|ξ|2 + s2 ) 2 , ⎪ ⎨  |Dx a (ξ, x)| ≤ C(n, p)Λ−1 (|ξ|2 + s2 )

⎪ ⎪ ⎩

Dξ a (ξ, x)ζ, ζ ≥ C(n, p)λ(|ξ|2 + s2 )

p−2 2

p−1 2

,

|ζ|2 ,

for every x, ξ, ζ ∈ Rn and for some constant C(n, p) > 0. Now, let w be the weak solution of  div a (Dw , x) = 0 in Q3 , w = w on ∂Q3 .

(57)

By the classical literature for the interior Lipschitz regularity of nonlinear elliptic equations, see for instance [21, 29, 43, 42], we find from (56) - (57) that Dw ∈ L∞ (Q2 ).

(58)

Since a(ξ, x) = a(ξ, x1 ), we find from (55) that a (ξ, x) = a (ξ, x1 ), and one can use the difference quotient method, see for instance [14, 34], to find (|Dw |2 + s2 )

p−2 4

DDx w ∈ L2 (Q2 )

⇒

DDx w ∈ L2 (Q2 ).

We apply (58) - (59) and Lemma 4.8 to w in (57) to discover that    1 (|Dw |2 + s2 ) 12  p+q ≤ C (|Dw |2 + s2 ) 2 Lp (Q3 ) . L (Q1 ) 17

(59)

(60)

In view of Lemma 3.5 and (60), we discover that 1

1

lim sup (|Dw |2 + s2 ) 2 Lp+q (Q1 ) ≤ C lim sup (|Dw |2 + s2 ) 2 Lp (Q3 ) 0

0

1

(61)

≤ C (|Dw|2 + s2 ) 2 Lp (Q3 ) , and we have a subsequence {j }∞ j=1 such that j 0 as j → ∞ and 

Dwj  Dw0 in Lp+q (Q1 ), Dw0 Lp+q (Q1 ) ≤ lim inf Dwj Lp+q (Q1 ) .

(62)

j→∞

Since strong convergence implies weak convergence, the uniqueness of weak limit, Lemma 3.5 and (62) imply that Dw0 = Dw a.e. in Q1 and  Dwj  Dw in Lp+q (Q1 ), (63) Dw Lp+q (Q1 ) ≤ lim inf Dwj Lp+q (Q1 ) . j→∞

Thus from (61) and (63), we see that 1

Dw Lp+q (Q1 ) ≤ C (|Dw|2 + s2 ) 2 Lp (Q3 ) , which completes the proof. 5. Comparison Estimates In this section, we obtain comparison estimates. We measure the oscillation of the nonlinearity a(ξ, x1 , x ) being averaged in x variables as in (5). So even though the nonlinearity a(ξ, x1 , x ) has no regularity assumption on x1 , one may follow the standard argument for obtaining comparison estimates to nonlinear elliptic equations with p-growth condition, see for instance [12]. Suppose that F ∈ Lp (Q2r , Rn ). Under the assumptions (2) and (3), let u be a weak solution of div a(Du, x1 , x ) = div (|F |p−2 F ) in Q2r , v be the weak solution of  div a(Dv, x1 , x ) = 0 in Q2r , v = u on ∂Q2r , and w be the weak solution of  div a ¯(Dw, x1 ) = 0 in Qr , w = v on ∂Qr , where

 a ¯(ξ, x1 ) = − a(ξ, x1 , z  ) dz  . Qr

18

(64)

(65)

(66)

To compare v and w, we use the well-known local higher integrability result, see for instance [49, Lemma 3.3] and the proof of [29, Lemma 3.5], which is 1  p+σ  0 p+σ0 2 2 2 − (|Dv| + s ) dx ≤C −

Qr

2

Q2r

2

p1

p 2

(|Dv| + s ) dx

,

(67)

for some constant σ0 = σ0 (n, p, λ, Λ) > 0. We next state the comparison estimate for 1 < p < 2. Lemma 5.1. Suppose that 1 < p < 2. Let κ0 = κ0 (n, p, λ, Λ) ∈ (0, 1) and σ0 = σ0 (n, p, λ, Λ) > 0 be the constants in Lemma 3.2 and (67), respectively. For any κ ∈ (0, κ0 ), if  p2 (p+σ0 ) − |θ(a, Qr )| dx ≤ κ σ0 (p−1) , (68) Qr

then for the weak solutions u and w of (64) and (66), we have     p(p−2)  p − |Du − Dw|p dx ≤ Cκp − (|Du|2 + s2 ) 2 dx + C 1 + κ p−1 − Qr

Q2r

Q2r

|F |p dx.

Proof. We test (64) and (65) by u − v to discover that   − a(Du, x) − a(Dv, x), Du − Dv dx = − |F |p−2 F, Du − Dv dx. Q2r

Q2r

Then we use (3) and Young’s inequality to find that  p−2 C1 − (|Du|2 + |Dv|2 + s2 ) 2 |Du − Dv|2 dx Q2r 2−p

≤

κ

4

 −

p

Q2r

|Du − Dv| + Cκ

p−2 p−1

 −

Q2r

|F |p dx,

where C1 is the constant in Lemma 3.2. In view of Lemma 3.2, we have p

|Du − Dv|p ≤ Cκp (|Du|2 + s2 ) 2 + C1 κp−2 (|Du|2 + |Dv|2 + s2 ) for any κ ∈ (0, κ0 ), and obtain   p p − |Du − Dv| dx ≤ Cκ − Q2r

Q2r

2

2 p

(|Du| + s ) dx + Cκ

p(p−2) p−1

 −

p−2 2

Q2r

|F |p dx, (69)

for any κ ∈ (0, κ0 ). Next, test (65) and (66) by v − w to find that  a(Dv, x1 ) − a ¯(Dw, x1 ), Dv − Dw dx − ¯ Qr  a(Dv, x1 ) − a(Dv, x), Dv − Dw dx. = − ¯ Qr

19

|Du − Dv|2 ,

Then (3), (5) and Young’s inequality imply that  p−2 C1 κp−2 − (|Dv|2 + |Dw|2 + s2 ) 2 |Dv − Dw|2 dx Qr  p−1 p−2 ≤ Cκ − θ(a, Qr )(|Dv|2 + s2 ) 2 |Dv − Dw| dx (70) Qr   p(p−2) p p 1 ≤ − |Dv − Dw|p dx + Cκ p−1 − |θ(a, Qr )| p−1 (|Dv|2 + s2 ) 2 dx. 4 Qr Qr Since 0 < κ < 1, we have from (6) and (68) that   p(p+σ0 ) p2 (p+σ0 ) p(p+σ0 ) − |θ(a, Qr )| σ0 (p−1) dx ≤ C − |θ(a, Qr )| dx ≤ Cκ σ0 (p−1) ≤ Cκ σ0 (p−1) . Qr

Qr

Then we discover from H¨ older’s inequality and (67) that  p(p−2) p p κ p−1 − |θ(a, Qr )| p−1 (|Dv|2 + s2 ) 2 dx Qr

≤ Cκ

p(p−2) p−1

 ≤ Cκ − p

Q2r

σ0   p p+σ p+σ  p(p+σ0 ) 0 0 p+σ0 − |θ(a, Qr )| σ0 (p−1) dx − (|Dv|2 + s2 ) 2 dx

Qr

Qr

2

2

p 2

(|Dv| + s ) dx,

(71)

In view of Lemma 3.2, we have   p − |Dv − Dw|p dx ≤ Cκp − (|Dv|2 + s2 ) 2 dx Qr Qr  p−2 + C1 κp−2 − (|Dv|2 + |Dw|2 + s2 ) 2 |Dv − Dw|2 dx, Qr

for any κ ∈ (0, κ0 ). Thus we discover from (70) and (71) that   p − |Dv − Dw|p dx ≤ Cκp − (|Dv|2 + s2 ) 2 dx Qr Q  2r p ≤ Cκp − (|Du|2 + s2 ) 2 + |Du − Dv|p dx,

(72)

Q2r

for any κ ∈ (0, κ0 ). Since 0 < κ < 1, we have from (69) and (72) that   − |Du − Dw|p dx ≤ C − |Du − Dv|p + |Dv − Dw|p dx Qr Qr   p ≤ Cκp − (|Du|2 + s2 ) 2 dx + C − |Du − Dv|p dx Q Q2r  2r  p(p−2) p ≤ Cκp − (|Du|2 + s2 ) 2 dx + Cκ p−1 − |F |p dx, Q2r

Q2r

for any κ ∈ (0, κ0 ). 20

We also have the comparison estimate for p ≥ 2. Lemma 5.2. Let p ≥ 2 and σ0 = σ0 (n, p, λ, Λ) > 0 be the constant in (67). For any κ ∈ (0, 1), if  p2 (p+σ0 ) (73) − |θ(a, Qr )| dx ≤ κ σ0 (p−1) , Qr

then for the weak solutions u and w of (64) and (66), we have     p(p−2)  − |Du − Dw|p dx ≤ Cκp − (|Du|2 + s2 )p dx + C 1 + κ p−1 − Qr

Q2r

Q2r

|F |p dx.

Proof. Test (64) and (65) by u − v to find that   − a(Du, x) − a(Dv, x), Du − Dv dx = − |F |p−2 F, Du − Dv dx. Q2r

Q2r

We have from (3) and Young’s inequality that   p−2 − (|Du|2 + |Dv|2 + s2 ) 2 |Du − Dv|2 dx ≤ C − Q2r

Q2r

|F |p dx.

(74)

We then test (65) and (66) by v − w to discover that  a(Dv, x1 ) − a ¯(Dw, x1 ), Dv − Dw dx − ¯ Qr  = − ¯ a(Dv, x1 ) − a(Dv, x), Dv − Dw dx. Qr

It follows from (3), (5) and Young’s inequality that  − |Dv − Dw|p dx Qr  p−2 ≤ C − (|Dv|2 + |Dw|2 + s2 ) 2 |Dv − Dw|2 dx Q  r p−1 ≤ C − |θ(a, Qr )|(|Dv|2 + s2 ) 2 |Dv − Dw| dx Q  r  p p 1 p ≤ − |Dv − Dw| dx + C − |θ(a, Qr )| p−1 (|Dv|2 + s2 ) 2 dx. 4 Qr Qr We then discover from H¨older’s inequality that  − |Dv − Dw|p dx Qr  p p ≤ C − |θ(a, Qr )| p−1 (|Dv|2 + s2 ) 2 dx Qr

(75)

σ0   p p+σ p+σ  p(p+σ0 ) 0 0 p+σ0 . − (|Dv|2 + s2 ) 2 dx ≤ C − |θ(a, Qr )| σ0 (p−1) dx

Qr

Qr

21

Since 0 < κ < 1, we derive from (6) and (73) that   p(p+σ0 ) p2 (p+σ0 ) p(p+σ0 ) − |θ(a, Qr )| σ0 (p−1) dx ≤ C − |θ(a, Qr )| dx ≤ Cκ σ0 (p−1) ≤ Cκ σ0 . (76) Qr

Qr

By combining (67), (75) and (76), we obtain   p − |Dv − Dw|p dx ≤ Cκp − (|Dv|2 + s2 ) 2 dx Qr Q  2r  p ≤ Cκp − (|Du|2 + s2 ) 2 dx + Cκp − Q2r

Q2r

|Du − Dv|p dx.

Then from the triangular inequality and the fact that 0 < κ < 1, we have   p − |Du − Dw| dx ≤ C − |Du − Dv|p + |Dv − Dw|p dx Qr Qr   p 2 2 p 2 ≤ Cκ − (|Du| + s ) dx + C − |Du − Dv|p dx, Q2r

Q2r

and the lemma follows from (74). 6. Proof of the main theorem We have a local higher integrability result for the limiting equations from Section 4, and the comparison estimates from Section 5. So we may follow the approach in [4, 5, 30, 50] to obtain the desired Calder´ on-Zygmund type estimate. We start this section with the following elementary lemma. Lemma 6.1. Let X, Y ∈ Rn and |X| ≥ κ for some κ > 0. Then we have |X|p ≤ C|X − Y |p + C(α)κp−α |Y |α

(α ≥ p).

Proof. From the triangular inequality, we have |X|p ≤ C|X − Y |p + C|Y |p .

(77)

If |Y | ≤ |X − Y |, then the lemma holds from (77). If |X − Y | ≤ |Y |, then κ ≤ |X| ≤ |X − Y | + |Y | ≤ 2|Y |. Since α ≥ p, we have |Y |p ≤ C(α)κp−α |Y |α , and (77) implies the lemma. We choose any R ≤ r1 < r2 ≤ 2R. We assume that Q2R ⊂⊂ Ω and n   p |F |p 60R − (|Du|2 + s2 ) 2 + p dx = γ0p , r2 − r1 τ Q2R

22

(78)

where τ ∈ (0, 1) is a constant chosen to be later. Then we have    p |F |p r2 − r1 y ∈ Q r1 , ≤ρ≤R . (|Du|2 + s2 ) 2 + p dx ≤ γ0p − τ 30 Qρ (y)

(79)

From (79), we also find that s ≤ γ0 .

(80)

Let σ0 = σ0 (n, p, λ, Λ) > 0 and κ0 = κ0 (n, p, λ, Λ) ∈ (0, 1) be the constants 1

in (67) and Lemma 3.2 respectively. Then we write τ0 = κ0p−1 ∈ (0, 1). Lemma 6.2. For any τ ∈ (0, τ0 ), suppose that  p2 (p+σ0 ) θ(a, Qρ (y))(x) dx ≤ τ σ0 . sup sup − 0<ρ≤R y∈Q2R

(81)

Qρ (y)

Then for any weak solution u of (4), γ ≥ γ0 and N > 1, we have  |Du|p dx Qr1 ∩{|Du|>N γ}

≤ C(τ

p−1

+N

−q

 )

p

Qr2 ∩{|Du|> γ2 }

|Du| dx +

 Qr2 ∩{|F |> τ2γ }

Proof. We define a set   E(γ) = x ∈ Qr1 : |Du|p + τ −p |F |p > γ p

|F |p dx . τp

(γ ≥ γ0 ).

We assume that E(γ) = ∅. Otherwise, we have {x ∈ Qr1 : |Du| > N γ} = ∅, and the lemma holds trivially.  r −r  2 1 Fix a point y ∈ E(γ). From (79), there exists ρy ∈ 0, such that 30 ⎧ |F |p ⎪ p p ⎪ ⎪ ⎨ −Q (y) |Du| + τ p dx = γ , ρy (82)  ⎪ |F |p ⎪ p p ⎪ |Du| + p dx ≤ γ (ρy < ρ ≤ R). ⎩− τ Qρ (y) From the next inequality, 1 |Qρy (y)| = p γ ≤

 Qρy (y)

|Du|p +

|Qρy (y)| 1 + p 2p γ



|F |p dx τp

Qρy (y)∩{|Du|> γ2

|Qρy (y)| 1 + p + p 2 γ

23



}

|Du|p dx

Qρy (y)∩{|F |> τ2γ }

|F |p dx, τp

we find that |Qρy (y)| ≤

C γp

 Qρy (y)∩{|Du|> γ2 }

|Du|p dx +

C γp

 Qρy (y)∩{|F |> τ2γ }

|F |p dx. (83) τp

Next, we obtain a comparison estimate. Let v be the weak solution of  div a(Dv, x1 , x ) = 0 in Q30ρy (y), v = u on ∂Q30ρy (y), and w be the weak solution of  div a ¯(Dw, x1 ) = 0 in Q15ρy (y), w = v on ∂Q15ρy (y),  a ¯(ξ, x1 ) = −

where

Qr (y  )

a(ξ, x1 , z  ) dz  .

1

Since 0 < τ < τ0 < 1 and τ0 = κ0p−1 , take κ = τ p−1 in (81), Lemma 5.1 and Lemma 5.2 to find that     p p |Du − Dw| dx ≤ C − τ p−1 (|Du|2 + s2 ) 2 + 1 + τ p(p−2) |F |p dx. − Q15ρy (y)

Q30ρy (y)

Since 30ρy ≤ r2 − r1 ≤ R and 0 < τ < 1, we have from (80) and (82) that  − |Du − Dw|p dx ≤ Cτ p−1 (γ p + sp ) + Cτ p γ p ≤ Cτ p−1 γ p . (84) Q15ρy (y)

Since 0 < τ < 1, we discover from Lemma 3.1, (80), (82) and (84) that   p 2 2 p 2 − (|Dw| + s ) dx ≤ C − (|Du|2 + s2 ) 2 + |Du − Dw|p dx ≤ Cγ p . Q15ρy (y)

Q15ρy (y)

It follows from Lemma 4.10 that 1 p+q   |Dw|p+q dx ≤C − − Q5ρy (y)

p

Q15ρy (y)

(|Dw|2 + s2 ) 2 dx

p1

≤ Cγ.

(85)

In view of Lemma 6.1, we have |Du(x)| ≥ N γ

⇒

|Du(x)|p ≤ C|Du(x) − Dw(x)|p + C(N γ)−q |Dw(x)|p+q ,

and we discover from (84) and (85) that  |Du|p dx Q5ρy (y)∩{|Du|>N γ}



≤C

Q5ρy (y)∩{|Du|>N γ}

|Du − Dw|p + (N γ)−q |Dw|p+q dx

≤ C(τ p−1 + N −q )γ p |Qρy (y)|. 24

(86)

Since y ∈ E(γ) was an arbitrary point in E(γ), there exists a Vitali covering {Q5ρm (ym )}∞ m=1 of {x ∈ Qr1 : |Du(x)| > N γ} ⊂ E(γ) such that ym ∈ E(γ) and

{Qρm (ym )}∞ m=1 are mutually disjoint.

(87)

Since 30ρm ≤ r2 − r1 , we have ∪∞ m=1 Qρm (ym ) ⊂ Qr2 . Thus we discover from (83), (86) and (87) that  |Du|p dx Qr1 ∩{|Du|>N γ}

≤

∞  

|Du|p dx

Q5ρm (ym )∩{|Du|>N γ}

m=1

≤ C(τ p−1 + N −q )γ p

∞ 

|Qρm (ym )|

m=1

≤ C(τ

p−1

+N

−q

 )



p

Qr2 ∩{|Du|> γ2 }

|Du| dx +

Qr2 ∩{|F |> τ2γ }

|F |p dx . τp

This completes the proof. We are now ready to prove Theorem 2.1. p2 (p+σ0 )

Proof. Take δ = τ σ0 , where τ ∈ (0, τ0 ) is a universal constant chosen to be later. Also let N > 1 be a universal constant chosen to be later. We claim that for any k ≥ γ0 and γ ≥ γ0 , we have  |Du|p dx Qr1 ∩{min{|Du(x)|,k}>N γ}

≤ C(τ

p−1

+N

−q



)

Qr2 ∩{min{|Du(x)|,k}> γ2 }  p

+ C(τ p−1 + N −q )

Qr2 ∩{|F |> τ2γ

}

|Du|p dx

(88)

|F | dx. τp

If k ≤ N γ, then we have Qr1 ∩ {min{|Du(x)|, k} > N γ} = ∅, and (88) holds trivially. So suppose that k > N γ > γ. Then we have 2|Du(x)| > γ

⇐⇒

which implies  Qr2 ∩{min{|Du(x)|,k}> γ2 }

2 min{|Du(x)|, k} > γ,

|Du|p dx =

 Qr2 ∩{|Du|> γ2 }

|Du|p dx.

Thus from Lemma 6.2 and the next inclusion     Qr1 ∩ min{|Du(x)|, k} > N γ ⊂ Qr1 ∩ |Du(x)| > N γ , 25

we discover that the claim (88) holds. In view of (88), we have  ∞  γ q−p−1 |Du|p dxdγ Qr1 ∩{min{|Du|,k}>N γ}   ∞ q−p−1

γ0

  ≤ C τ p−1 + N −q

γ

γ0

  + C τ p−1 + N −q



Qr2 ∩{min{|Du|,k}> γ2

∞

γ q−p−1

γ0



Qr2 ∩{|F |> τ2γ

We use Fubini’s theorem to find that  ∞  q−p q−p−1 N γ

Qr1 ∩{min{|Du|,k}>N γ}

γ0 ∞

 

N γ0

= Qr1

|Du|

∞

γ q−p−1

γ0



= Qr2



∞ γ0

=τ

−p

|Du|p



q−p−1

−1



γ0



Qr2



Qr2 ∩{|F |> τ2γ }

(q − p)

(90)

dγdx  − (N γ0 )q−p dx,

|Du|p dxdγ

γ q−p−1 dγdx

(91)

 q−p  |Du|p 2 min{|Du|, k} − γ0q−p dx,



|F |

}

|F | dxdγ. τp

|Du|p dxdγ

q−p

2 min{|Du|,k}

(89)

|Du|p dxdγ

min{|Du|, k}

Qr2 ∩{min{|Du|,k}> γ2

Qr2 −p

γ

N γ0

Qr1



γ q−p−1

=τ

min{|Du|,k}

|Du|p

= (q − p) and

Qr1 ∩{min{|Du|,k}>γ}



p



= (q − p)−1 



γ q−p−1

=

}

} p

|Du|p dxdγ

p

−1





|F |p dxdγ τp

2τ −1 |F |

γ0

Qr2

γ q−p−1 dγdx

(92)

  |F |p (2τ −1 |F |)q−p − γ0q−p dx.

By combining (89), (90), (91) and (92), we discover that   q−p |Du|p min{|Du|, k} dx Qr1



≤C τ

p−1

+ Cτ

N

 −q

q−p

+N

−p



 Qr2

τ p−1 N q−p + N −p

 q−p |Du|p min{|Du|, k} dx 

 Qr2

26

|F |q dx +

 Qr1

|Du|p (N γ0 )q−p dx.

By taking the constant N = N (n, p, q, λ, Λ) > 1 sufficiently large and the corresponding constant τ = τ (n, p, q, λ, Λ) ∈ (0, τ0 ) sufficiently small, we have    q−p 1 |Du|p (min{|Du|, k})q−p dx ≤ |Du|p min{|Du|, k} dx 2 Qr2 Qr1   + Cγ0q−p |Du|p dx + C |F |q dx. Q2R

Q2R

Since R ≤ r1 < r2 ≤ 2R was arbitrary, we find from Lemma 3.3 and (78) that  −

QR

p

|Du| (min{|Du|, k})

q−p

 dx ≤ C −

Q2R

2

2

p 2

(|Du| + s ) dx

pq

 +C−

Q2R

|F |q dx.

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