Higher integrability near the initial boundary for nonhomogeneous parabolic systems of 𝑝-Laplacian type

Sun-Sig Byun; Wontae Kim; Minkyu Lim

doi:10.1515/forum-2020-0068

Artikel Open Access

Higher integrability near the initial boundary for nonhomogeneous parabolic systems of 𝑝-Laplacian type

Sun-Sig Byun , Wontae Kim und Minkyu Lim

Veröffentlicht/Copyright: 6. August 2020

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Aus der Zeitschrift Forum Mathematicum Band 32 Heft 6

Abstract

We establish a sharp higher integrability near the initial boundary for a weak solution to the following p-Laplacian type system:

{ut-div⁡𝒜⁢(x,t,∇⁡u)=div|F|p-2F+fin⁢ΩT,u=u0on⁢Ω×{0},

by proving that, for given δ∈(0,1), there exists ε>0 depending on δ and the structural data such that

|∇⁡u0|p+ε∈Lloc1⁢(Ω) and |F|p+ε,|f|(δ⁢p⁢(n+2)n)′+ε∈L1⁢(0,T;Lloc1⁢(Ω))⟹|∇⁡u|p+ε∈L1⁢(0,T;Lloc1⁢(Ω)).

Our regularity results complement established higher regularity theories near the initial boundary for such a nonhomogeneous problem with f≢0 and we provide an optimal regularity theory in the literature.

Keywords: Higher integrability; initial boundary data; non-divergence data

MSC 2010: 35K20; 35K92

1 Introduction

In this paper, we are interested in finding a sharp higher integrability near the initial boundary to a weak solution to the parabolic system

{ut-div⁡𝒜⁢(x,t,∇⁡u)=div|F|p-2F+fin⁢ΩT,u=u0on⁢Ω×{0}.

Here, 2⁢nn+2<p<∞, Ω is a bounded domain in ℝn with n≥2, 𝒜⁢(x,t,ζ) is modeled after the p-Laplacian operator, u0∈W1,p⁢(Ω,ℝN), F∈Lp⁢(ΩT;ℝN⁢n) and f∈Lq′⁢(ΩT,ℝN) for some N≥1, where q=p⁢(n+2)n is the parabolic Sobolev conjugate of p and q′ is Hölder conjugate of q.

In the case f≡0, interior higher integrability results were proved by Kinnunen and Lewis in [13, 14] by providing a suitable application of DiBenedetto’s intrinsic geometry method from [9] to the setting of Gehring type estimates. Higher integrability results near the initial and lateral boundary were proved by Parviainen in [19, 20]. These regularity results were extended to higher-order systems by Bögelein and Parviainen in [5].

On the other hand, in the case f≢0, a nonlinear relation between the gradient of a weak solution and the non-divergence data f coming from parabolic embeddings naturally occurs. To be specific, there is an exponent α>1 such that ∥∇⁡u∥Lpp is related to ∥f∥Lq′q′⁢α. Indeed, there have been regularity estimates coming from such a nonlinear relation rather than Gehring type estimates. In particular, Kuusi and Mingione in [15] proved gradient L∞ regularity and gradient continuity of a weak solution with an nonlinear exponent α based on the intrinsic geometry method. The recent paper [2] provides a new representation of the nonlinear relation. It replaces the exponent α by 1 from the intrinsic geometry method, obtaining interior higher integrability results for a system of p⁢(x,t)-Laplacian type with scaling invariant estimates and extending gradient continuity results in [15] to the p⁢(x,t)-Laplacian system. For the elliptic case, there is a divergence representation for the non-divergence data, and Gehring type estimates directly follow from [10, 17, 11].

As already mentioned, the main purpose of this paper is to establish higher integrability results near the initial boundary. In addition, we are considering a class of the nonlinearities whose structures are associated with the divergence data F. Regarding the non-divergence data, f∈L(δ⁢p⁢(n+2)n)′ with δ∈(0,1) is sharp in obtaining the reverse Hölder inequality by the parabolic Sobolev embedding theorem. A noteworthy feature of the present paper is to find the optimality and sharpness of the initial data u0 from the intrinsic geometry method. Due to a suitable application of the Poincaré inequality in a type of the Caccioppoli inequality, ∥∇⁡u0∥Ls⁢(Bρ)2 appears for some s>1. The optimal exponent s can be found in the process of the use of the Fubini theorem. In fact, (⨍⨍|∇⁡u0|s⁢d⁢z)ps appears after using the reverse Hölder inequality, and the Minkowski integral inequality enforces the condition s=p. On the other hand, there is an alternative for the non-divergence data f to avoid applying the Minkowski integral inequality, thanks to the presence of radius ρ (see (4.3) below). Therefore, p is the optimal exponent for s, and we can find the sharp exponent β=min⁡{2,p} in ∥∇⁡u0∥Lpβ from the stopping time argument.

Our work can be applicable to different problems including obstacle problem and system of p⁢(x,t)-Laplacian type problem as in [6, 4] as well as Calderón–Zygmund type estimates as in [1, 3, 7, 8, 16] both when f≢0 and when ∇⁡u0≢0.

The paper is organized as follows. In Section 2, we introduce basic notation and definitions to state our main results. Section 3 is devoted to proving reverse Hölder inequality. Finally, in Section 4, we prove our main results.

2 Basic notations and results

2.1 Notations

We shall clarify all the notations that will be used in this paper.

We use ∇ to denote derivatives with respect the space variable x and ∂t to denote the time derivative.
In what follows, we always assume the bounds 2⁢nn+2<p<∞.
Let z0=(x0,t0)∈ℝn+1 be a point, ρ,s>0 two given parameters, and let λ∈[1,∞). We use the following notations:
Is(t0):=(t0-s2,t0+s2)⊂ℝ,Qρ,s(z0):=Bρ(x0)×Is(t0)⊂ℝn+1,
Isλ(t0):=(t0-λ2-ps2,t0+λ2-ps2)⊂ℝ,Qρ,sλ(z0):=Bρ(x0)×Isλ(t0)⊂ℝn+1,
Qρλ(z0):=Qρ,ρλ(z0),Qρ(z0):=Qρ1(z0).
We use ∫ to denote the integral with respect to either space variable or time variable and use ∬ to denote the integral with respect to both space and time variables simultaneously.
Analogously, we use ⨍ and ⨍⨍ to denote the integral averages as defined below: for any set A×B⊂ℝn×ℝ, we define
(f)A:=⨍Af(x)dx=1|A|∫Af(x)dx,(f)A×B:=⨍⨍A×Bf(x,t)dxdt=1|A×B|∬A×Bf(x,t)dxdt.
We use the notation ≲(a,b,…) to denote an inequality with a constant depending on a,b,….

Definition 2.1.

Let Ω be a bounded domain in ℝn with n≥2 and u0∈L2⁢(Ω,ℝN), f∈L(p⁢(n+2)n)′⁢(ΩT;ℝN) for some N≥1. A weak solution u∈C⁢(0,T;L2⁢(Ω,ℝN))∩Lp⁢(0,T;W1,p⁢(Ω,ℝN)) to

(2.1){ut-div⁡𝒜⁢(x,t,∇⁡u)=fin⁢ΩT,u=u0on⁢Ω×{0}

is a distributional energy solution in the sense

∬ΩT-u⁢ϕt+〈𝒜⁢(x,t,∇⁡u),∇⁡ϕ〉⁢d⁢z=∬ΩTf⁢ϕ⁢d⁢z for all⁢ϕ∈C0∞⁢(ΩT,ℝN),

and

(2.2)limh→0+⁡⨍0h∫Ω|u⁢(x,t)-u0⁢(x)|2⁢d⁢x⁢d⁢t=0.

2.2 Structures of the operator

We now describe the assumptions on the nonlinear structures in (2.1). Assume 𝒜⁢(x,t,∇⁡u) is a Carathéodory function, i.e., we have that (x,t)↦𝒜⁢(x,t,ζ) is measurable for every ζ∈ℝn and ζ↦𝒜⁢(x,t,ζ) is continuous for almost every (x,t)∈ΩT.

We further assume that, for a.e. (x,t)∈ΩT and for any ζ∈ℝn, there exist two positive constants Λ0 and Λ1 such that the following bounds are satisfied by the nonlinear structures:

(2.3)〈𝒜⁢(x,t,ζ),ζ〉≥Λ0⁢|ζ|p-|F⁢(x,t)|p and |𝒜⁢(x,t,ζ)|≤Λ1⁢|ζ|p-1+|F⁢(x,t)|p-1,

where F∈Lp⁢(ΩT,ℝN⁢n).

2.3 Main results

Before stating our main theorem, we fix some constants which will be frequently used in this paper.

Definition 2.2.

For fixed constants

max{nn+2,n+1(n+2)⁢p}<δ<1 and d:={p2if⁢p≥2,2⁢pp⁢(n+2)-2⁢nif⁢p<2,

we denote

𝔞:=δ⁢p⁢(n+2)n,ν:=𝔞′(min⁡{2,p}2-p𝔞) and α:=(1-dpν)-1.

Remark 2.3.

(i) 0<pd⁢α holds, and (ii) 𝔞 is a parabolic Sobolev conjugate of δ⁢p.

To see Remark 2.3 (i), we split two cases.

Case p≥2: There hold pd⁢α=pd-𝔞′⁢(1-p𝔞) and
pd-𝔞′⁢(1-p𝔞)=2-𝔞′⁢(1-p𝔞)>0⇔2𝔞′-(1-p𝔞)>0⇔2-2𝔞>1-p𝔞⇔1>1𝔞⁢(2-p).
Case p<2: There hold pd⁢α=pd-𝔞′⁢(p2-p𝔞) and
0<pd⁢α=p⁢(n+2)-2⁢n2-𝔞′⁢(p2-p𝔞)⇔0<n⁢(p-2)𝔞′+p⇔2⁢n-p⁢n-p<2⁢n-p⁢n𝔞.
Recall 𝔞=δ⁢p⁢(n+2)n. Since δ∈(0,1) and
(n+2)⁢p⁢[2⁢n-(n+1)⁢p]≤n2⁢(2-p)⇔((n+2)⁢p-2⁢n)⁢((n+1)⁢p-n)≥0,
the remark follows.

To apply the intrinsic geometry method developed in [13], let us define the following notations.

Definition 2.4.

Let u∈C⁢(0,T;L2⁢(Ω,ℝN))∩Lp⁢(0,T;W1,p⁢(Ω,ℝN)) be a weak solution of (2.1) under the assumption of (2.2), (2.3) and f∈L𝔞′⁢(ΩT,ℝN). For any (x0,t0)=z0∈ΩT and B2⁢r⁢(x0)⊂Ω, we define

λ0pd:=⨍⨍Q2⁢r⁢(z0)∩ΩT(|∇u|+|F|+1)pdz+(⨍⨍Q2⁢r⁢(z0)∩ΩT(2r)𝔞′|f|𝔞′dz)α+(⨍B2⁢r⁢(x0)|∇u0|pdx)min⁡{1,2p}.

Now, we state the main theorem.

Theorem 2.5.

Let u∈C⁢(0,T;L2⁢(Ω,RN))∩Lp⁢(0,T;W1,p⁢(Ω,RN)) be a weak solution of (2.1) under the assumption of (2.2), (2.3) and f∈La′⁢(ΩT,RN). Then there exists ε0⁢(n,N,p,Λ0,Λ1,δ) such that, for any ε∈(0,ε0) and for any (x0,t0)=z0∈ΩT such that B2⁢r⁢(x0)⊂Ω, there holds

⨍⨍Qr⁢(z0)∩ΩT|∇⁡u|p+ε⁢d⁢z≲(n,N,p,Λ0,Λ1,δ)λ0ε⁢⨍⨍Q2⁢r⁢(z0)∩ΩT|∇⁡u|p⁢d⁢z+⨍⨍Q2⁢r⁢(z0)∩ΩT(|F|+1)p+ε⁢d⁢z+⨍B2⁢r⁢(x0)|∇⁡u0|p+ε⁢d⁢x+λ0ν⁢(1+εp)⁢⨍⨍Q2⁢r⁢(z0)∩ΩT(2⁢r⁢|f|)𝔞′+𝔞′p⁢ε⁢d⁢z.

Here, λ0 is defined in Definition 2.4.

Remark 2.6.

As a consequence of Theorem 2.5, we can also obtain global estimates of the weak solution to

{ut-div⁡𝒜⁢(x,t,∇⁡u)=fin⁢ΩT,u=ϕon⁢∂⁡Ω×(0,T),u=u0on⁢Ω×{0},

where 𝒜, f and u0 are assumed as in (2.1) and ϕ∈Lp⁢(0,T;W1,p⁢(ΩT,ℝN)) such that ϕt∈L(p⁢(n+2)n)′⁢(ΩT,ℝN). Especially, ∇⁡ϕ behaves like divergence data F does while ϕt behaves exactly in the same way as non-divergence data f does.

Before ending this section, we provide some important lemmas which will be used later in the proof of the main theorem. Let us state Gagliardo–Nirenberg’s inequality (see [18]).

Lemma 2.7.

Let Bρ⁢(x0)⊂Rn with 0<ρ≤1, σ,q,r∈[1,∞) and ϑ∈(0,1) such that -nσ≤ϑ⁢(1-nq)-(1-ϑ)⁢nr. Then, for any u∈W1,q⁢(Bρ⁢(x0)), there holds

⨍Bρ⁢(x0)|u|σρσ⁢d⁢x≲(n,σ,q)(⨍Bρ⁢(x0)|u|qρq+|∇⁡u|q⁢d⁢x)ϑ⁢σq⁢(⨍Bρ⁢(x0)|u|rρr⁢d⁢x)(1-ϑ)⁢σr.

The following iteration lemma can be found in [12, Lemma 6.1].

Lemma 2.8.

Let 0<r<R<∞ be given, and let h:[r,R]→R be a non-negative and bounded function. Furthermore, let θ∈(0,1) and A,B,γ1,γ2≥0 be fixed constants, and suppose that

h⁢(ρ1)≤θ⁢h⁢(ρ2)+A(ρ2-ρ1)γ1+B(ρ2-ρ1)γ2,

holds for all r≤ρ1<ρ2≤R. Then the following conclusion holds:

h⁢(r)≲(θ,γ1,γ2)A(R-r)γ1+B(R-r)γ2.

3 Estimates near the initial boundary

In this section, we assume that B4⁢ρ⁢(x0)⊂Ω, 0<t0 and λ≥1. Also, note that, in the case 0∈I4⁢ρλ⁢(t0), for any ρ≤ρ1≤ρ2≤4⁢ρ, there holds

(3.1)|Iρ1λ⁢(t0)|≤|Iρ2λ⁢(t0)|≤42⁢|Iρ1λ⁢(t0)| and 12⁢|Iρ1λ⁢(t0)|≤|Iρ1λ⁢(t0)∩(0,T)|≤|Iρ1λ⁢(t0)|.

We will show a reverse Hölder inequality in intrinsic cylinders under the following assumptions.

Assumption 3.1.

We assume

(3.2)⨍⨍Q4⁢ρλ⁢(z0)∩ΩT(|∇⁡u|+|F|+1)p⁢d⁢z+λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q4⁢ρλ⁢(z0)∩ΩT(4⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z+⨍B4⁢ρ⁢(x0)|∇⁡u0|p⁢d⁢x≤λp,

(3.3)⨍⨍Qρλ⁢(z0)∩ΩT(|∇⁡u|+|F|+1)p⁢d⁢z+λ𝔞′⁢(1-p𝔞)⁢⨍⨍Qρλ⁢(z0)∩ΩTρ𝔞′⁢|f|𝔞′⁢d⁢z+⨍Bρ⁢(x0)|∇⁡u0|p⁢d⁢x≥λp,

Note that, by a choice of δ∈(0,1) in Definition 2.2, 0<𝔞′⁢(1-p𝔞)<p holds, and this exponent 𝔞′⁢(1-p𝔞) is chosen to preserve a nonlinear relation between the gradient of a weak solution and the non-divergence data. Indeed, using Young’s inequality, (3.3) becomes

⨍⨍Qρλ⁢(z0)∩ΩT(|∇⁡u|+|F|+1)p⁢d⁢z+(⨍⨍Qρλ⁢(z0)∩ΩTρ𝔞′⁢|f|𝔞′⁢d⁢z)(1-𝔞′p⁢(1-p𝔞))-1+⨍Bρ⁢(x0)|∇⁡u0|p⁢d⁢x≳λp.

3.1 Caccioppoli inequality and Poincaré type inequality

Let us first state a Caccioppoli type inequality.

Lemma 3.2.

Let u be a weak solution of (2.1). Suppose B4⁢ρ⁢(x0)⊂Ω and 0<t0. Then, for any ρ≤ρa<ρb≤4⁢ρ, there holds

Proof.

The proof basically follows from [5, Lemma 5.1]. For the sake of completeness, we present the details in our setting.

In the space direction, take a cut-off function η satisfying

0≤η=η⁢(x)∈Cc∞⁢(Bρb⁢(x0)),η≡1⁢on⁢Bρa⁢(x0),|∇⁡η|≤cρb-ρa.

For the time direction, we divide two cases. In the case that 0∉I4⁢ρλ⁢(t0), consider a cut-off function ζ satisfying

0≤ζ=ζ⁢(t)∈Cc∞⁢(Iρbλ⁢(t0)),ζ≡1⁢on⁢Iρaλ⁢(t0),|∂t⁡ζ|≤cλ2-p⁢(ρb-ρa)2.

On the other hand, in the case that 0∈I4⁢ρλ⁢(t0), consider a cut-off function ζ satisfying

0≤ζ=ζ⁢(t)∈Cc∞⁢(Iρbλ⁢(t0)),ζ≡1⁢on⁢ 0≤t≤t0+λ2-p⁢ρa2,|∂t⁡ζ|≤cλ2-p⁢(ρb-ρa)2.

Taking (u-(u0)Bρb⁢(x0))⁢ηp⁢ζ2 as a test function in (2.1), we obtain

I+II:=⨍⨍Qρbλ⁢(z0)∩ΩT∂tu(u-(u0)Bρb⁢(x0))ηpζ2dz+⨍⨍Qρbλ⁢(z0)∩ΩT〈𝒜(z,∇u),∇u〉ηpζ2dz≲⨍⨍Qρbλ⁢(z0)∩ΩT|𝒜⁢(z,∇⁡u)|⁢|∇⁡η|⁢|u-(u0)Bρb⁢(x0)|⁢ηp-1⁢ζ2⁢d⁢z+⨍⨍Qρbλ⁢(z0)∩ΩT|f|⁢|u-(u0)Bρb⁢(x0)|⁢ηp⁢ζ2⁢d⁢z=:III+IV.

Estimate of I: We use integration by parts and (2.2) to find that

Estimate of I1: Using the triangle inequality along with the fact that ρ≤ρa≤ρb≤4⁢ρb and (3.1), we obtain

1|Iρaλ⁢(t0)|⁢supt∈Iρaλ⁢(t0)⁡⨍Bρa⁢(x0)|u-(u0)Bρa⁢(x0)|2⁢d⁢x≲I1.

Estimate of I2: Applying Poincaré’s inequality, we have

Estimate of I3: We get

I3≲λp-2⁢⨍⨍Qρbλ⁢(z0)∩ΩT[|u-(u0)Bρb⁢(x0)|ρb-ρa]2⁢d⁢z.

Therefore, we obtain

Estimate of II: Applying (2.3), we discover

II≳⨍⨍Qρbλ⁢(z0)∩ΩT|∇⁡u|p⁢ηp⁢ζ2⁢d⁢z-⨍⨍Qρbλ⁢(z0)∩ΩT|F|p⁢d⁢z.

Estimate of III:

III≲(2.3)⁢⨍⨍Qρbλ⁢(z0)∩ΩT(|∇⁡u|p-1+|F|p-1)⁢[|u-(u0)Bρb⁢(x0)|ρb-ρa]⁢ηp-1⁢ζ2⁢d⁢z≤(a)⁢γ⁢⨍⨍Qρbλ⁢(z0)∩ΩT|∇⁡u|p⁢ηp⁢ζ2⁢d⁢z+C⁢(γ)⁢⨍⨍Qρbλ⁢(z0)∩ΩT[|u-(u0)Bρb⁢(x0)|ρb-ρa]p+|F|p⁢d⁢z.

Here, to obtain (a), we used Young’s inequality with γ∈(0,1).

Estimate of IV: Clearly, there holds

IV≤⨍⨍Qρbλ⁢(z0)∩ΩT|f|⁢|u-(u0)Bρb⁢(x0)|⁢d⁢z.

Combining all the above calculations and taking γ=γ⁢(n,N,p,Λ0,Λ1) small enough, the conclusion follows. ∎

In our definition of a weak solution to (2.1), there is no differentiability assumption on u with respect to time. Therefore, we cannot apply Poincaré’s inequality directly. Nevertheless, we shall use (2.1) to estimate continuity of u with respect to time.

Lemma 3.3.

Let u be a weak solution of (2.1). Suppose that Br⁢(x0)⊂Ω and 0<t0. Then, for all θ∈[1,p], there holds

⨍⨍Qrλ⁢(z0)∩ΩT[|u-(u0)Br⁢(x0)|r]θ⁢d⁢z≲(n,N,p,Λ1)⨍⨍Qrλ⁢(z0)∩ΩT|∇⁡u|θ⁢d⁢z+(λ2-p⁢⨍⨍Qrλ⁢(z0)∩ΩT|∇⁡u|p-1+|F|p-1⁢d⁢z)θ+λθ⁢(λ1-p⁢r⁢⨍⨍Qrλ⁢(z0)∩ΩT|f|⁢d⁢z)θ+(⨍Br⁢(x0)|∇⁡u0|⁢d⁢x)θ.

Proof.

Let 0≤η∈Cc∞⁢(Br⁢(x0)) such that ⨍Br⁢(x0)η⁢d⁢z=1 and ∥η∥∞+r⁢∥∇⁡η∥∞≤C⁢(n). For a.e. t, we denote weighted integral averages of u by

(u)η(t):=⨍Br⁢(x0)u(x,t)η(x)dx.

The triangle inequality gives

Estimate of I: Applying Poincaré’s inequality in space direction, we obtain

Estimate of II: Similarly, we have

II≲r-θ⁢⨍⨍Qrλ⁢(z0)∩ΩT|u-(u)Br⁢(x0)⁢(t)|θ⁢d⁢z≲⨍⨍Qrλ⁢(z0)∩ΩT|∇⁡u|θ⁢d⁢z.

Estimate of III: Again, the triangle inequality implies

To estimate the second term, we test η to (2.1) in Br⁢(x0)×(t1,t2)⊂Qrλ⁢(z0)∩ΩT. Then (2.3) implies that

|(u)η⁢(t1)-(u)η⁢(t2)|≲(Λ1)∫t1t2⨍Br⁢(x0)|∇⁡u|p-1⁢|∇⁡η|+|F|p-1⁢|∇⁡η|+|f|⁢|η|⁢d⁢x⁢d⁢t≲(3.1)⁢λ2-p⁢r⁢⨍⨍Qrλ⁢(z0)∩ΩT|∇⁡u|p-1+|F|p-1⁢d⁢z+λ2-p⁢r2⁢⨍⨍Qrλ⁢(z0)∩ΩT|f|⁢d⁢z.

Using (2.2), we obtain

supt∈Irλ⁢(t0)⁡|(u)η⁢(t)-(u0)η|≲λ2-p⁢r⁢⨍⨍Qrλ⁢(z0)∩ΩT|∇⁡u|p-1+|F|p-1⁢d⁢z+λ2-p⁢r2⁢⨍⨍Qrλ⁢(z0)∩ΩT|f|⁢d⁢z.

Therefore, we get

III≲(n,N,p,Λ1)⨍⨍Qrλ⁢(z0)∩ΩT|∇⁡u|θ⁢d⁢z+(λ2-p⁢⨍⨍Qrλ⁢(z0)∩ΩT|∇⁡u|p-1+|F|p-1⁢d⁢z)θ+λθ⁢(λ1-p⁢r⁢⨍⨍Qrλ⁢(z0)∩ΩT|f|⁢d⁢z)θ+(⨍Br⁢(x0)|∇⁡u0|⁢d⁢x)θ.

This completes the proof. ∎

3.2 Some crucial estimates

The purpose of this subsection is to refine the estimate u in C(Irλ(t0)∩(0,T);L2(Br(x0),ℝN). To this end, we first estimate the right-hand side of the inequality in Lemma 3.2 using Lemma 3.3 and assumption (3.2), and then return to the left-hand side of the inequality.

Lemma 3.4.

Let u be a weak solution of (2.1). Suppose B4⁢ρ⁢(x0)⊂Ω and 0<t0. Also, assume (3.2) for some λ≥1. Then, for all θ∈[1,p], there holds

⨍⨍Q4⁢ρλ⁢(z0)∩ΩT[|u-(u0)B4⁢ρ⁢(x0)|4⁢ρ]θ⁢d⁢z≲(n,N,p,Λ0,Λ1)λθ.

Proof.

Applying Lemma 3.3 and Hölder’s inequality, we find

⨍⨍Q4⁢ρλ⁢(z0)∩ΩT[|u-(u0)B4⁢ρ⁢(x0)|4⁢ρ]θ⁢d⁢z≲(⨍⨍Q4⁢ρλ⁢(z0)∩ΩT|∇⁡u|p⁢d⁢z)θp+[λ2-p⁢(⨍⨍Q4⁢ρλ⁢(z0)∩ΩT|∇⁡u|p+|F|p⁢d⁢z)p-1p]θ+λθ⁢(λ𝔞′⁢(1-p)⁢⨍⨍Q4⁢ρλ⁢(z0)∩ΩT(4⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z)θ𝔞′+(⨍B4⁢ρ⁢(x0)|∇⁡u0|p⁢d⁢x)θp.

Note that

(3.4)λ𝔞′⁢(1-p)⁢⨍⨍Q4⁢ρλ⁢(z0)∩ΩT(4⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z=λ𝔞′⁢(1-p𝔞)-p⁢⨍⨍Q4⁢ρλ⁢(z0)∩ΩT(4⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z⁢≤(3.2)⁢1.

Therefore, making use of (3.2), this completes the proof. ∎

Lemma 3.5.

Under the assumptions and the conclusion in Lemma 3.4, we further have

supI2⁢ρλ⁢(t0)∩(0,T)⁡⨍B2⁢ρ⁢(x0)[|u-(u0)B2⁢ρ⁢(x0)|2⁢ρ]2⁢d⁢x≲(n,N,p,Λ0,Λ1)λ2.

Proof.

Take 2⁢ρ≤ρa<ρb≤4⁢ρ in Lemma 3.2 to get

Estimate of I: Note that, under the restriction 2⁢ρ≤ρa<ρb≤4⁢ρ, we see that

Here, to obtain (a), we used Lemma 3.4 and Poincaré’s inequality with (3.2) for initial data.

Estimate of II: There holds that

where, to obtain (a), we used Sobolev’s inequality with respect to space direction and then Hölder’s inequality with respect to the time integral, and to obtain (b), we used (3.5) and (3.2).

Applying Young’s inequality, for any γ∈(0,1), there holds

II≲γ⁢(λp-2⁢supIρbλ⁢(t0)∩(0,T)⁡⨍Bρb⁢(x0)[|u-(u0)Bρb⁢(x0)|ρb]2⁢d⁢x)+C⁢(γ)⁢(ρbρb-ρa)4⁢λp.

Estimate of III: After applying Hölder’s inequality, we get

Here, to obtain (a), we used Lemma 2.7 with σ=𝔞, q=δ⁢p, r=2 and ϑ=nn+2. To obtain (b), we used Hölder’s inequality with respect to the time integral.

Also, the restriction ρ≤ρb≤4⁢ρ and (3.1) imply

(⨍⨍Qρbλ⁢(z0)∩ΩTρb𝔞′⁢|f|𝔞′⁢d⁢z)1𝔞′≲(⨍⨍Q4⁢ρλ⁢(z0)∩ΩT(4⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z)1𝔞′⁢≤(3.2)⁢λp𝔞′-p⁢(1p-1𝔞)=λp⁢(1𝔞′+1𝔞)-1=λp-1.

Thus, along with (3.5) and (3.2), (3.6) becomes

III≲λp-1+nn+2⁢(supIρbλ⁢(t0)∩(0,T)⁡⨍Bρbλ⁢(x0)[|u-(u0)Bρb⁢(x0)|ρb]2⁢d⁢x)1n+2=λp⁢(1-1n+2)⁢(λp-2⁢supIρbλ⁢(t0)∩(0,T)⁡⨍Bρbλ⁢(x0)[|u-(u0)Bρb⁢(x0)|ρb]2⁢d⁢x)1n+2,

and Young’s inequality gives that

III≲γ⁢(λp-2⁢supIρbλ⁢(t0)∩(0,T)⁡⨍Bρ2λ⁢(x0)[|u-(u0)Bρb⁢(x0)|ρb]2⁢d⁢x)+C⁢(γ)⁢λp,

where γ∈(0,1).

Estimate of IV and V:

IV+V⁢≲(3.2)⁢λp.

Combining estimates and taking γ=γ⁢(n,N,p,Λ0,Λ1)∈(0,1) small enough, there holds

supIρaλ⁢(t0)∩(0,T)⁡⨍Bρa⁢(x0)[|u-(u0)Bρa⁢(x0)|ρa]2⁢d⁢x≤12⁢supIρbλ⁢(t0)∩(0,T)⁡⨍Bρbλ⁢(x0)[|u-(u0)Bρb⁢(x0)|ρb]2⁢d⁢x+C⁢(ρbρb-ρa)4⁢λ2+C⁢(ρbρb-ρa)p⁢λ2,

where C=C⁢(n,N,p,Λ0,Λ1). Hence, from Lemma 2.8, the conclusion follows. ∎

We will not use following corollary in further estimates for our purpose, but it is worthwhile observing the estimate of the corollary.

Corollary 3.6.

Under the assumptions and the conclusion in Lemma 3.5, we derive

⨍⨍Q2⁢ρλ⁢(z0)∩ΩT[|u-(u0)B2⁢ρ⁢(x0)|2⁢ρ]p⁢(n+2)n⁢d⁢x≲(n,N,p,Λ0,Λ1)λp⁢(n+2)n.

Proof.

Constants σ=p⁢(n+2)n, q=p, r=2 and ϑ=nn+2∈(0,1) satisfy the condition in Lemma 2.7. Therefore, we obtain

where, to obtain (a), we used Lemma 3.4, (3.2), (3.5) and Lemma 3.5. ∎

3.3 Reverse Hölder inequality in intrinsic cylinders

We are now ready to obtain a reverse Hölder inequality under assumptions (3.2) and (3.3). We shall again estimate the right-hand side of Lemma 3.2 with ρa=ρ and ρb=2⁢ρ using Lemma 3.5. Let us fix a constant

(3.7)q:=max{2⁢nn+2,p-1,n⁢pn+2,δp}.

Lemma 3.7.

Let u be a weak solution of (2.1). Suppose B4⁢ρ⁢(x0)⊂Ω and 0<t0. Also, assume (3.2) for some λ≥1. Then, for any γ∈(0,1), there holds

λp-2⁢⨍⨍Q2⁢ρλ⁢(z0)[|u-(u0)B2⁢ρ⁢(x0)|2⁢ρ]2⁢d⁢z≲(n,N,p,Λ0,Λ1,δ)γ⁢λp+C⁢(γ)⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q⁢d⁢z)pq+C⁢(γ)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|F|p⁢d⁢z+C⁢(γ)⁢λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z+C⁢(γ)⁢⨍B2⁢ρ⁢(x0)|∇⁡u0|p⁢d⁢x.

Proof.

We apply Lemma 2.7 with σ=2, q as defined in (3.7), r=2, and any ϑ∈(0,1) is admissible since

-n2≤ϑ⁢(1-nq)-(1-ϑ)⁢n2⟸2⁢nn+2≤q.

Let us take ϑ∈(0,1) such that

0<ϑ<min⁡{q2,p2⁢(p-1),12}.

Then we get

where, to obtain (a), we used Hölder’s inequality along with the fact that 2⁢ϑq≤1, and to obtain (b), we used Lemma 3.5.

Thus, applying Lemma 3.3 with θ=q to estimate the second term, we obtain

λp-2⁢⨍⨍Q2⁢ρλ⁢(z0)[|u-(u0)B2⁢ρ⁢(x0)|2⁢ρ]2⁢d⁢z≲λp-2⁢ϑ⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q⁢d⁢z)2⁢ϑq+λp-2⁢ϑ⁢(λ2-p⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|p-1+|F|p-1⁢d⁢z)2⁢ϑ+λp⁢(λ1-p⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)⁢|f|⁢d⁢z)2⁢ϑ+λp-2⁢ϑ⁢(⨍B2⁢ρ⁢(x0)|∇⁡u0|⁢d⁢x)2⁢ϑ=:I+II+III+IV.

Estimate of I: Applying Young’s inequality along with the fact 2⁢ϑp<2⁢ϑq≤1, we get

I≤γ⁢λp+C⁢(γ)⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q⁢d⁢z)pq.

Estimate of II: Similarly, since 2⁢ϑ⁢(p-1)p<1, Hölder’s inequality and Young’s inequality give

II≤λp-2⁢ϑ⁢(p-1)⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q+|F|q⁢d⁢z)pq⁢2⁢ϑ⁢(p-1)p

≤γ⁢λp+C⁢(γ)⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q⁢d⁢z)pq+C⁢(γ)⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|F|q⁢d⁢z)pq

≤γ⁢λp+C⁢(γ)⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q⁢d⁢z)pq+C⁢(γ)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|F|p⁢d⁢z.

Estimate of III: Since 2⁢ϑ𝔞′<1, Young’s inequality gives

III≤λp⁢(λ𝔞′⁢(1-p)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z)2⁢ϑ𝔞′=λp⁢(1-2⁢θ𝔞′)⁢(λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z)2⁢ϑ𝔞′≤γ⁢λp+C⁢(γ)⁢λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z.

Estimate of IV: Since 2⁢ϑp<1, Hölder’s inequality and Young’s inequality give

IV≤λp-2⁢ϑ⁢(⨍B2⁢ρ⁢(x0)|∇⁡u0|p⁢d⁢x)2⁢ϑp≤γ⁢λp+C⁢(γ)⁢⨍B2⁢ρ⁢(x0)|∇⁡u0|p⁢d⁢x.

Therefore, combining all the estimates, the proof is completed. ∎

Lemma 3.8.

Under the assumptions and the conclusion in Lemma 3.7, we further have

⨍⨍Q2⁢ρλ⁢(z0)[|u-(u0)B2⁢ρ⁢(x0)|2⁢ρ]p⁢d⁢z≲(n,N,p,Λ0,Λ1,δ)γ⁢λp+C⁢(γ)⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q⁢d⁢z)pq+C⁢(γ)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|F|p⁢d⁢z+C⁢(γ)⁢λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z+C⁢(γ)⁢⨍B2⁢ρ⁢(x0)|∇⁡u0|p⁢d⁢x.

Proof.

We apply Lemma 2.7 with σ=p, q as defined in (3.7), r=2 and ϑ=qp. Note that (3.7) implies

n⁢pn+2≤q⇔-np≤qp⁢(1-nq)-(1-qp)⁢n2.

Therefore, we have

Here, to obtain (a), we used p⁢ϑq=1, and to obtain (b), we used Lemma 3.5. Now, we apply Lemma 3.3 with θ=q to the first term on the right-hand side to get

⨍⨍Q2⁢ρλ⁢(z0)[|u-(u0)B2⁢ρ⁢(x0)|2⁢ρ]p⁢d⁢z≲λp-q⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q⁢d⁢z+λp-q⁢(λ2-p⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|p-1+|F|p-1⁢d⁢z)q+λp⁢(λ1-p⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)⁢|f|⁢d⁢z)q+λp-q⁢(⨍B2⁢ρ⁢(x0)|∇⁡u0|⁢d⁢x)q=:I+II+III+IV.

Estimate of I: Applying Young’s inequality, there holds

I≤γ⁢λp+C⁢(γ)⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q⁢d⁢z)pq.

Estimate of II: Applying Hölder’s inequality and Young’s inequality, we get

II≤λp+q-p⁢q⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q+|F|q⁢d⁢z)pq⁢q⁢(p-1)p≲λp+q-p⁢q⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q+|F|q⁢d⁢z)pq⁢p-1p⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|p+|F|p⁢d⁢z)(q-1)⁢(p-1)p≤(3.2)⁢λ⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q+|F|q⁢d⁢z)pq⁢p-1p≤γ⁢λp+C⁢(γ)⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q⁢d⁢z)pq+C⁢(γ)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|F|p⁢d⁢z.

Estimate of III: We observe

III≤λp⁢(λ𝔞′⁢(1-p)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z)q𝔞′=λp⁢(λ𝔞′⁢(1-p)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z)1𝔞′⁢(λ𝔞′⁢(1-p)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z)q-1𝔞′≤(3.4)⁢λp⁢(λ𝔞′⁢(1-p)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z)1𝔞′=λp⁢(1-1𝔞′)⁢(λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z)1𝔞′.

Therefore, Young’s inequality gives

III≲γ⁢λp+C⁢(γ)⁢λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z.

Estimate of IV: Hölder’s inequality and Young’s inequality imply

IV≤λp-q⁢(⨍B2⁢ρ⁢(x0)|∇⁡u0|p⁢d⁢x)qp≤γ⁢λp+C⁢(γ)⁢⨍B2⁢ρ⁢(x0)|∇⁡u0|p⁢d⁢x.

We combine all the estimates to complete the proof. ∎

Lemma 3.9.

Under the assumptions and the conclusion in Lemma 3.7, we further have

⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|f|⁢|u-(u0)|⁢d⁢z≲(n,N,p,Λ0,Λ1,δ)γ⁢λp+C⁢(γ)⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q⁢d⁢z)pq+C⁢(γ)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|F|p⁢d⁢z+C⁢(γ)⁢λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z+C⁢(γ)⁢⨍B2⁢ρ⁢(x0)|∇⁡u0|p⁢d⁢x.

Proof.

Apply Lemma 2.7 as in (3.6) and Lemma 3.5 to get

⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|f|⁢|u-(u0)|⁢d⁢z≲λ2n+2⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z)1𝔞′⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT[|u-(u0)B2⁢ρ⁢(x0)|2⁢ρ]δ⁢p+|∇⁡u|δ⁢p⁢d⁢z)1𝔞≲(3.4)⁢λp-nn+2⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT[|u-(u0)B2⁢ρ⁢(x0)|2⁢ρ]δ⁢p+|∇⁡u|δ⁢p⁢d⁢z)1𝔞≤λp-nn+2⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT[|u-(u0)B2⁢ρ⁢(x0)|2⁢ρ]q+|∇⁡u|q⁢d⁢z)nq⁢(n+2).

Using Lemma 3.3 with θ=q to the first term on the right-hand side, we have

⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|f|⁢|u-(u0)|⁢d⁢z≲λp-nn+2⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q⁢d⁢z)pq⁢np⁢(n+2)+λp-nn+2⁢(λ2-p⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|p-1+|F|p-1⁢d⁢z)pq⁢q⁢np⁢(n+2)+λp⁢(λ1-p⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)⁢|f|⁢d⁢z)nn+2+λp-nn+2⁢(⨍B2⁢ρ⁢(x0)|∇⁡u0|⁢d⁢x)nn+2=:I+II+III+IV.

Estimate of I: Applying Young’s inequality, we obtain

I≤γ⁢λp+C⁢(γ)⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q⁢d⁢z)pq.

Estimate of II: Note that (p-1)⁢np⁢(n+2)<1. Apply Hölder’s inequality and Young’s inequality to get

II≤λp+n⁢(1-p)n+2⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q+|F|q⁢d⁢z)pq⁢(p-1)⁢np⁢(n+2)≤γ⁢λp+C⁢(γ)⁢(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q⁢d⁢z)pq+C⁢(γ)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|F|p⁢d⁢z.

Estimate of III: Applying Hölder’s inequality and Young’s inequality, we have

III≤λp⁢(1-n𝔞′⁢(n+2))⁢(λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z)n𝔞′⁢(n+2)≤γ⁢λp+C⁢(γ)⁢λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z.

Estimate of IV: Again, Hölder’s inequality and Young’s inequality give

IV≤γ⁢λp+C⁢(γ)⁢⨍B2⁢ρ⁢(x0)|∇⁡u0|p⁢d⁢x.

The proof follows.∎

Lemma 3.10.

Let u be a weak solution of (2.1). Suppose B4⁢ρ⁢(x0)⊂Ω and 0<t0. Also, assume (3.2) and (3.3) for some λ≥1. Then there holds

⨍⨍Qρλ⁢(z0)∩ΩT|∇⁡u|p⁢d⁢z≲(n,N,p,Λ0,Λ1,δ)(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q⁢d⁢z)pq+⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|F|p⁢d⁢z+⨍B2⁢ρ⁢(x0)|∇⁡u0|p⁢d⁢x+λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z.

Proof.

From Lemma 3.2 with ρa=ρ and ρb=2⁢ρ, there holds

⨍⨍Qρλ⁢(z0)∩ΩT|∇⁡u|p⁢d⁢z≲⨍⨍Q2⁢ρλ⁢(z0)∩ΩT[|u-(u0)B2⁢ρ⁢(x0)|2⁢ρ]p⁢d⁢z+λp-2⁢⨍⨍Q2⁢ρλ⁢(z0)∩ΩT[|u-(u0)B2⁢ρ⁢(x0)|2⁢ρ]2⁢d⁢z+⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|f|⁢|u-(u0)B2⁢ρ⁢(x0)|⁢d⁢z+λp-1⁢(⨍B2⁢ρ⁢(x0)|∇⁡u0|p⁢d⁢z)1p+⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|F|p⁢d⁢z=:I+II+III+IV+⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|F|p⁢d⁢z.

We apply Lemma 3.7, Lemma 3.8 and Lemma 3.9 to I, II and III. Use Young’s inequality to estimate IV. Then there holds

λp≤(???)⁢⨍⨍Qρλ⁢(z0)∩ΩT|∇⁡u|p⁢d⁢z+⨍⨍Qρλ⁢(z0)∩ΩT|F|p⁢+⁡d⁢z⨍Bρ⁢(x0)|∇⁡u0|p⁢d⁢x+λ𝔞′⁢(1-p𝔞)⁢⨍⨍Qρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z≲γ⁢λp+(⨍⨍Q2⁢ρλ⁢(z0)∩ΩT|∇⁡u|q⁢d⁢z)pq+⨍⨍Qρλ⁢(z0)∩ΩT|F|p⁢+⁡d⁢z⨍Bρ⁢(x0)|∇⁡u0|p⁢d⁢x+λ𝔞′⁢(1-p𝔞)⁢⨍⨍Qρλ⁢(z0)∩ΩT(2⁢ρ)𝔞′⁢|f|𝔞′⁢d⁢z

Taking γ=γ⁢(n,N,p,Λ0,Λ1,δ) small enough, we finish the proof. ∎

4 Proof of Theorem 2.5

In this section, we prove the main results. First of all, we shall find intrinsic cylinders such that (3.2) and (3.3) hold.

Definition 4.1.

Let z0∈ΩT, r≤r1<r2≤2⁢r and 𝔷=(𝔵,𝔱)∈Qr1⁢(z0), and let d and α be defined in Definition 2.2.

Eλ:={z∈Qr1(z0)∩ΩT:|∇u|p>λp},

𝔹:=[2(20⁢rr2-r1)](n+2)⁢dp+[2(20⁢rr2-r1)](n+2)⁢d⁢αp+(20⁢rr2-r1)np+(20⁢rr2-r1)2⁢np⁢(n+2)-2⁢n,

G(Qρλ(𝔷)):=⨍⨍Qρλ⁢(𝔷)∩ΩT|∇u|pdz+⨍⨍Qρλ⁢(𝔷)∩ΩT|F|pdz+⨍Bρ⁢(𝔵)|∇u0|pdx+λ𝔞′⁢(1-p𝔞)⨍⨍Qρλ⁢(𝔷)∩ΩTρ𝔞′|f|𝔞′dz.

Lemma 4.2.

Let λ0 be defined in Definition 2.4. Then, for λ>3⁢B⁢λ0 and z∈Eλ, there exists ρz∈(0,r2-r110) such that

(4.1)G⁢(Qρ𝔷λ⁢(𝔷))=λp 𝑎𝑛𝑑 G⁢(Qρλ⁢(𝔷))<λp for all⁢ρ∈(ρ𝔷,r2-r1).

Proof.

Due to intrinsic geometry, we split the proof into two cases.

Case p≥2: For any r2-r110<ρ<r2-r1, there holds

G⁢(Qρλ⁢(𝔷))≤|Q2⁢r⁢(z0)∩ΩT||Qρλ⁢(𝔷)∩ΩT|⁢⨍⨍Q2⁢r⁢(z0)∩ΩT(|∇⁡u|+|F|+1)p⁢d⁢z+|Q2⁢r⁢(z0)∩ΩT||Qρλ⁢(𝔷)∩ΩT|⁢λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢r⁢(z0)∩ΩT(2⁢r)𝔞′⁢|f|𝔞′⁢d⁢z+|B2⁢r⁢(x0)||Bρ⁢(𝔷)|⁢⨍B2⁢r⁢(x0)|∇⁡u0|p⁢d⁢x.

Note that (3.1) implies

(4.2)|Q2⁢r⁢(z0)∩ΩT||Qρλ⁢(𝔷)∩ΩT|≤|Q2⁢r⁢(z0)|12⁢|Qρλ⁢(𝔷)|≤2⁢(2⁢rρ)n+2⁢λp-2≤2⁢(20⁢rr2-r1)n+2⁢λp-2.

From Remark 2.3, we have pd⁢α=2-𝔞′⁢(1-p𝔞)>0, d=p2,

⨍⨍Q2⁢r⁢(z0)∩ΩT(|∇⁡u|+|F|+1)p⁢d⁢z≤λ02,⨍B2⁢r⁢(x0)|∇⁡u0|p⁢d⁢x≤λ0p,λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢r⁢(z0)∩ΩT(2⁢r)𝔞′⁢|f|𝔞′⁢d⁢z≤λ𝔞′⁢(1-p𝔞)⁢λ0pd⁢α=λ𝔞′⁢(1-p𝔞)⁢λ02-𝔞′⁢(1-p𝔞).

It follows that

G⁢(Qρλ⁢(𝔷))≤2⁢(20⁢rr2-r1)n+2⁢λp-2⁢(λ02+λ𝔞′⁢(1-p𝔞)⁢λ02-𝔞′⁢(1-p𝔞))+(20⁢rr2-r1)n⁢λ0p<λp.

On the other hand, since 𝔷∈Eλ, there exists ρ𝔷∈(0,r2-r110) such that (4.1) holds.

Case p<2: Let us denote ρλ:=λp-22ρ. Then, for any r2-r110<ρ<r2-r1, there holds

G⁢(Qρλλ⁢(𝔷))≤|Q2⁢r⁢(z0)∩ΩT||Qρλλ⁢(𝔷)∩ΩT|⁢⨍⨍Q2⁢r⁢(z0)∩ΩT(|∇⁡u|+|F|+1)p⁢d⁢z+|Q2⁢r⁢(z0)∩ΩT||Qρλλ⁢(𝔷)∩ΩT|⁢λ𝔞′⁢(p2-p𝔞)⁢⨍⨍Q2⁢r⁢(z0)∩ΩT(2⁢r)𝔞′⁢|f|𝔞′⁢d⁢z+|B2⁢r⁢(x0)||Bρλ⁢(𝔷)|⁢⨍B2⁢r⁢(x0)|∇⁡u0|p⁢d⁢x.

Here, we used the fact that ρλ≤λp-22⁢r for the second term. Note that

|Q2⁢r⁢(z0)∩ΩT||Qρλλ⁢(𝔷)∩ΩT|≤|Q2⁢r⁢(z0)|12⁢|Qρλλ⁢(𝔷)|≤2⁢(20⁢rr2-r1)n+2⁢λ(2-p)⁢n2.

Again, from Remark 2.3, we have pd⁢α=p⁢(n+2)-2⁢n2-𝔞′⁢(p2-p𝔞)>0, pd=p⁢(n+2)-2⁢n2,

⨍⨍Q2⁢r⁢(z0)∩ΩT(|∇⁡u|+|F|+1)p⁢d⁢z+⨍B2⁢r⁢(x0)|∇⁡u0|p⁢d⁢x≤λ0(p-2)⁢n+2⁢p2,λ𝔞′⁢(p2-p𝔞)⁢⨍⨍Q2⁢r⁢(z0)∩ΩT(2⁢r)𝔞′⁢|f|𝔞′⁢d⁢z≤λ𝔞′⁢(p2-p𝔞)⁢λ0pd⁢α=λ𝔞′⁢(p2-p𝔞)⁢λ0(p-2)⁢n+2⁢p2-𝔞′⁢(p2-p𝔞).

It follows that

G⁢(Qrλλ⁢(z0))≤2⁢(20⁢rr2-r1)n+2⁢λ(2-p)⁢n2⁢(λ0(p-2)⁢n+2⁢p2+λ𝔞′⁢(p2-p𝔞)⁢λ0(p-2)⁢n+2⁢p2-𝔞′⁢(p2-p𝔞))+(20⁢rr2-r1)n⁢λ(2-p)⁢n2⁢λ0p⁢(n+2)-2⁢n2<λp.

On the other hand, since 𝔷∈Eλ, there exists ρ𝔷∈(0,r2-r110) such that (4.1) holds. The lemma follows. ∎

We now define upper level sets.

Definition 4.3.

Let η>0 and λ>3⁢𝔹⁢λ0. We define the following:

Φη⁢λρ:={z∈Qρ(z0)∩ΩT:|∇u|p(z)>ηλp},
Ψη⁢λρ:={z∈Qρ(z0)∩ΩT:Hp(z)>ηλp} where H(z):=|F(z)|+|∇u0(x)|+1,
Ση⁢λρ:={z∈Qρ(z0)∩ΩT:|f~η|(z)>ηλp}, where f~η:=2𝔞′η-𝔞′λ0ν(2r|f|(z))𝔞′.

Our covering argument is divided into three steps.

Step 1

Let λ>3⁢𝔹⁢λ0 and 𝔷∈Eλ. Assumptions (3.2) in Q4⁢ρ𝔷λ⁢(𝔷) and (3.3) in Qρ𝔷λ⁢(𝔷) are satisfied by (4.1). Applying Lemma 3.10, there exists q<p defined in (3.7) such that

G⁢(Qρ𝔷λ⁢(𝔷))≲(⨍⨍Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT|∇⁡u|q⁢d⁢z)pq+⨍⨍Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT(|F|+1)p⁢d⁢z+⨍⨍Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT|∇⁡u0|p⁢d⁢x+λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT(2⁢ρ𝔷)𝔞′⁢|f|𝔞′⁢d⁢z=:I+II+III+IV.

Estimate of I+II+III: Let η∈(0,1) to be chosen later. There holds

Here, to obtain (a), we used Hölder’s inequality and (4.1). Therefore, we get

I+II+III≲η⁢λp+1|Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT|⁢∬Φη⁢λr2∩Q2⁢ρ𝔷λ⁢(𝔷)λp-q⁢|∇⁡u|q⁢d⁢z+1|Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT|⁢∬Ψη⁢λr2∩Q2⁢ρ𝔷λ⁢(𝔷)|H|p⁢d⁢z.

Estimate of IV: Let us consider the alternative

(4.3)λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT(2⁢ρ𝔷)𝔞′⁢|f|𝔞′⁢d⁢z≤η⁢λp or λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT(2⁢ρ𝔷)𝔞′⁢|f|𝔞′⁢d⁢z≥η⁢λp.

Leaving the first case of (4.3), suppose the second case holds.

Case p≥2: Applying (4.2), we have

(4.4)η⁢λp≤2⁢λ𝔞′⁢(1-p𝔞)+p-pd⁢(rρ𝔷)n+2-𝔞′⁢⨍⨍Q2⁢r⁢(z0)∩ΩT(2⁢r⁢|f|)𝔞′⁢d⁢z≤2⁢λ𝔞′⁢(1-p𝔞)+p-pd⁢λ0pd⁢α⁢(rρ𝔷)n+2-𝔞′.

Since Definition 2.2 implies the inequality

(4.5)n+2-𝔞′>1⇔n+1>𝔞′⇔n+1n=(n+1)′<𝔞=δ⁢p⁢(n+2)n⇔n+1p⁢(n+2)<δ,

we see that (4.4) becomes

ρ𝔷r≤2⁢η-1⁢λ𝔞′⁢(1-p𝔞)-pdn+2-𝔞′⁢λ01n+2-𝔞′⁢pd⁢α.

Therefore, we get

λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT(2⁢ρ𝔷)𝔞′⁢|f|𝔞′⁢d⁢z=λ𝔞′⁢(1-p𝔞)⁢(ρ𝔷r)𝔞′⁢⨍⨍Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT(2⁢r)𝔞′⁢|f|𝔞′⁢d⁢z≤2𝔞′⁢η-𝔞′⁢λ𝔞′⁢(1-p𝔞)+𝔞′n+2-𝔞′⁢(𝔞′⁢(1-p𝔞)-pd)⁢λ0𝔞′n+2-𝔞′⁢pd⁢α⁢⨍⨍Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT(2⁢r)𝔞′⁢|f|𝔞′⁢d⁢z≤(a)⁢2𝔞′⁢η-𝔞′⁢λ0𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT(2⁢r)𝔞′⁢|f|𝔞′⁢d⁢z.

Here, to obtain (a), we used λ≥λ0 and

𝔞′⁢nn+2-𝔞′⁢(1-1δ)+𝔞′n+2-𝔞′⁢pd⁢α=𝔞′⁢(1-p𝔞)+𝔞′n+2-𝔞′⁢(𝔞′⁢(1-p𝔞)-pd)+𝔞′n+2-𝔞′⁢pd⁢α=𝔞′⁢(1-p𝔞).

It follows that both cases of (4.3) give

λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρ𝔷λ⁢(z0)∩ΩT(2⁢ρ𝔷)𝔞′⁢|f|𝔞′⁢d⁢z≤η⁢λp+2𝔞′⁢η-𝔞′⁢λ0𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρ𝔷λ⁢(z0)∩ΩT(2⁢r)𝔞′⁢|f|𝔞′⁢d⁢z≤2⁢η⁢λp+2𝔞′⁢η-𝔞′⁢λ0ν|Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT|⁢∬Ση⁢λr2∩Q2⁢ρ𝔷λ⁢(𝔷)(2⁢r)𝔞′⁢|f|𝔞′⁢d⁢z.

Case p<2: Since ρ𝔷=λp-22⁢ρ~𝔷 for some ρ~𝔷∈(0,r2-r110), (4.4) becomes

η⁢λp≤2⁢λ𝔞′⁢(1-p𝔞)+n⁢(2-p)2+𝔞′⁢(p-22)⁢λ0pd⁢α⁢(rρ~𝔷)n+2-𝔞′=2⁢λ𝔞′⁢(p2-p𝔞)+n⁢(2-p)2⁢λ0pd⁢α⁢(rρ~𝔷)n+2-𝔞′,

and thus (4.5) gives

(ρ~𝔷r)≤2⁢η-1⁢λ𝔞′⁢(p2-p𝔞)-pdn+2-𝔞′⁢λ01n+2-𝔞′⁢pd⁢α.

Analogously, the second case of (4.3) implies

λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT(2⁢ρ𝔷)𝔞′⁢|f|𝔞′⁢d⁢z=λ𝔞′⁢(p2-p𝔞)⁢(ρ~𝔷r)𝔞′⁢⨍⨍Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT(2⁢r)𝔞′⁢|f|𝔞′⁢d⁢z≤2𝔞′⁢η-𝔞′⁢λ𝔞′⁢(p2-p𝔞)+𝔞′n+2-𝔞′⁢(𝔞′⁢(p2-p𝔞)-pd)⁢λ0𝔞′n+2-𝔞′⁢pd⁢α⁢⨍⨍Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT(2⁢r)𝔞′⁢|f|𝔞′⁢d⁢z≤(a)⁢2𝔞′⁢η-𝔞′⁢λ0𝔞′⁢(p2-p𝔞)⁢⨍⨍Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT(2⁢r)𝔞′⁢|f|𝔞′⁢d⁢z.

Here, to obtain (a), we used λ≥λ0 and

𝔞′⁢nn+2-𝔞′⁢(1-1δ)+𝔞′n+2-𝔞′⁢pd⁢α=𝔞′⁢(p2-p𝔞)+𝔞′n+2-𝔞′⁢(𝔞′⁢(p2-p𝔞)-pd)+𝔞′n+2-𝔞′⁢pd⁢α=𝔞′⁢(p2-p𝔞).

Therefore, we have

λ𝔞′⁢(1-p𝔞)⁢⨍⨍Q2⁢ρ𝔷λ⁢(z0)∩ΩT(2⁢ρ𝔷)𝔞′⁢|f|𝔞′⁢d⁢z≤2⁢η⁢λp+2𝔞′⁢η-𝔞′⁢λ0ν|Q2⁢ρ𝔷λ⁢(𝔷)∩ΩT|⁢∬Ση⁢λr2∩Q2⁢ρ𝔷λ⁢(𝔷)(2⁢r)𝔞′⁢|f|𝔞′⁢d⁢z.

Combining all the estimates, we get

Taking η=η⁢(n,N,p,Λ0,Λ1,δ)∈(0,1) small enough, we obtain

⨍⨍Q10⁢ρ𝔷λ⁢(𝔷)∩ΩT|∇⁡u|p⁢d⁢z≲1|Q2⁢ρ𝔷λ⁢(z0)∩ΩT|⁢∬Φη⁢λr2∩Q2⁢ρ𝔷λ⁢(𝔷)λp-q⁢|∇⁡u|q⁢d⁢z+1|Q2⁢ρ𝔷λ⁢(z0)∩ΩT|⁢∬Ψη⁢λr2∩Q2⁢ρ𝔷λ⁢(𝔷)|H|p⁢d⁢z+2𝔞′⁢η-𝔞′⁢λ0ν|Q2⁢ρ𝔷λ⁢(z0)∩ΩT|⁢∬Ση⁢λr2(2⁢r)𝔞′⁢|f|𝔞′⁢d⁢z.

Therefore, for any λ>3⁢𝔹⁢λ0 and 𝔷∈Eλ, there holds

(4.6)∬Q10⁢ρ𝔷λ⁢(𝔷)∩ΩT|∇⁡u|p⁢d⁢z≲∬Φη⁢λr2∩Q2⁢ρ𝔷λ⁢(𝔷)λp-q⁢|∇⁡u|q⁢d⁢z+∬Ψη⁢λr2∩Q2⁢ρ𝔷λ⁢(𝔷)|H|p⁢d⁢z+∬Ση⁢λr2∩Q2⁢ρ𝔷λ⁢(𝔷)|f~η|⁢d⁢z.

Step 2

We apply Vitali’s covering lemma to {Q2⁢ρ𝔷λ⁢(𝔷)}𝔷∈Eλ and to obtain a disjoint countable subfamily {Q2⁢ρ𝔷iλ⁢(𝔷i)}i∈ℕ such that

Eλ⊂⋃𝔷∈EλQ2⁢ρ𝔷λ⁢(𝔷)⊂⋃1≤i<∞Q10⁢ρ𝔷iλ⁢(𝔷i)⊂Qr2⁢(z0).

Therefore, we get

∬Eλ|∇⁡u|p⁢d⁢z≤∑1≤i<∞∬Q10⁢ρ𝔷iλ⁢(𝔷)∩ΩT|∇⁡u|p⁢d⁢z⁢≲(a)⁢∬Φη⁢λr2λp-q⁢|∇⁡u|q⁢d⁢z+∬Ψη⁢λr2|H|p⁢d⁢z+∬Ση⁢λr2|f~η|⁢d⁢z.

where, to obtain (a), we used (4.6) for each i and disjointness of {Q2⁢ρ𝔷iλ⁢(𝔷i)}i∈ℕ in ℝn+1.

Also, since there holds

∬Φη⁢λr1∖Eλ|∇⁡u|p⁢d⁢z≤∬Φη⁢λr1∖Eλλp-q⁢|∇⁡u|q⁢d⁢z,

it follows

∬Φη⁢λr1|∇⁡u|p⁢d⁢z≲∬Φη⁢λr2λp-q⁢|∇⁡u|q⁢d⁢z+∬Ψη⁢λr2|H|p⁢d⁢z+∬Ση⁢λr2|f~η|⁢d⁢z.

Letting λ1:=3η1p𝔹λ0, for any λ>λ1, we have

(4.7)∬Φλr1|∇⁡u|p⁢d⁢z≲∬Φλr2λp-q⁢|∇⁡u|q⁢d⁢z+∬Ψλr2|H|p⁢d⁢z+∬Σλr2|f~η|⁢d⁢z.

Step 3

For k>λ1, let us define

|∇u|k:=min{|∇u|,k} and Φλ,kρ:={z∈Qρ(z0)∩ΩT:|∇u|kp>λp}.

We see that if λ>k, then Φλ,kρ=∅, and if λ≤k, then Φλ,kρ=Φλρ. From (4.7), we deduce

(4.8)∬Φλ,kr1|∇⁡u|kp-q⁢|∇⁡u|q⁢d⁢z≲∬Φλ,kr2λp-q⁢|∇⁡u|q⁢d⁢z+∬Ψλr2|H|p⁢d⁢z+∬Σλr2|f~η|⁢d⁢z.

Let ε>0 to be chosen later. Multiply (4.8) by λε-1 and integrate over (λ1,∞) to get

(4.9)I:=∫λ1∞λε-1∬Φλ,kr1|∇u|kp-q|∇u|qdzdλ≲∫λ1∞λε-1⁢∬Φλ,kr2λp-q⁢|∇⁡u|q⁢d⁢z⁢d⁢λ+∫λ1∞λε-1⁢∬Ψλr2|H|p⁢d⁢z⁢d⁢λ+∫λ1∞λε-1⁢∬Σλr2|f~η|⁢d⁢z⁢d⁢λ=:II+III+IV.

Estimate of I: Applying Fubini’s theorem, we get

I=∬Φλ1,kr1|∇⁡u|kp-q⁢|∇⁡u|q⁢∫λ1|∇⁡u|kλε-1⁢d⁢λ⁢d⁢z=1ε⁢∬Φλ1,kr1|∇⁡u|kp-q+ε⁢|∇⁡u|q⁢d⁢z-1ε⁢λ1ε⁢∬Φλ1,kr1|∇⁡u|kp-q⁢|∇⁡u|q⁢d⁢z.

Estimate of II: Again, using Fubini’s theorem, we obtain

II=∬Φλ1,kr2|∇⁡u|q⁢∫λ1|∇⁡u|kλp-q+ε-1⁢d⁢λ⁢d⁢z≤1p-q⁢∬Φλ1,kr2|∇⁡u|kp-q+ε⁢|∇⁡u|q⁢d⁢z.

Estimate of III: Again, by Fubini’s theorem, we have

III=∬Ψλ1r2|H|p⁢∫λ1|H|λε-1⁢d⁢λ⁢d⁢z≤1ε⁢∬Ψλ1r2|H|p+ε⁢d⁢z.

Estimate of IV: Similarly, we have

IV=∬Σλ1r2|f~η|⁢∫λ1|f~η|1pλε-1⁢d⁢λ⁢d⁢z≤1ε⁢∬Σλ1r2|f~η|1+εp⁢d⁢z.

It follows that (4.9) becomes

∬Φλ1,kr1|∇⁡u|kp-q+ε⁢|∇⁡u|q⁢d⁢z≤𝐂⁢εp-q⁢∬Φλ1,kr2|∇⁡u|kp-q+ε⁢|∇⁡u|q⁢d⁢z+λ1ε⁢∬Φλ1,kr1|∇⁡u|kp-q⁢|∇⁡u|q⁢d⁢z+𝐂⁢∬Ψλ1r2|H|p+ε⁢d⁢z+𝐂⁢∬Σλ1r2|f~η|1+εp⁢d⁢z,

where 𝐂=𝐂⁢(n,N,p,Λ0,Λ1,δ).

Since there holds

∬Qr1⁢(z0)∩ΩT∖Φλ1,kr1|∇⁡u|kp-q+ε⁢|∇⁡u|q⁢d⁢z≤λ1ε⁢∬Qr1⁢(z0)∩ΩT∖Φλ1,kr1|∇⁡u|kp-q⁢|∇⁡u|q⁢d⁢z,

λ1:=3η1p𝔹λ0 and 𝔹≥1, we obtain

∬Qr1⁢(z0)∩ΩT|∇⁡u|kp-q+ε⁢|∇⁡u|q⁢d⁢z≤𝐂⁢εp-q⁢∬Qr2⁢(z0)∩ΩT|∇⁡u|kp-q+ε⁢|∇⁡u|q⁢d⁢z+𝐂⁢𝔹⁢λ0ε⁢∬Q2⁢r⁢(z0)∩ΩT|∇⁡u|kp-q⁢|∇⁡u|q⁢d⁢z+𝐂⁢∬Q2⁢r⁢(z0)∩ΩT|H|p+ε⁢d⁢z+𝐂⁢∬Q2⁢r⁢(z0)∩ΩT|f~η|1+εp⁢d⁢z.

Take ε0=ε0⁢(n,N,p,Λ0,Λ1,δ) so that 𝐂⁢ε0p-q=12. Then, for any ε∈(0,ε0), it follows that

∬Qr1⁢(z0)∩ΩT|∇⁡u|kp-q+ε⁢|∇⁡u|q⁢d⁢z≤12⁢∬Qr2⁢(z0)∩ΩT|∇⁡u|kp-q+ε⁢|∇⁡u|q⁢d⁢z+𝐂⁢𝔹⁢λ0ε⁢∬Q2⁢r⁢(z0)∩ΩT|∇⁡u|kp-q⁢|∇⁡u|q⁢d⁢z+𝐂⁢∬Q2⁢r⁢(z0)∩ΩT|H|p+ε⁢d⁢z+𝐂⁢∬Q2⁢r⁢(z0)∩ΩT|f~η|1+εp⁢d⁢z.

Hence, applying Lemma 2.8, we get

∬Qr⁢(z0)∩ΩT|∇⁡u|kp-q+ε⁢|∇⁡u|q⁢d⁢z≲λ0ε⁢∬Q2⁢r⁢(z0)∩ΩT|∇⁡u|kp-q⁢|∇⁡u|q⁢d⁢z+∬Q2⁢r⁢(z0)∩ΩT|H|p+ε⁢d⁢z+∬Q2⁢r⁢(z0)∩ΩT|f~η|1+εp⁢d⁢z.

Let k→∞ to derive the desired estimate. This completes the proof.

Communicated by Frank Duzaar

Funding source: National Research Foundation of Korea

Award Identifier / Grant number: NRF-2017R1A2B2003877

Award Identifier / Grant number: NRF-2019R1C1C1003844

Funding statement: S. Byun was supported by the National Research Foundation of Korea (NRF-2017R1A2B2003877). W. Kim and M. Lim were supported by the National Research Foundation of Korea (NRF-2019R1C1C1003844).

Acknowledgements

The authors are grateful to Karthik Adimurthi for valuable conversations and constant support throughout all this work. The authors wish to thank the referees for careful reading of the early version of this manuscript and providing many valuable suggestions and comments.

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Received: 2020-03-16

Revised: 2020-06-29

Published Online: 2020-08-06

Published in Print: 2020-11-01

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Higher integrability; initial boundary data; non-divergence data

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