Harnack inequality for a class of functionals with non-standard growth via De Giorgi’s method

Jihoon Ok

doi:10.1515/anona-2016-0083

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Harnack inequality for a class of functionals with non-standard growth via De Giorgi’s method

Jihoon Ok

Published/Copyright: May 27, 2016

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From the journal Advances in Nonlinear Analysis Volume 7 Issue 2

Abstract

We study the regularity theory of quasi-minimizers of functionals with $Lp⁢(⋅)⁢log⁡L$ -growth. In particular, we prove the Harnack inequality and, in addition, the local boundedness and the Hölder continuity of the quasi-minimizers. We directly prove our results via De Giorgi’s method.

Keywords: Non-standard growth; quasi-minimizer; Hölder continuity; Harnack inequality, variable exponent

MSC 2010: 35J92; 35J20; 35B65

1 Introduction

In this paper, we are interested in regularity problems for a class of functionals of the type

(1.1)

$w∈W1,1⁢(Ω)↦ℱ⁢(w):=∫ΩF⁢(x,D⁢w)⁢𝑑x,$

with the so-called non-standard growth conditions in the sense that $F:Ω×ℝn→ℝ$ is a Carathéodory function satisfying

$|z|p≲F⁢(x,z)≲|z|q+1,1<p≤q.$

Here, Ω is a bounded open subset in $ℝn$ , $n≥2$ . The standard case is when $p=q$ , while the genuinely non-standard case is when $p<q$ . This has been treated at length by Marcellini in a series of seminal papers [33, 34]. A peculiar feature of these functionals, that appears not to be present when $p=q$ , is that when the integrand also depends on x, a subtle interplay between the regularity of the integrand $F⁢(⋅)$ with respect to x and the size of the gap ration $q/p$ comes into play. This has been first observed in [38, 39, 22], and later on widely exploited, for instance, in [1, 8, 12, 13, 15, 14, 21], when considering several different structure conditions.

In this paper, we shall focus on a special functional, which is in some sense a borderline case of classical Orlicz type functionals of the type

$w∈W1,1(Ω)↦∫Ω|Dw|plog(e+|Dw|)dx.$

The functionals and equations defined in the Orlicz setting have been considered in [4, 9, 18, 20, 25, 32, 36], see also the references therein. We shall consider a class of functionals with non-standard growth conditions modelled by

(1.2)

$w∈W1,1(Ω)↦∫Ω|Dw|p⁢(x)log(e+|Dw|)dx=:∫ΩΦ(x,|Dw|)dx,$

by defining the function $Φ:Ω×[0,∞)→ℝ$ as follows:

(1.3)

$Φ⁢(x,t):=tp⁢(x)⁢log⁡(e+t),$

where $p⁢(⋅):Ω→(1,∞)$ is a continuous function and such that

(1.4)

$1<γ1≤p⁢(⋅)≤γ2<∞.$

For this Φ, we investigate the regularity of its quasi-minimizers. We say $u∈W1,1⁢(Ω)$ is a quasi-minimizer of Φ with $Q>0$ , or Q-minimizer of Φ, if for any $v∈Wloc1,1⁢(Ω)$ with $K:=supp⁢(u-v)⋐Ω$ , we have

$∫KΦ⁢(x,|D⁢u|)⁢𝑑x≤Q⁢∫KΦ⁢(x,|D⁢v|)⁢𝑑x.$

In particular, if $Q=1$ we say u is a minimizer of Φ.

The main feature of the functional in (1.2) is that its integrand changes its growth and ellipticity properties according to x. We call functionals with this property non-autonomous functionals. They describe strongly anisotropic materials and are connected to Lavrentiev’s phenomenon. Zhikov proposed various examples of non-autonomous functionals, see, for instance, [37, 38, 39]. In this respect, two basic functionals are the following:

(1.5)

$w∈W1,1(Ω)↦∫Ω|Dw|p⁢(x)dx,$

where $p⁢(⋅)$ satisfies (1.4), and

(1.6)

$w∈W1,1(Ω)↦∫Ω|Dw|p+a(x)|Dw|qdx,0≤a(x)≤L.$

The one in (1.5) is a functional to which a large literature has been devoted over the years, and that closely relates to the one in (1.2). Its Euler–Lagrange equation $div⁢(|D⁢u|p⁢(x)-2⁢D⁢u)=0$ is called the $p⁢(x)$ -Laplace equation. Basic regularity results have been given in [1, 17, 23, 24, 28]. Calderón–Zygmund estimates have been given in [2, 11, 10] while potentials estimates have been given in [3, 7], also following the methods of [30]. Note that in order to remove Lavrentiev’s phenomenon, Zhikov gave an important condition on the variable exponent $p⁢(⋅)$ , that is, log-Hölder continuity. We say $p⁢(⋅):Ω→ℝ$ is log-Hölder continuous if there exists $C>0$ such that

$|p⁢(x)-p⁢(y)|≤C-log⁡|x-y| for any ⁢x,y∈Ω⁢ with ⁢|x-y|<12.$

We remark that the log-Hölder continuity is equivalent to the following condition:

(1.7)

$L:=sup0<r<12⁡ω⁢(r)⁢log⁡(1r)<∞,$

where $ω:[0,∞)→[0,∞)$ is the modulus of continuity of $p⁢(⋅)$ . Hence, $ω⁢(0)=0$ , ω is nondecreasing and concave and satisfies

$|p⁢(x)-p⁢(y)|≤ω⁢(|x-y|) for any ⁢x,y∈Ω.$

This condition has turned out to be a very important condition in the analysis of functionals and equations with $p⁢(x)$ -growth. Indeed, if $p⁢(⋅)$ is log-Hölder continuous then the quasi-minimizers of (1.5) is Hölder continuous [24] and satisfies the Harnack inequality [28]. In addition, the log-Hölder continuity also plays a crucial role in the analysis of variable exponent spaces, see the monograph by Diening et al. [19].

As for the functional in (1.6), basic conditions for regularity have been given in [22], while more recently Baroni, Colombo and Mingione [5, 15, 14] have established a complete regularity theory for its minimizers, eventually establishing the suitable Calderón–Zygmund theory [16]. A significant borderline variant of (1.6), that relates to the functional in (1.2) is

(1.8)

$w∈W1,1(Ω)↦∫Ω(|Dw|p+a(x)|Dw|plog(1+|Dw|))dx,$

which is in fact the limiting case of the functional in (1.6) when $q→p$ . This has been considered in [6] and we shall come back on it later on.

The functional in (1.2) considered in this paper is first treated by Giannetti and Passarelli di Napoli [26], in which it has been proved that if $p⁢(⋅)$ satisfies

$limr→0⁡ω⁢(r)⁢log⁡(1r)=0,$

then the minimizer of (1.2) is $Cα$ for all $α∈(0,1)$ and that the gradient of the minimizer is Hölder continuous if so is $p⁢(⋅)$ . We point out that the conditions of $p⁢(⋅)$ to obtain desired regularities are exactly the same as the ones for the functionals in (1.5). Recently, the author [35] has proved global Calderón–Zygmund estimates in non-smooth domains of the type established in [2, 11, 16].

In this paper, we investigate the boundedness (Theorem 3.2), the Hölder continuity (Theorem 4.4) and finally the Harnack inequality (Theorem 5.5) of quasi-minimizers of Φ. We also show that the conditions on $p⁢(⋅)$ to obtain such regularities are exactly same to the case of functionals in (1.5) established in [24, 28]. More precisely, we will prove that the quasi-minimizers are locally bounded if $p⁢(⋅)$ is plain continuous, and Hölder continuous and satisfies the Harnack inequality if $p⁢(⋅)$ is log-Hölder continuous. Note that if we have the Harnack inequality then the Hölder continuity follows automatically. Hence, we provide a directly approach obtaining Hölder continuity.

We point out that the result of Hölder continuity in [26] does not cover our result. In fact, one can deduce from the result in [26] that for each $α∈(0,1)$ , there exists sufficiently small $L>0$ satisfying (1.7) such that the quasi-minimizers are $Cα$ , hence their result does not include arbitrary $L>0$ in (1.7). This difference is originated by the different methods of the proof. In [26] Giannetti and Passarelli di Napoli use a freezing argument and well known regularity results of a frozen functional which has Orlicz growth. However, unlike the argument in [26], we use the De Giorgi method directly. Therefore, the main object considered in [26] is the gradient of quasi-minimizers, on the other hand, the one in this paper is the quasi-minimizers itself.

The Harnack inequality for the functionals in (1.2) is a new result. We remark that the Harnack inequality for the functionals in (1.5) has been first proved in [28], in which it has been proved that

$supBr⁡u≤c⁢(infBr⁡u+r)$

for any nonnegative quasi-minimizer u of the functionals in (1.5). Here the radius $r>0$ is sufficiently small and the constant c depends on the quasi-minimizer itself. in this paper, we prove the same estimate for the functionals in (1.2). However, in our estimate, the constant c depends only on the structure constants, that is, independent of the quasi-minimizer. Instead, the radius $r>0$ is sufficiently small depending on the quasi-minimizer. We also remark that in [5], Baroni, Colombo and Mingione proved the Harnack inequality for the functionals (1.6) and (1.8), by using a freezing argument.

We observe that the functional is a model case of functionals with so-called $Lp⁢(⋅)⁢log⁡L$ -growth. In fact, our results still hold for the functionals (1.1) with

$ν⁢|z|p⁢(x)⁢log⁡(e+|z|)≤F⁢(x,z)≤Λ⁢|z|p⁢(x)⁢log⁡(e+|z|)+Λ0$

for some $0<ν≤Λ$ and $Λ0>0$ and relevant equations, in the same proofs with minor modifications. Moreover, without considering energy functional or equation, we can also prove that the functions satisfying a Caccioppoli-type estimates for Φ are Hölder continuous and satisfy the Harnack inequality, see Remark 5.6.

Finally, we would like to mention a few analogies between the problem treated here and other ones already appearing in the literature. The results obtained here closely relate to those considered in [5, 6] for the functional in (1.8), that can actually be obtained under a similar log-Hölder continuity assumption on $a⁢(x)$ . More general cases can be considered starting from the methods used here, and these regard instances as

$w∈W1,1⁢(Ω)⁢ ↦∫Ω[Ψ0⁢(|D⁢w|)]p⁢(x)⁢log⁡(e+|D⁢w|)⁢𝑑x,$

$⁢and w∈W1,1⁢(Ω)⁢ ↦∫Ω(Ψ1⁢(|D⁢w|)+a⁢(x)⁢Ψ1⁢(|D⁢w|)⁢log⁡(1+|D⁢w|))⁢𝑑x,$

where $Ψ0⁢(⋅),Ψ1⁢(⋅)$ are Orlicz functions.

The rest of the paper is organized as follows. In the next section, we introduce notations and preliminary lemmas. In Section 3, we obtain Caccioppoli-type estimates and prove the local boundedness. In Section 4, we prove Hölder continuous without the Harnack inequality. In the final section, we prove the Harnack inequality.

2 Preliminaries

We first introduce notations that will be used in this paper. Recall that $p⁢(⋅):Ω→(1,∞)$ is a continuous function satisfying (1.4), Φ is a function given by (1.3). $Bρ$ is a ball in $ℝn$ with radius $ρ>0$ . For a function w, set

$w±:=max⁡{±w,0} and osc⁢(u,Bρ):=supBρ⁡u-infBρ⁡u.$

For a quasi-minimizer u of Φ, $k∈ℝ$ and ρ, we write

$Ak,ρ:={x∈Bρ:u>k} and Ak,ρ-:={x∈Bρ:u<k}.$

We define a convex function $Φp:[0,∞)→[0,∞)$ , $1<p<∞$ , by

$Φp⁢(t):=tp⁢log⁡(e+t).$

Note that we have

$Φ⁢(x,t)=Φp⁢(x)⁢(t).$

We then state simple properties of $Φp$ .

Proposition 2.1.

Let $1<γ1≤p≤γ2<∞$ .

If $θ>1$ , then for any $t>0$ ,
(2.1) $Φp⁢(θ⁢t)≤θγ2+1⁢Φp⁢(t).$
If $0<θ<1$ , then for any $t>0$ ,
(2.2) $Φp⁢(θ⁢t)≥θγ2+1⁢Φp⁢(t).$
For any $t,τ>0$ we have
(2.3) $Φp⁢(t+τ)≤12⁢(Φp⁢(2⁢t)+Φp⁢(2⁢τ))≤2γ2⁢(Φp⁢(t)+Φp⁢(τ)).$

Proof.

For $0<θ<∞$ , let $f⁢(t):=θ⁢log⁡(e+t)-log⁡(e+θ⁢t)$ , $t≥0$ . Then we have

$f′⁢(t)=θ⁢(1e+t-1e+θ⁢t).$

From this, we see that $f⁢(t)≥0$ if $θ>1$ and that $f⁢(t)≤0$ if $0<θ<1$ . These imply (2.1) and (2.2), and (2.3) follows directly from (2.1) and the convexity of Φ. ∎

As the properties above are rather elementary, we shall use them without any mention of this proposition except when they are crucially used.

For Φ, we define a generalized Lebesgue space $LΦ⁢(Ω)=Lp⁢(⋅)⁢log⁡L⁢(Ω)$ by the set of all measurable functions $f:Ω→ℝ$ satisfying

$∫ΩΦ⁢(x,|f|)⁢𝑑x=∫Ω|f|p⁢(x)⁢log⁡(e+|f|)⁢𝑑x<∞.$

Note that $Lp⁢(⋅)⁢log⁡L$ space is of the Musielak–Orlicz space, see [19]. We further define a generalized Sobolev space by

$W1,Φ⁢(Ω):={f∈W1,1⁢(Ω):f,|D⁢f|∈Lp⁢(⋅)⁢log⁡L⁢(Ω)}.$

Remark 2.2.

From now on, we shall only consider quasi-minimzers u of Φ belonging to $W1,Φ⁢(Ω)$ instead of $W1,1⁢(Ω)$ . Therefore, we have

$∫ΩΦ⁢(x,|u|)⁢𝑑x<∞ and ∫ΩΦ⁢(x,|D⁢u|)⁢𝑑x<∞.$

2.1 Preliminary lemmas

We start by stating Poincaré–Sobolev type inequalities for $Φp$ .

Lemma 2.3.

Let $1<γ1≤p≤γ2$ . There exists $θ=θ⁢(n,γ1,γ2)∈(0,1)$ such that for any $w∈W01,1⁢(Br)$ ,

(2.4)

$⨍BrΦp⁢(|w|r)⁢𝑑x≤c⁢(⨍BrΦp⁢(|D⁢w|)θ⁢𝑑x)1θ$

for some $c=c⁢(n,γ1,γ2)>0$ . Moreover, if $0<r<1$ , then we have

(2.5)

$∫BrΦp⁢(|w|)⁢𝑑x≤c⁢(∫BrΦp⁢(|D⁢w|)θ⁢𝑑x)1θ$

for some $c=c⁢(n,γ1,γ2)>0$ .

Proof.

In view of [18, Theorem 7] and [26, Theorem 2.5], we see that there exists $θ=θ⁢(n,γ1,γ2)∈(0,1)$ such that for any $w∈W1,1⁢(Bρ)$ with $ρ>0$ ,

(2.6)

$⨍BρΦp⁢(|w|ρ)⁢𝑑x≤c⁢(1+|Bρ||E|)⁢(⨍BρΦp⁢(|D⁢w|)θ⁢𝑑x)1θ$

for some $c⁢(n,γ1,γ2)>0$ , where $E:={x∈Bρ:w⁢(x)=0}$ . We extend the function $w∈W01,1⁢(Br)$ to the hole space $ℝn$ by the zero. Since $w≡0$ in $B2⁢r∖Br$ , applying (2.6) to $ρ=2⁢r$ , we have (2.4). On the other hand, if $0<r<1$ , then applying (2.6) to $ρ=2$ and using the fact that $w≡0$ in $B2∖Br$ , we have (2.5). ∎

The next three technical lemmas can be found as Lemma 3.5, Lemma 4.7 and Lemma 4.8 in [31, Chapter 2].

Lemma 2.4.

Let $w∈W1,1⁢(Bρ)$ . For $l>k$ and $ρ>0$ , we have

$(l-k)|Bρ∩{w>l}|1-1n≤c⁢|Bρ||Bρ∖{w>k}|∫Bρ∩{k<w≤l}|Dw|dx$

for some $c=c⁢(n)>0$ .

Lemma 2.5.

Let ${yi}i=0∞$ be a sequence of nonnegative numbers such that

$yi+1≤b1⁢b2i⁢yi1+β,i=0,1,2,…$

for some $b1,β>0$ and $b2>1$ . If

$y0≤b1-1β⁢b2-1β2,$

then $yi→0$ as $i→∞$ .

Lemma 2.6.

Let $u∈L∞⁢(Bρ0)$ , $ρ0>0$ and $b>1$ . Suppose that for arbitrary $ρ∈(0,b-1⁢ρ0)$ , we have either

$osc⁢(u,Bρ)≤b1⁢ρα1 𝑜𝑟 osc⁢(u,Bρ)≤θ1⁢osc⁢(u,Bb⁢ρ)$

for some $b1>0$ , $α1∈(0,1]$ and $θ1∈(0,1)$ . Then, for all $ρ≤ρ0$ ,

$osc⁢(u,Bρ)≤bα⁢max⁡{osc⁢(u,Bρ0),b1⁢ρ0α1}⁢(ρρ0)α,$

where

$α:=min⁡{-logb⁡θ1,α1}.$

3 Caccioppoli inequality and boundedness

We prove a Caccioppoli-type inequality and the boundedness of quasi-minimizers of Φ. Let us start with a Caccioppoli-type inequality.

Lemma 3.1 (Caccioppoli estimates).

Let $u∈W1,Φ⁢(Ω)$ be a Q-minimizer of Φ. There exists $c=c⁢(Q,γ2)>0$ such that for any concentric balls $Bρ′⊂Bρ⊂Ω$ , $0<ρ′<ρ<∞$ , and $k∈R$ , we have

(3.1)

$∫Bρ′Φ⁢(x,|D⁢(u-k)±|)⁢𝑑x≤c⁢∫BρΦ⁢(x,(u-k)±ρ-ρ′)⁢𝑑x.$

Note that we shall write (3.1) $±$ , (3.1) $+$ , (3.1) $-$ as estimate (3.1) for $(u-k)±$ , $(u-k)+$ , $(u-k)-$ , respectively.

Proof.

Note that since $-u$ is also a Q-minimizer, it suffices to prove (3.1) $+$ . Let $η∈C0∞⁢(Bρ)$ be a cut-off function with $0≤η≤1$ , $η≡1$ in $Bρ′$ and $|D⁢η|≤2/(ρ-ρ′)$ , and consider $v=u-η⁢(u-k)+$ . Note that $supp⁢(u-v)⊂Ak,ρ$ . Then, by the definition of Q-minimizer we have

$∫Ak,ρ′Φ⁢(x,|D⁢u|)⁢𝑑x≤Q⁢∫Ak,ρΦ⁢(x,|D⁢v|)⁢𝑑x$

$=Q⁢∫Ak,ρΦ⁢(x,|(1-η)⁢D⁢u+(u-k)+⁢D⁢η|)⁢𝑑x$

$≤c0⁢(∫Ak,ρ∖Ak,ρ′Φ⁢(x,|D⁢u|)⁢𝑑x+∫Ak,ρΦ⁢(x,u-kρ-ρ′)⁢𝑑x)$

for some $c0=c0⁢(Q,γ2)≥1$ . By adding

$c0⁢∫Ak,ρ′Φ⁢(x,|D⁢u|)⁢𝑑x$

to both sides of the previous inequality, we have

(3.2)

$∫Ak,ρ′Φ⁢(x,|D⁢u|)⁢𝑑x≤ϑ⁢∫Ak,ρΦ⁢(x,|D⁢u|)⁢𝑑x+∫Ak,ρΦ⁢(x,u-kρ-ρ′)⁢𝑑x,$

where $ϑ=c0c0+1∈(0,1)$ , for any $0<ρ′<ρ<∞$ with $Bρ⊂Ω$ .

Now fix $ρ′<ρ$ and set $ρ0:=ρ′$ and $ρi+1=(1-λ)⁢λi⁢(ρ-ρ′)+ρi$ , $i=0,1,2,…$ , where $λ∈(0,1)$ will be determined later. Applying (3.2) inductively and using (2.1), we have

$∫Ak,ρ′Φ⁢(x,|D⁢u|)⁢𝑑x≤ϑ⁢∫Ak,ρ1Φ⁢(x,|D⁢u|)⁢𝑑x+∫Ak,ρ1Φ⁢(x,u-k(1-λ)⁢(ρ-ρ′))⁢𝑑x$

$≤ϑ2⁢∫Ak,ρ2Φ⁢(x,|D⁢u|)⁢𝑑x+ϑ⁢∫Ak,ρ2Φ⁢(x,u-k(1-λ)⁢λ⁢(ρ-ρ′))⁢𝑑x+∫Ak,ρΦ⁢(x,u-k(1-λ)⁢(ρ-ρ′))⁢𝑑x$

$≤ϑi⁢∫Ak,ρiΦ⁢(x,|D⁢u|)⁢𝑑x+∑k=0i-1(ϑ⁢λ-(γ2+1))k⁢∫Ak,ρΦ⁢(x,u-k(1-λ)⁢(ρ-ρ′))⁢𝑑x.$

Consequently, choosing $λ=λ⁢(Q,γ2)∈(0,1)$ such that $ϑ⁢λ-(γ2+1)=(θ+1)/2∈(0,1)$ and letting $i→∞$ , we have

$∫Ak,ρ′Φ⁢(x,|D⁢u|)⁢𝑑x≤22-θ⁢∫Ak,ρΦ⁢(x,u-kρ-ρ′)⁢𝑑x,$

which implies (3.1) $+$ . ∎

Now we prove the boundedness of quasi-minimizers of Φ.

Theorem 3.2 (Local boundedness).

Assume that $p⁢(⋅):Ω→(1,∞)$ is continuous and satisfies (1.4), and let $u∈W1,Φ⁢(Ω)$ be a Q-minimizer of Φ. Then u is locally bounded. Moreover, if $r>0$ satisfies

(3.3)

$r≤14 𝑎𝑛𝑑 ω⁢(4⁢r)≤min⁡{γ1⁢(1-θ)2⁢θ,1},$

where θ is determined in Lemma 2.3 and $B4⁢r⊂Ω$ , then we have

(3.4)

$Φp2⁢(∥u±∥L∞⁢(Br/2)r)≤c⁢(⨍BrΦp2⁢(u++rr)⁢𝑑x)1+2⁢(p2-p1)γ1⁢(1-θ)≤c⁢(⨍B4⁢r[Φ⁢(x,u±r)+1]⁢𝑑x)1+β0⁢(p2-p1)$

for some $c=c⁢(n,Q,γ1,γ2)≥1$ and $β0=β0⁢(n,Q,γ1,γ2)>0$ , where $p1:=infx∈B2⁢r⁡p⁢(x)$ and $p2:=supx∈B2⁢r⁡p⁢(x)$ .

Proof.

Note that since $-u$ is also a Q-minimizer of Φ, it suffices to show (3.4) only for $u+$ . Since

$p2-p1≤ω⁢(4⁢r)≤γ1⁢(1-θ)2⁢θ,$

by (3.3) we have

(3.5)

$p1-θ⁢p2≥γ1⁢(1-θ)-(p2-p1)⁢θ≥γ1⁢(1-θ)2$

and

(3.6)

$p1θ⁢p2≥1+p1-θ⁢p2θ⁢γ2≥1+(1-θ)⁢γ12⁢θ⁢γ2>1.$

By (3.1) $+$ , for $0<ρ′<ρ≤r$ and $k≥0$ , we have

$∫Bρ′Φp1⁢(|D⁢(u-k)+|)⁢𝑑x≤c⁢∫BρΦp2⁢((u-k)+ρ-ρ′)⁢𝑑x+c⁢|Ak,ρ|.$

Now, we consider the scaled function $u~⁢(x):=r-1⁢u⁢(r⁢x)$ . Then, from the previous estimate, we have that

(3.7)

$∫Bρ′Φp1⁢(|D⁢(u~-k)+|)⁢𝑑x≤c⁢∫BρΦp2⁢((u~-k)+ρ-ρ′)⁢𝑑x+c⁢|A~k,ρ|$

for all $0<ρ′<ρ≤1$ and all $k≥0$ , where

$A~k,ρ:={x∈Bρ:u~⁢(x)>k}.$

Let us set the following sequences:

$ki:=k0⁢(2-12i) and ρi:=12⁢(1+12i),i=0,1,2,….$

Here, $k0≥1$ will be determined later. We further define

$ρ¯i:=ρi+ρi+12,A~i:=A~ki,ρi and yi:=1Φp2⁢(k0)⁢∫A~iΦp2⁢(u~-ki)⁢𝑑x.$

Let $ηi∈C0∞⁢(Bρ¯i)$ be a cut off function such that $ηi≡1$ on $Bρi+1$ and $|D⁢ηi|≤4/(ρi-ρi+1)$ . Then applying (2.5), Hölder’s inequality with (3.6), (3.7) with $(k,ρ,ρ′)=(ki+1,ρi,ρ¯i)$ and (2.1), we have

$Φp2⁢(k0)⁢yi+1≤∫Bρ¯iΦp2⁢((u~-ki+1)+⁢ηi)⁢𝑑x$

$≤c⁢(∫Bρ¯iΦp2⁢(|D⁢[(u~-ki+1)+⁢ηi]|)θ⁢𝑑x)1θ$

$≤c⁢(∫Bρ¯iΦp2⁢(|D⁢[(u~-ki+1)+⁢ηi]|)p1p2⁢𝑑x)p2p1⁢|A~ki+1,ρi|1-θ⁢p2p1$

$≤c⁢(2(i+1)⁢(γ1+1)⁢∫A~ki+1,ρiΦp2⁢(u~-ki+1)⁢𝑑x+|A~ki+1,ρi|)p2p1⁢|A~ki+1,ρi|1-θ⁢p2p1.$

On the other hand, from the definitions of the sequences and (2.2), we see that

$|A~ki+1,ρi|≤∫A~ki+1,ρiΦp2⁢(u~-ki)Φp2⁢(ki+1-ki)⁢𝑑x=∫A~ki+1,ρiΦp2⁢(u~-ki)Φp2⁢(k0/2i+1)⁢𝑑x≤2(i+1)⁢(γ2+1)⁢yi$

and

$∫A~ki+1,ρiΦp2⁢(u~-ki+1)⁢𝑑x≤∫A~iΦp2⁢(u~-ki)⁢𝑑x.$

Combining the three estimates above, using (2.1) and the fact that $k0≥1$ , we obtain

$yi+1≤c⁢Φp2⁢(k0)-1⁢(Φp2⁢(k0)⁢2i⁢(γ2+1)⁢yi)p2p1⁢(2i⁢(γ2+1)⁢yi)1-θ⁢p2p1$

$≤c1⁢Φp2⁢(k0)(p2-p1)p1⁢2i⁢(1+β)⁢(γ2+1)⁢yi1+β,$

where $β:=(1-θ)⁢p2p1$ and $c1≥1$ depends only on $n,Q,γ1,γ2$ . Consequently, in view of Lemma 2.5, we see that $yi→0$ as $i→∞$ if

(3.8)

$yi≤1Φp2⁢(k0)⁢∫B1Φp2⁢(u~+)⁢𝑑x≤c1-1β⁢2-(1+β)⁢(γ2+1)β2⁢Φp2⁢(k0)-p2-p1p1⁢β.$

At this point, we take $k0≥1$ such that

$Φp2⁢(k0)1-p2-p1p1⁢β=c01β⁢2(1+β)⁢(γ2+1)β2⁢∫B1Φp2⁢(u~++1)⁢𝑑x.$

Then (3.8) holds, and hence we have $u~⁢(x)≤2⁢k0$ for a.e. $x∈B1/2$ , which together with (3.5) yields

$Φp2⁢(∥u~+∥L∞⁢(B1/2))=∥Φp2⁢(u~+)∥L∞⁢(B1/2)≤c⁢(∫B1Φp2⁢(u~++1)⁢𝑑x)1+p2-p1p1-θ⁢p2≤c⁢(⨍B1Φp2⁢(u~++1)⁢𝑑x)1+2⁢(p2-p1)γ1⁢(1-θ).$

Therefore, by the definition of $u~$ , that is, $u~⁢(x)=r-1⁢u⁢(r⁢x)$ , we obtain the first inequality in (3.4).

To prove the second inequality in (3.4), let $η~∈C0∞⁢(B2⁢r)$ be such that $0≤η~≤1$ , $η~≡1$ in $Br$ and $|D⁢η~|≤2r$ . Then, from (2.4) and Hölder’s inequality, we have that

$⨍BrΦp2⁢(u+r)⁢𝑑x≤⨍B2⁢rΦp2⁢(u+⁢η~r)⁢𝑑x$

$≤c⁢(⨍B2⁢rΦp2⁢(|D⁢[u+⁢η~]|)θ⁢𝑑x)1θ$

$≤c⁢(⨍B2⁢rΦp2⁢(|D⁢[u+⁢η~]|)p1p2⁢𝑑x)p2p1$

$≤(⨍B2⁢r[Φ⁢(x,D⁢u+)+Φ⁢(x,u+r)+1]⁢𝑑x)p2p1.$

Finally, applying (3.1) $+$ with $(k,ρ,ρ′)=(0,4⁢r,2⁢r)$ , we have

(3.9)

$⨍BrΦp2⁢(u+r)⁢𝑑x≤c⁢(⨍B4⁢r[Φ⁢(x,u+r)+1]⁢𝑑x)p2p1.$

This implies the second inequality in (3.4). ∎

We again point out that in order to obtain the boundedness of quasi-minimizers, it suffices to assume that $p⁢(⋅)$ is just plain continuous. We point out that estimate (3.4) is not homogenous which means the exponent on the right-hand side of (3.4) is not 1 but larger than 1. In the next corollary, however, if $p⁢(⋅)$ is log-Hölder continuous we can obtain a homogeneous estimate. Let us define

(3.10)

$M:=∫Ω[Φ⁢(x,|u|)+1]⁢𝑑x+1.$

Corollary 3.3.

Suppose $p⁢(⋅):Rn→(1,∞)$ satisfies (1.4) and (1.7), and let u be a Q-minimizer of Φ. For any $s>0$ , there exists a constant $cb⁢(s)=cb⁢(n,Q,γ1,γ2,L,s)≥1$ such that if $r>0$ satisfies

$r≤min⁡{18,1M} 𝑎𝑛𝑑 ω⁢(4⁢r)≤min⁡{γ1⁢(1-θ)2⁢θ,1}$

and $B4⁢r⊂Ω$ , then we have

(3.11)

$∥u±∥L∞⁢(Br/2)≤cb⁢(s)⁢{(⨍Bru±s⁢𝑑x)1s+r}.$

Proof.

We recall the results of the previous theorem. By (1.7) and (3.10), we observe

$(⨍B4⁢r[Φ⁢(x,u±r)+1]⁢𝑑x)p2-p1≤c⁢(Mrn+γ2+1)ω⁢(4⁢r)≤c⁢(4⁢r)-(n+γ2+2)⁢ω⁢(4⁢r)≤c.$

Then, from the first inequality in (3.4), (3.9) and the previous estimate, we have that

$Φp2⁢(∥u±∥L∞⁢(Br/2)r)≤c⁢⨍BrΦp2⁢(u±+rr)⁢𝑑x.$

Since $Φp2⁢(t1γ2+1)$ is a concave function of t, by Jensen’s inequality we see that

$⨍BrΦp2⁢(u±r)⁢𝑑x≤Φp2⁢((⨍Br[u±+rr]γ2+1⁢𝑑x)1γ2+1).$

Therefore, we obtain estimate (3.11) for $q=γ2+1$ , and so $q>γ2+1$ by Hölder’s inequality. On the other hand, when $q<γ2+1$ , estimate (3.11) follows from a standard interpolation method, see [27, Theorem 7.3]. This completes the proof. ∎

Remark 3.4.

In the proofs of Theorem 3.2 and Corollary 3.3 for $u+$ , we used the fact that the quasi-minimizer u satisfies (3.1) $+$ with $k≥0$ . Therefore, we see that if $u∈W1,Φ⁢(Ω)$ satisfies (3.1) $+$ for any $k≥0$ , then all results in Theorem 3.2 and Corollary 3.3 still hold for $u+$ .

4 Hölder continuity

In this section, we investigate the Hölder continuity of quasi-minimizers of Φ. We assume that $p⁢(⋅):Ω→(1,∞)$ satisfies (1.4) and is log-Hölder continuous, hence it satisfies (1.7). Let u be a Q-minimizer of Φ. Then, from the previous section, we know that u is locally bounded. Using this, in this section we will prove an oscillation decay estimate.

Suppose $r>0$ satisfies

(4.1)

$r≤min⁡{116,12⁢M} and ω⁢(8⁢r)≤min⁡{γ1⁢(1-θ)2⁢θ,1},$

and $B8⁢r⊂Ω$ , and set

$p1=infx∈Br⁡p⁢(x) and p2=supx∈Br⁡p⁢(x).$

Then, in view of Corollary 3.3, we have

(4.2)

$∥u∥L∞⁢(Br)≤cb⁢(s)⁢{(⨍B2⁢r|u|s⁢𝑑x)1s+2⁢r},$

where $s∈(0,∞)$ . Without loss of generality, we assume that

(4.3)

$|{x∈Br/2:u⁢(x)>supBr⁡u-12⁢osc⁢(u,Br)}|≤12⁢|Br/2|.$

If this is not true, then inequality (4.3) holds with u replaced by $-u$ . Therefore, in this case it suffices to consider $-u$ instead of u. We also recall the Caccioppoli-type inequality (3.1) $+$ :

(4.4)

$∫Ak,ρ′Φ⁢(x,|D⁢u|)⁢𝑑x≤c⁢∫Ak,ρΦ⁢(x,u-kρ-ρ′)⁢𝑑x,$

where $0<ρ′<ρ≤r$ . Finally, from (1.7), (3.10), (4.1) and (4.2) with $s=γ1$ , we observe that

$(∥u∥L∞⁢(Br)r)p2-p1≤c⁢(1rγ1⁢⨍B2⁢r|u|γ1⁢𝑑x)p2-p1γ1+c$

$≤c⁢(2⁢r)-ω⁢(2⁢r)⁢(n+γ1+1)γ1+c$

(4.5)

$≤c.$

Lemma 4.1.

For any $τ∈(0,1)$ , there exists a large $m=m⁢(n,Q,γ1,γ2,τ)∈N$ with $m>2$ such that if

(4.6)

$Φp2⁢(osc⁢(u,Br)2m+1⁢r)>1,$

then

$|Ak⁢(m),r/2|:=|{x∈Br/2:u(x)>supBru-2-mosc(u,Br)}|≤τ|Br/2|,$

where $k⁢(m):=supBr⁡u-2-m⁢osc⁢(u,Br)$ .

Proof.

Let $m>2$ be a large natural number that will be determined later. Define for $j=0,1,…,m-1$ ,

$kj:=supBr⁡u-osc⁢(u,Br)2j+1 and Cj:=Akj,r/2∖Akj+1,r/2$

Then, applying Lemma 2.4 to $(w,l,k,ρ)=(u,kj+1,kj,r/2)$ , we obtain

$∫Cj|Du|dx≥c(kj+1-kj)|Akj+1,r/2|1-1n|Br/2∖Akj,r/2|r-n$

(4.7)

$≥c⁢2-j⁢osc⁢(u,Br)⁢|Akj+1,r/2|1-1n.$

Here, we have used inequality (4.3), and so $|Br/2∖Akj,r/2|≥12⁢|Br/2|$ . On the other hand, by using (4.4) with $(k,ρ,ρ′)=(kj,r,r/2)$ , (4.6) and (4) with the fact $osc⁢(u,Br)≤2⁢∥u∥L∞⁢(Br)$ , we have

$∫CjΦp1⁢(|D⁢u|)⁢𝑑x≤c⁢∫Akj,rΦp2⁢(osc⁢(u,Br)2j⁢r)⁢𝑑x+c⁢|Akj,r|$

$≤c⁢Φp2⁢(osc⁢(u,Br)2j⁢r)⁢|Akj,r|$

$=c⁢(osc⁢(u,Br)2j⁢r)p2-p1⁢Φp1⁢(osc⁢(u,Br)2j⁢r)⁢|Akj,r|$

$≤c⁢Φp1⁢(osc⁢(u,Br)2j⁢r)⁢|Akj,r|.$

Since $Φp1$ is strictly increasing and convex, its inverse $Φp1-1$ is well defined and concave. Therefore, from Jensen’s inequality and the previous inequality, it follows that

$⨍Cj|Du|dx≤Φp1-1(⨍CjΦp1(|Du|)dx)$

$≤Φp1-1⁢(c⁢|Akj,r||Cj|⁢Φp1⁢(osc⁢(u,Br)2j⁢r))$

$≤Φp1-1⁢(Φp1⁢([c⁢|Akj,r||Cj|]1p1⁢osc⁢(u,Br)2j⁢r))$

(4.8)

$≤c⁢rnp1-1⁢|Cj|-1p1⁢2-j⁢osc⁢(u,Br).$

Note that in the third inequality above we have used the fact that $c⁢|Akj,r||Cj|≥1$ . Therefore, combining (4) and (4), we see for $j=0,1,…,m-2$ ,

$|Akm-1,r/2|(n-1)⁢p1n⁢(p1-1)≤|Akj+1,r/2|(n-1)⁢p1n⁢(p1-1)≤c⁢rn-p1p1-1⁢|Cj|.$

Hence, summing up the previous inequalities for $j=0,1,…,m-2$ , we have

$|Akm-1,r/2|(n-1)⁢p1n⁢(p1-1)≤c⁢r(n-1)⁢p1p1-1m-1,$

which finally implies

$|Ak⁢(m),r/2|=|Akm-1,r/2|≤(cm-1)(n-1)⁢p1n⁢(p1-1)⁢rn≤τ⁢|Br/2|,$

by taking sufficiently large $m=m⁢(n,Q,γ1,γ2,L,τ)$ . ∎

Lemma 4.2.

There exists a small $τ0=τ0⁢(n,Q,γ1,γ2,L)∈(0,2-n-1)$ such that for $0<λ<osc⁢(u,Br)/2$ , if

(4.9)

$Φp2⁢(λr)>1 𝑎𝑛𝑑 |Ak0,r/2|≤τ0⁢|Br/2|,where ⁢k0=supBr⁡u-λ,$

then

(4.10)

$supBr/4⁡u≤k0+λ2=supBr⁡u-λ2.$

Proof.

For $i=0,1,2,…$ , define

$ρi:=r4⁢(1+12i),ki:=k0+λ2⁢(1-12i),Ai:=Aki,ρi,Ci+1:=Aki,ρi+1∖Ai+1 and yi=|Ai|/|Br/2|.$

Note that, by the definitions of $k0$ and $ki$ , we have $(u-ki)+≤λ$ and $λ≤∥u∥L∞⁢(Br)$ . In view of (4.4) with $(k,ρ,ρ′)=(ki,ρi,ρi+1)$ , (2.1), the first assumption on (4.9) and (4) with $λ≤∥u∥L∞⁢(Br)$ , we have

$∫Aki,ρi+1Φp1⁢(|D⁢u|)⁢𝑑x≤c⁢∫AiΦp2⁢(2i+3⁢(u-ki)+r)⁢𝑑x+c⁢|Ai|≤c⁢2i⁢(γ2+1)⁢Φp2⁢(λr)⁢|Ai|≤c⁢2i⁢(γ2+1)⁢Φp1⁢(λr)⁢|Ai|.$

On the other hand, in the same way we estimated inequalities (4) and (4), we derive

$⨍Ci+1|Du|dx≤[c|Ai||Ci+1|2i⁢(γ2+1)]1p2λr≤c|Ai||Ci+1|2i⁢(γ2+1)λr$

$and ∫Ci+1|Du|dx≥c(ki+1-ki)|Ai+1|1-1n|Bρi+1∖Aki,ρi+1|ρi+1-n$

$≥c⁢2-i⁢λ⁢r-n⁢|Ai+1|1-1n⁢(|Br/4|-τ0⁢|Br/2|)$

$≥c⁢2-i⁢λ⁢rn-1⁢yi+11-1n.$

In the last inequality, we have used the fact that $τ0<2-n-1$ . From those estimates, we have

$yi+11-1n≤c2iλ-1r1-n∫Ci+1|Du|dx≤c2i⁢(γ2+2)r-n|Ai|≤c2i⁢(γ2+2)yi,$

and so

$yi+1≤c2⁢2i⁢n⁢(γ2+2)n-1⁢yi1+1n-1$

for some $c2=c2⁢(n,Q,γ1,γ2,L)≥1$ . Finally, in view of Lemma 2.5, by taking $τ0>0$ sufficiently small such that

$y0=|Ak0,r/2|/|Br/2|≤τ0≤c2-(n-1)⁢2-n⁢(n-1)⁢(γ2+2),$

we obtain $|Ak0+λ/2,r/4|=0$ , which implies (4.10). ∎

From the two lemmas above we obtain the following decay estimate for the oscillation of quasi-minimizers.

Proposition 4.3.

Let $m=m⁢(n,Q,γ1,γ2,L,τ0)=m⁢(n,Q,γ1,γ2,L)$ be the positive integer determined in Lemma 4.1, where $τ0∈(0,2-n-1)$ is given in Lemma 4.2. Then we have either

(4.11)

$osc⁢(u,Br)≤2m+1⁢r$

(4.12)

$osc⁢(u,Br/4)≤(1-12m+1)⁢osc⁢(u,Br).$

Proof.

If inequality (4.6) dose not hold, that is,

$osc⁢(u,Br)≤2m+1⁢Φp2-1⁢(1)⁢r,$

then, we obtain (4.11) since $Φp2-1⁢(1)≤1$ . Therefore, we suppose that (4.6) holds and then we will show (4.12). Recall the notations and the results in Lemma 4.1 and Lemma 4.2. Set

$λ:=osc⁢(u,Br)2m.$

Then we see that $k⁢(m)=supBr⁡u-λ=k0$ ,

$|Ak0,r/2|=|Ak⁢(m),r/2|≤τ0⁢|Br/2|,$

and by (4.6),

$Φp2⁢(λr)=Φp2⁢(osc⁢(u,Br)2m⁢r)>Φp2⁢(osc⁢(u,Br)2m+1⁢r)>1.$

Therefore, in view of Lemma 4.2, we have

$supBr/4⁡u≤supBr⁡u-λ2=supBr⁡u-osc⁢(u,Br)2m+1,$

and so

$osc⁢(u,Br)≤2m+1⁢(supBr⁡u-supBr/4⁡u)≤2m+1⁢(osc⁢(u,Br)-osc⁢(u,Br/4)),$

which implies (4.12). ∎

Finally, applying Lemma 2.6 to the previous proposition, we obtain the Hölder inequality for the quasi-minimizers of Φ.

Theorem 4.4 (Hölder continuity).

Suppose $p⁢(⋅):Rn→(1,∞)$ satisfies (1.4) and (1.7), and let u be a Q-minimizer of Φ. There exists $α=α⁢(n,Q,γ1,γ2,L)∈(0,1)$ such that $u∈Cloc0,α⁢(Ω)$ . Moreover, if $r>0$ satisfies (4.1) and $B8⁢r⊂Ω$ , then for any $x,y∈Br$ ,

$|u(x)-u(y)|≤c(|x-y|r)α(∥u∥L∞⁢(Br)+r)≤c~(s)(|x-y|r)α{(⨍B2⁢r|u|sdx)1s+r}$

for some $c⁢(n,Q,γ1,γ2,L)>0$ and $c~⁢(s)=c~⁢(n,Q,γ1,γ2,L,s)>0$ with $s∈(0,∞)$ .

5 The Harnack inequality

We prove the Harnack inequality for quasi-minimizers of Φ. Suppose $p⁢(⋅):Ω→(1,∞)$ satisfies (1.4) and is log-Hölder continuous, hence satisfies (1.7), and let $u∈W1,Φ⁢(Ω)$ be a nonnegative Q-minimizer of Φ. We assume that $r>0$ satisfies (4.1) and $B10⁢r⊂Ω$ , and shall use notations and results of the previous section. The next proposition is our main result in this section.

Proposition 5.1.

For any $τ1∈(0,1)$ , there exists a small $δ=δ⁢(n,Q,γ1,γ2,τ1)>0$ such that for $0<λ≤∥u∥L∞⁢(Br)$ if

(5.1)

$|{x∈Br:u⁢(x)≥λ}|≥τ1⁢|Br|,$

then

(5.2)

$infBr⁡u+r≥δ⁢λ.$

To prove Proposition 5.1, we first prove the following lemma.

Lemma 5.2.

For any $τ1,τ2∈(0,1)$ , there exists a large positive integer $m1=m1⁢(n,Q,γ1,γ2,L,τ1,τ2)$ such that for any $0<λ≤∥u∥L∞⁢(Br)$ if

(5.3)

$r≤2-m1⁢λ,$

(5.4)

$μ:=infB4⁢r⁡u≤2-m1⁢λ$

and (5.1) holds, then

(5.5)

$|{x∈B2⁢r:u⁢(x)<μ+2-m1⁢λ}|≤τ2⁢|B2⁢r|.$

Proof.

Fix a positive integer $m1$ that will be determined later. For $j=1,2,…,m1$ , define

$kj:=μ+2-j⁢λ and Dj:=Aj,2⁢r-∖Aj+1,2⁢r-.$

Note that from (5.1) and (5.4), we have

$|{x∈B2⁢r:u⁢(x)≥kj}|≥|{x∈Br:u⁢(x)≥(2-m1+2-j)⁢λ}|≥τ1⁢|Br|.$

Then, applying Lemma 2.4 to $(w,l,k,ρ)=(u,kj,kj+1,2⁢r)$ , from the previous inequality we have that

$τ12-jλ|Akj,2⁢r-|1-1n≤c∫Dj|Du|dx=c∫Dj|D(u-kj)-|dx.$

On the other hand, by (3.1) $-$ with $(k,ρ,ρ′)=(kj,4⁢r,2⁢r)$ , (5.3) and (5.4), we have

$∫B2⁢rΦp1⁢(|D⁢(u-kj)-|)⁢𝑑x≤c⁢∫B4⁢rΦp2⁢((u-kj)-r)⁢𝑑x+c⁢|Akj,4⁢r-|$

$≤c⁢Φp2⁢(2-j⁢λr)⁢|B4⁢r|+c⁢(2-j⁢λr)⁢|B4⁢r|$

$≤c⁢Φp2⁢(2-j⁢λr)⁢|B4⁢r|$

$≤c⁢Φp1⁢(2-j⁢λr)⁢|B4⁢r|.$

In the last inequality above, we have used (4). Therefore, from Lemma 2.4, the two estimates above and Jensen’s inequality, we obtain

$τ1⁢2-j⁢λ|Dj||Akj,2⁢r-|1-1n≤c⨍Dj|D(u-kj)-|dx$

$≤c⁢Φp1-1⁢(⨍DjΦp1⁢(|D⁢(u-kj)-|)⁢𝑑x)$

$≤c⁢Φp1-1⁢(Φp1⁢(2-j⁢λr)⁢c⁢|B4⁢r||Dj|)$

$≤c⁢2-j⁢λ⁢|B4⁢r|1p1r⁢|Dj|1p1,$

and so

$|Akm1,2⁢r-|(n-1)⁢p1n⁢(p1-1)≤|Akj,2⁢r-|(n-1)⁢p1n⁢(p1-1)≤c⁢τ1-p1p1-1⁢rn-p1p1-1⁢|Dj|.$

Summing up this estimate for $j=1,2,…,m1$ , we have

$|Akm1,2⁢r-|(n-1)⁢p1n⁢(p1-1)≤c⁢τ1-p1p1-1m1⁢rn-p1p1-1⁢|B2⁢r|≤c3⁢τ1-p1p1-1m1⁢|B2⁢r|(n-1)⁢p1n⁢(p1-1)$

for some $c3=c3⁢(n,Q,γ1,γ2,L)≥1$ . At this point, we choose $m1$ large so that

$c3⁢τ1-p1p1-1/m1≤τ2n⁢(p1-1)(n-1)⁢p1.$

Hence, we obtain (5.5). ∎

Now, we prove Proposition 5.1.

Proof of Proposition 5.1.

Let $m1$ be the positive number determined in Lemma 5.2 with $τ2=(4⁢cb⁢(1))-1$ , where $cb⁢(1)≥1$ is the constant determined in Corollary 3.3 when $s=1$ . Hence, $m1$ depends only on $n,Q,γ1,γ2$ and $τ1$ . We first observe that if

$r>cb⁢(1)-1⁢2-(m1+2)⁢λ or infB4⁢r⁡u>2-m1⁢λ,$

then (5.2) is automatically satisfied with $δ=cb⁢(1)-1⁢2-(m1+2)⁢λ$ . Now we assume that

(5.6)

$r≤cb⁢(1)-1⁢2-(m1+3)⁢λ and infB4⁢r⁡u≤2-m1⁢λ.$

From these, we see that (5.3) and (5.4) are satisfied, and so by Lemma 5.2 we have (5.5). Since $-u+μ+2-m1⁢λ$ is also a Q-minimizer of Φ, applying (4.2) of Corollary 3.3 with $s=1$ and r replaced by $2⁢r$ , we have

(5.7)

$infBr⁡u≥μ+2-m1⁢λ-cb⁢(1)⁢{⨍B2⁢r(-u+μ+2-m1⁢λ)+⁢𝑑x+2⁢r}.$

Inserting (5.5) with $τ2=(4⁢cb⁢(1))-1$ and the first inequality in (5.6) into the previous inequality, we have

$infBr⁡u≥μ+2-m1⁢λ-cb⁢(1)⁢{(4⁢cb⁢(s))-1⁢(μ+2-m1⁢λ)+(4⁢cb⁢(1))-1⁢2-m1⁢λ}≥2-(m1+1)⁢λ≥cb⁢(1)-1⁢2-(m1+2)⁢λ,$

which implies (5.2) with $δ=cb⁢(1)-1⁢2-(m1+2)$ . ∎

From Proposition 5.1 and the covering theorem of Krylov and Safonov [29], see also [27, Proposition 7.2], we have the weak Harnack inequality for the quasi-minimizers of Φ. Since the proof is standard and exactly the same to that of [28, Theorem 5.7], we omit it here.

Theorem 5.3 (The weak Harnack inequality).

Let $p⁢(⋅):Ω→(1,∞)$ satisfy (1.4) and (1.7), and let $u∈W1,Φ⁢(Ω)$ be a nonnegative Q-minimizer of Φ. There exists $s0=s0⁢(n,Q,γ1,γ2,L)>0$ such that if $r>0$ satisfies (4.1) and $B10⁢r⊂Ω$ , then we have

$(⨍Br/nus0⁢𝑑x)1s0≤c⁢(⨍Q2⁢r/nus0⁢𝑑x)1s0≤c⁢(infBr⁡u+r)$

for some $c=c⁢(n,Q,γ1,γ2,L)>0$ , where $Q2⁢r/n$ is the cube with the center same to $Br/n$ and the side length $2⁢r/n$ , hence $Br/n⊂Q2⁢r/n⊂Br$ .

Remark 5.4.

To prove Theorem 5.3, we use the fact that the nonnegative quasi-minimizer u satisfies (4) and (3.1) $-$ with $k∈ℝ$ . Note that this condition also implies estimate (5.7), since $-u+μ+2m1⁢λ$ satisfies (3.1) $+$ , see Remark 3.4. Therefore, we see that if $u∈W1,Φ⁢(Ω)$ is nonnegative and satisfies (3.1) $-$ for any $k∈ℝ$ and (4), then we have the same results as in Theorem 5.3.

Finally, in view of Corollary 3.3 and Theorem 5.3 with a standard covering argument, we obtain the Harnack inequality.

Theorem 5.5 (The Harnack inequality).

Let $p⁢(⋅):Ω→(1,∞)$ satisfy (1.4) and (1.7), and let $u∈W1,Φ⁢(Ω)$ be a nonnegative Q-minimizer of Φ. There exist $ϵ0=ϵ0⁢(n,γ1,γ2,ω⁢(⋅))∈(0,1)$ and $c=c⁢(n,Q,γ1,γ2,L)>1$ such that if $r>0$ satisfies

$r≤ϵ0⁢M-1,𝑤ℎ𝑒𝑟𝑒 M:=∫Ω[Φ⁢(x,u)+1]⁢𝑑x+1,$

and $B2⁢r⊂Ω$ , then we have

$supBr⁡u≤c⁢(infBr⁡u+r).$

We end this paper with the following remark.

Remark 5.6.

Define the De Giorgi class of Φ, $D⁢GΦ±⁢(Ω)$ , by the set of all $u∈W1,Φ⁢(Ω)$ satisfying (3.1) $±$ . Note that if $w∈D⁢GΦ±⁢(Ω)$ then $-w∈D⁢GΦ∓⁢(Ω)$ , and that if $u∈W1,Φ⁢(Ω)$ is a quasi-minimizer of Φ then we have $u∈D⁢GΦ+⁢(Ω)∩D⁢GΦ-⁢(Ω)$ . From the proofs of the theorems presented in this paper, one can deduce that if $u∈D⁢GΦ+⁢(Ω)∩D⁢GΦ-⁢(Ω)$ , then it is Hölder continuous and satisfies the Harnack inequality.

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Received: 2016-4-7

Revised: 2016-4-25

Accepted: 2016-4-25

Published Online: 2016-5-27

Published in Print: 2018-5-1

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Articles in the same Issue

https://doi.org/10.1515/anona-2016-0083

Keywords for this article

Non-standard growth; quasi-minimizer; Hölder continuity; Harnack inequality, variable exponent

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