Density of C inW 1,p (x ) with discontinuous exponent p (x )

Math. Nachr. 279, No. 1–2, 142 – 149 (2006) / DOI 10.1002/mana.200310351

Density of C ∞ (Ω) in W 1,p(x)(Ω) with discontinuous exponent p(x) Xianling Fan∗1 , Suyun Wang∗∗1 , and Dun Zhao∗∗∗1 1

Department of Mathematics, Lanzhou University, Lanzhou, 730000, P. R. of China Received 31 May 2003, revised 25 August 2004, accepted 21 September 2004 Published online 20 December 2005 Key words Density, generalized Sobolev space, Lavrentiev phenomenon MSC (2000) 46E30, 46E35 In this paper, we present a sufficient condition for the density of C ∞ (Ω) in W 1,p(x) (Ω) with p(x) being discontinuous. Roughly speaking, we can get the density under the following assumptions: 1) Ω can be divided into some open pieces Ωi such that p(x) is “fine”on each Ωi , for example, p(x) satisfies the Dini–Lipschitz condition on each Ωi . 2) The corresponding pieces “get along with each other regularly ”. c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 

1

Introduction

The generalized Sobolev spaces W 1,p(x) (Ω) with variable exponent p(x) have been investigated by many authors in recent years, for references, we refer to [1–16]. In this paper, we consider the density of C ∞ (Ω) in W 1,p(x) (Ω). It is known that in general, C ∞ (Ω) may not be dense in W 1,p(x) (Ω). In [5], Edmunds and R´akosn´ık have presented a sufficient condition to guarantee the density of C ∞ (Ω) in W 1,p(x) (Ω), which is called Condition (C) in this paper (see Remark 2.13). The condition does not require the continuity of p(x) but requires a kind of local cone-monotonicity. Another type of sufficient condition to ensure the density of C ∞ (Ω) in W 1,p(x) (Ω) is the Dini–Lipschitz condition, which is expressed as C , − ln |x − y|

|p(x) − p(y)| ≤

for all x , y ∈ Ω ,

|x − y| ≤

1 2

(see [4, 9, 16]). In [7], the first named author has obtained the density under the hypothesis that p(x) is H¨older continuous. In fact the method used in [7] is also valid for the case of the Dini–Lipschitz condition. Zhikov ([17, 18]) has presented an example that C ∞ (Ω) is not dense in W 1,p(x) (Ω) as follows.
 Zhikov’s example Let Ω = x = (x1 , x2 ) ∈ R2 : |x| < 1 ,  p(x) =

α1 α2

if if

x1 x2 > 0 , x1 x2 < 0 ,

where 1 < α1 < 2 < α2 , then C ∞ (Ω) is not dense in W 1,p(x) (Ω) and the corresponding integral functional admits the Lavrentiev phenomenon. Zhikov has also pointed out that, in the example, if Ω is replaced by Ωε = Ω\B(0, ε), where ε ∈ (0, 1), then  the Lavrentiev phenomenon will not appear in fact, in this situation, C ∞ (Ωε ) is dense in W 1,p(x) (Ωε ) . Acerbi ([1]) has considered the case of that p(x) takes only two values and jumps across a Lipschitz surface and gives the regularity of the minimizer of the correspondence integral functional. ∗ ∗∗ ∗∗∗

Corresponding author: e-mail: [email protected], Phone: +869318911173, Fax: +869318912444 e-mail: [email protected], Phone: +869317679111, Fax: +869318912444 e-mail: [email protected], Phone: +869318911971, Fax: +869318912444

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In this paper, we will give a sufficient condition for the density of C ∞ (Ω) in W 1,p(x) (Ω) with discontinuous p(x). The idea is that, assuming that p(x) is discontinuous on Ω, if Ω can be divided into some open pieces Ωi , p(x) is “fine” on each Ωi (for example, the Dini–Lipschitz condition is satisfied on each Ωi ), and the corresponding pieces get along with each other regularly, then the density can be guaranteed.

2

The results on density

Let Ω be an open subset of RN , set   ∞ ∞ L+ (Ω) = p ∈ L (Ω) : ess inf p(x) ≥ 1 . x∈Ω

For p ∈ L∞ + (Ω), we define   Z u | u : Ω → R is measurable and |u(x)|p(x) dx < ∞ , Ω  W 1,p(x) (Ω) = u ∈ Lp(x) (Ω) : |∇u| ∈ Lp(x) (Ω) .

Lp(x) (Ω) =

On the basic properties of the space W 1,p(x) (Ω), we refer to [3–16]. In this paper, the norm on the space W 1,p(x) (Ω) is defined by uW 1,p(x) (Ω) = uLp(x) (Ω) + ∇uLp(x) (Ω) , where

(

uLp(x) (Ω) = inf

) ˛ Z ˛ ˛ u(x) ˛p(x) ˛ ˛ dx ≤ 1 . λ>0: ˛ λ ˛ Ω

For p ∈ L∞ + (Ω), set p− (Ω) = ess inf p(x) , x∈Ω

p+ (Ω) = ess sup p(x) . x∈Ω

Let Q be an open subset of Ω. We denote W 1,p(x) (Ω|Q ) =


 v = u|Q : u ∈ W 1,p(x) (Ω) .

Obviously, W 1,p(x) (Ω|Q ) ⊂ W 1,p(x) (Q), and when every element in W 1,p(x) (Q) has an extension to W 1,p(x) (Ω), we have W 1,p(x) (Ω|Q ) = W 1,p(x) (Q). It is easy to see that, if C ∞ (Q) ∩ W 1,p(x) (Q) is dense in W 1,p(x) (Ω|Q ), C ∞ (V ) ∩ W 1,p(x) (V ) is also dense in W 1,p(x) (Ω|V ) for any open subset V of Q. Definition 2.1 Let Ω be an open subset of RN , p ∈ L∞ (Ω), and x ∈ Ω. x is called a p(x)-regular point of Ω  +  ∞ if there is an open neighborhood Q of x such that C Q ∩ Ω ∩ W 1,p(x) (Q ∩ Ω) is dense in W 1,p(x) (Ω|Q∩Ω )   in the sense of the norm of W 1,p(x) (Q ∩ Ω) . The following theorem shows that the problem on the density of C ∞ (Ω) in W 1,p(x) (Ω) can be localized, i.e., we can deal with it locally for each point in Ω. Theorem 2.2 Let Ω be an open subset of RN , p ∈ L∞ + (Ω). ◦ 1 . If each point in Ω is a p(x)-regular point of Ω, then C ∞ (Ω) ∩ W 1,p(x) (Ω) is dense in W 1,p(x) (Ω). 1,p(x) 2◦ . If each point in Ω is a p(x)-regular point of Ω, then C ∞ ( Ω ) ∩ W 1,p(x)  N (Ω)  is dense in W  (Ω).  ◦ N N ∞ 1,p(x) 3 . If Ω = R and each point in R is a p(x)-regular point, then C0 R is dense in W RN . P r o o f. 1◦ . Taking any x ∈ Ω, since x is a p(x)-regular point of Ω, we can find a bounded open neighborhood Qx ⊂ Ω of x such that C ∞ ( Qx ) is dense in W 1,p(x) (Ω|Qx ). As {Qx : x ∈ Ω} is an open cover of Ω, we can www.mn-journal.com

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choose a countable subcover U = {Qi : i= 1, 2, . . . } with locally finite property

and a smooth unit partition {λi } subordinated to U such that λi ∈ C0∞ RN , suppλi ⊂ Qi , λi (x) ≥ 0, and ∞ i=1 λi (x) = 1 for x ∈ Ω. Set   Mi = max max |∇λi (x)|, 1 . x∈Ω

For any u ∈ W 1,p(x) (Ω) and ε > 0, taking a positive sequence {εi } such that C ∞ ( Qi ) in W 1,p(x) (Ω|Qi ), there are ϕi ∈ C ∞ ( Qi ) such that εi , Mi

ϕi − uW 1,p(x) (Qi ) ≤ Set ϕ(x) =

∞ 

λi (x)ϕi (x) ,

∞

i=1 εi

≤ ε, by the density of

i = 1, 2, ...

for all x ∈ Ω ,

i=1

then ϕ ∈ C ∞ (Q) and ϕ(x) − u(x) =

∞ 

λi (x)(ϕi (x) − u(x)) ,

for all x ∈ Ω ,

i=1

which shows

∞ 

 εi ≤ ε, Mi i=1 i   ∇ϕ − ∇uLp(x) (Ω) ≤ ∇ϕi − ∇uLp(x) (Qi ) + Mi ϕi − uLp(x) (Qi )

ϕ − uLp(x) (Ω) ≤

ϕi − uLp(x) (Qi ) ≤

i

 εi  ≤ + εi ≤ 2ε . Mi i i

i

Therefore, ϕ − uW 1,p(x) (Ω) ≤ 3ε, which reaches conclusion 1◦ . 2◦ . The proof of conclusion 2◦ issimilar to that of 1◦ .  N  N ∞ 1,p(x) R ∩ W R 3◦ . Given any u ∈ W 1,p(x) RN and ε > 0, by the conclusion 1◦, there is v ∈ C  ∞ N be such that ϕk (x) = 1 if |x| ≤ k, such that u − vW 1,p(x) (RN ) ≤ ε/2. For each k > 0, let ϕk ∈ C0 R ϕk (x) = 0 if |x| ≥ k + 2, ϕk (x) ∈ (0, 1) and |∇ϕk (x)| ≤ 1 if k < |x| < k + 2. Setting wk (x) = ϕk (x)v(x) for   x ∈ RN , then wk ∈ C0∞ RN . It is easy to see that wk − vW 1,p(x) (RN ) → 0 as k → ∞. Take k ∗ sufficiently large such that wk∗ − vW 1,p(x) (RN ) ≤ ε/2, then wk∗ − uW 1,p(x) (RN ) ≤ ε, which reaches conclusion 3◦ . In the following we will consider the local properties at each point in Ω. For simplicity, we assume that Ω is a bounded open subset of RN , thus we have C ∞ ( Ω ) ⊂ W 1,p(x) (Ω). The closure of C0∞ (Ω) in W 1,1 (Ω) and the closure of C0∞ (Ω) in W 1,p(x) (Ω) are denoted by W01,1 (Ω) and by 1,p(x) W0 (Ω), respectively. As we always have W 1,p(x) (Ω) ⊂ W 1,1 (Ω), when Ω has Lipschitz boundary, for any u ∈ W 1,p(x) (Ω), we can talk about the trace u|∂Ω of u on ∂Ω. Definition 2.3 Let Q be a bounded open subset of Ω, and p ∈ L∞ + (Ω). 1◦ . (Ω, p(x)) is said to satisfy Condition (E) if there is an extension p (x) of p(x) to RN such that     p − RN = p− (Ω) , p + RN = p+ (Ω) , N 1,e p(x) and for any open subset (Q) can be extended to an  N Q of R with Lipschitz boundary, every element in W 1,e p(x) element in W R . 2◦ . (Ω, p(x)) is said to satisfy Condition (D) if C ∞ (Ω) is dense in W 1,p(x) (Ω). 3◦ . (Ω, p(x)) is said to satisfy Condition (D0 ) if (Ω, p(x)) satisfies Condition (D) and for any u ∈ W 1,p(x) (Ω) and ε > 0, there is a function v ∈ C ∞ ( Ω ) such that v − uW 1,p(x) (Ω) ≤ ε and v|S = 0 provided u|S = 0, where S is an (N − 1)-dimensional Lipschitz manifold in ∂Ω (with or without boundary). c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 

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4◦ . (Ω, p(x)) is said to satisfy Condition (F ) (or be “fine”), if (Ω, p(x)) satisfies Condition (E), and there is an extension of p(x) to RN mentioned in Condition (E), which is written as p (x), such that for any open subset Q of RN with Lipschitz boundary, (Q, p (x)) satisfies Condition (D0 ). Theorem 2.4 Let Ω be a bounded open subset of RN with Lipschitz boundary, p ∈ L∞ + (Ω). If p(x) satisfies the Dini–Lipschitz condition on Ω, then (Ω, p(x)) satisfies Condition (F ). P r o o f. For the fact that (Ω, p(x)) satisfies Condition (E), we refer to [3, 4, 6] in which it is pointed out that there is an extension p (x) of p(x) to RN such that p (x) satisfies the Dini–Lipschitz condition on RN . From this we can see that for any open subset Q of RN with Lipschitz boundary, (Q, p (x)) satisfies Condition (D) (see [3, 4, 6, 7, 9, 16]). Below we will use the method of outward cone-mollifier (see [7]) to prove that (Q, p (x)) satisfies Condition (D0 ). For brevity, we will write p(x) instead of p (x). Let S be an (N − 1)-dimensional Lipschitz manifold in ∂Q, and u ∈ W 1,p(x) (Q) with u|S = 0. Given any x0 ∈ S, there is a ball B(x0 , 2r) and an open cone K ⊂ RN with vertex at the origin such that for a sufficiently small δ > 0, (x + Kδ ) ∩ Q = ∅ ,

for all x ∈ B(x0 , 2r) ∩ ∂Q ,

where Kδ = K ∩ B(0, δ). We define additionally u(x) = 0 for x ∈ y∈S∩B(x0 ,2r) (y + Kδ ) and take an    N 1,p(x) N ∞ extension of u to RN , which R . Take J ∈ C R such that is still written as u, such that u ∈ W 0  J ≥ 0, suppJ ⊂ K1 and RN J(x) dx = 1. For sufficiently small δ > 0, define Jδ u : B(x0 , r) → R by  u(x + δz)J(z) dz , for all x ∈ B(x0 , r) . (Jδ u)(x) = RN

Then Jδ u ∈ C ∞ (B(x0 , r)) and Jδ u(x) = 0 for x ∈ S ∩ B(x0 , r). Relying on the Dini–Lipschitz condition and using the standard arguments (see e.g. [4, 7, 9, 16]), we can prove Jδ u → u in W 1,p(x) (B(x0 , r)) as δ → 0. From this we can conclude that (Q, p(x)) satisfies Condition (D0 ), therefore (Ω, p(x)) satisfies Condition (F ). The theorem is proved. The authors do not know whether the Edmunds–R´akosn´ık’s Condition (C) implies Condition (F ). Hereafter p (x) always denotes a given extension of p(x) to RN satisfying the requirement of Condition (E) and Condition (F ). Definition 2.5 Let Ω1 and Ω2 be two nonempty disjoint open subsets of RN . Ω1 and Ω2 are said to be get along with each other regularly if either Ω1 ∩Ω2 = ∅ or ∂Ω1 ∩∂Ω2 is an (N −1)-dimensional Lipschitz manifold. Lemma 2.6 Assume that Ω1 and Ω2 are two nonempty disjoint bounded open subsets of RN with Lipschitz boundary, Ω1 and Ω2 get along with each other regularly, Ω = int ( Ω1 ∪ Ω2 ), and Ω has Lipschitz boundary. Let p ∈ L∞ + (Ω), p1 = p|Ω1 , p2 = p|Ω2 . If − (1) p+ 1 (Ω1 ) ≤ p2 (Ω2 ), (2) (Ω1 , p1 (x)) satisfies Condition (D0 ), (3) (Ω2 , p2 (x)) satisfies Condition (E) with an extension p 2 (x) of p2 (x) to RN . Then we have p2 (x) 1◦ . For any u ∈ W 1,p(x) (Ω) and ε > 0, there is v ∈ W 1,e (Ω) such that v|Ω2 = u|Ω2

and v − uW 1,p(x) (Ω) = v − uW 1,p1 (x) (Ω1 ) ≤ ε .

2◦ . If u ∈ W 1,p(x) (Ω) with u|S = 0, where S is an (N − 1)-dimensional Lipschitz manifold in ∂Ω, then for p2 (x) any given ε > 0, there is v ∈ W 1,e (Ω) such that v|Ω2 = u|Ω2 ,

v|S = 0

and v − uW 1,p(x) (Ω) ≤ ε .

P r o o f. 1◦ . Taking any u ∈ W 1,p(x) (Ω) and ε > 0, since (Ω2 , p2 (x)) satisfies the Condition (E), there p2 (x) p2 (x) exists u ∈ W 1,e (Ω) such that u (x) = u(x) for x ∈ Ω2 . Note that the condition (1) implies W 1,e (Ω) ⊂ 1,p(x) W (Ω). Set w(x) = u (x) − u(x) , www.mn-journal.com

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then w ∈ W 1,p(x) (Ω) and w|Ω2 = 0. Since (Ω1 , p1 (x)) satisfies the Condition (D0 ), for w|Ω1 , we can find ϕ ∈ C ∞ ( Ω1 ) such that ϕ|∂Ω1 ∩∂Ω2 = 0 and ϕ − wW 1,p1 (x) (Ω1 ) ≤ ε . p2 (x) Making an additional definition ϕ|Ω2 = 0 and setting v = u − ϕ, one has v ∈ W 1,e (Ω), v|Ω2 = u|Ω2 and

u − ϕ − uW 1,p(x) (Ω) v − uW 1,p(x) (Ω) =  =

w − ϕW 1,p(x) (Ω)

= w − ϕW 1,p1 (x) (Ω1 ) ≤ ε ,

this completes the proof of conclusion 1◦ . 2◦ . Assuming that u ∈ W 1,p(x) (Ω) with u|S = 0, where S is an (N − 1)-dimensional Lipschitz manifold in ∂Ω, we can take a bounded open subset G of RN with Lipschitz boundary, such that G ∩ Ω = ∅ and ∂G ∩ ∂Ω = S .   p2 (x) 2 = int( Ω2 ∪ G ), then u| e ∈ W 1,e Ω . By condition (3), Additionally defining u|G = 0 and setting Ω Ω 2  N  N 2 − − 1,e p2 (x) R . Note that u |S = 0 and p 2 R = p2 (Ω2 ) ≥ p+ there is u , an extension of u|Ω e 2 to W 1 (Ω1 ). ◦ 1,p(x) Similar to the proof of 1 , letting w = u − u, then w|Ω = 0, w| = 0 and w ∈ W (Ω). Given any S e2 ∞ ε > 0, since (Ω1 , p1 (x)) satisfies Condition (D0 ), for w|Ω1 , we can find ϕ ∈ C ( Ω1 ) such that ϕ|∂Ω1 ∩∂Ω2 = 0, ϕ|∂Ω1 ∩S = 0 and ϕ − wW 1,p1 (x) (Ω1 ) ≤ ε . p2 (x) Making an additional definition ϕ|Ω2 = 0 and setting v = u − ϕ, one has v ∈ W 1,e (Ω), v|Ω2 = u|Ω2 , ◦ v|S = 0 and v − uW 1,p(x) (Ω) ≤ ε. The conclusion 2 is proved.

Theorem 2.7 Let {Qi : i = 1, 2, . . . , k} be nonempty disjoint bounded open subsets of RN and Q =   k

k

int i=1 Qi such that Q\ i=1 Qi has zero measure and Qi = Q\ j=i Qj for i = 1, 2, . . . , k. Let p ∈ L∞ + (Ω). For each i = 1, 2, . . . , k, set pi = p|Qi . Write Ωi = Q



Qj ,

Ωi = Q\Ωi ,

i = 1, 2, ... , k − 1.

j>i

Assume that (1) Every (Ωi , pi (x)) satisfies Condition (F ). − + − + − (2) p+ 1 (Q1 ) ≤ p2 (Q2 ) ≤ p2 (Q2 ) ≤ p3 (Q3 ) ≤ . . . ≤ pk−1 (Qk−1 ) ≤ pk (Qk ).  (3) For each i = 1, 2, . . . , k − 1, Ωi and Ωi get along with each other regularly, and the boundaries of Ωi , Ωi and Q are Lipschitz. Then we have 1◦ . C ∞ ( Q ) is dense in W 1,p(x) (Q).   2◦ . C0∞ ( Q ) is dense in W01,1 (Ω) ∩ W 1,p(x) (Ω) in the sense of the norm of W 1,p(x) (Ω) , and consequently there holds the following equality 1,p(x)

W01,1 (Q) ∩ W 1,p(x) (Q) = W0

(Q) .

(R)

P r o o f. Let p i , i = 2, 3, . . . , be a given extension of pi to RN satisfying the requirement of Condition (F ). 1◦ . Taking any u ∈ W 1,p(x) (Q) and ε > 0, by condition (2) we have u|Ω1 ∈ W 1,p1 (x) (Ω1 )

p2 (x) and u|Ω1 ∈ W 1,e (Ω1 ) .

p2 (x) For (Ω1 , p1 (x)) and (Ω1 , p 2 (x)), by Lemma 2.1, we know that there is u2 ∈ W 1,e (Q) such that

u2 |Ω1 = u|Ω1

and u2 − uW 1,p1 (x) (Ω1 ) ≤ ε .

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p2 (x) p3 (x) Moreover u2 |Ω2 ∈ W 1,e (Ω2 ) and u2 |Ω2 ∈ W 1,e (Ω2 ). Using Lemma 2.1 for (Ω2 , p 2 (x)) and (Ω2 , p 3 (x)), p3 (x) we know that there is u3 ∈ W 1,e (Q) such that

u3 |Ω2 = u2 |Ω2

and u3 − u2 W 1,pe2 (x) (Ω2 ) ≤ ε .

Repeat the above process, we get u2 , u3 , . . . , uk , where uj , j = 2, 3, . . . , k, satisfies (set u = u1 ) pj (x) uj ∈ W 1,e (Q) ,

uj |Ωj−1 = uj−1 |Ωj−1 ,

uj − uj−1 W 1,pej−1 (x) (Ωj−1 ) ≤ ε .

Observing that p(x) ≤ p j−1 (x) for x ∈ Ωj−1 , we have uk − uW 1,p(x) (Q) ≤

k 

uj − uj−1 W 1,p(x) (Q)

j=2

=

k 

uj − uj−1 W 1,p(x) (Ωj−1 ) ≤ c(k − 1)ε ,

j=2 pj−1 (x) where c is a positive constant determined by the embeddings W 1,e (Ωj−1 ) → W 1,p(x) (Ωj−1 ) and is independent of u and ε. pk (x) pk (x) Note that uk ∈ W 1,e (Q), which implies that W 1,e (Q) is dense in W 1,p(x) (Q). By the assumptions in Theorem 2.7, (Qk , pk (x)) satisfies Condition (F ) and Q has Lipschitz boundary, so C ∞ ( Q ) is dense in pk (x) W 1,e (Q). Since p(x) ≤ p k (x) for x ∈ Q, we can assert that C ∞ ( Q ) is dense in W 1,p(x) (Q). Conclusion ◦ 1 is proved. 1,p(x) (Q) ⊂ W01,1 (Q) ∩ W 1,p(x) (Q). 2◦ . It is obvious that W0 1,1 Now let u ∈ W0 (Q) ∩ W 1,p(x) (Q), then u ∈ W 1,p(x) (Q) and u|∂Q = 0. Similar to the proof of 1◦ but each time using conclusion 2◦ in Lemma 2.1 for Ωi and Ωi , we can get in turn the approximate functions u2 , u3 , . . . , uk of u with uj |∂Q = 0. In particular, from uk |∂Q = 0, we get the smooth approximate function ϕ ∈ C ∞ ( Q ) of u satisfying ϕ|∂Q = 0. So C0∞ (Q) is dense in W01,1 (Q) ∩ W 1,p(x) (Q), and the relation (R) holds. This ends the proof.

Remark 2.8 Consider the following Lagrangian f : Ω × RN → R with p(x) growth condition c1 |ξ|p(x) − b1 (x) ≤ f (x, ξ) ≤ c2 |ξ|p(x) + b2 (x) ,

(x, ξ) ∈ Ω × RN ,

where c1 and c2 are positive constants, b1 (x) and b2 (x) are non-negative integrable functions. It is well-known that the relation (R) implies the regularity of the Lagrangian f (x, ξ), i.e., the corresponding integral functional does not admit the Lavrentiev phenomenon (see [7, 17, 18]). Remark 2.9 A special case of Theorem 2.2 is that pi (x) is identical with a constant αi
on Qi . In the Zhikov’s
 2 example on non-regularity, Q = x = (x , x ) ∈ R : |x| < 1, x x > 0 , Q = x = (x1 , x2 ) ∈ R2 : 1 1 2 1 2 2   |x| < 1, x1 x2 < 0 . In this situation, Ω1 = Q1 , Ω1 = Q2 , neither the boundaries of Ω1 and Ω1 is Lipschitz, nor Ω1 and Ω1 get along with each other regularly. Note that in Theorem 2.3, we do not require every Qi to have Lipschitz boundary. 
Example 2.10 : Let Q = x = (x1 , x2 ) ∈ R2 : |x| < 1 , and let Q1 = {x ∈ Q : x1 > 0, x2 > 0} , Q2 = {x ∈ Q : x1 x2 < 0} , Q3 = {x ∈ Q : x1 < 0, x2 < 0} . Let p ∈ L∞ + (Q) be such that pi = p|Qi (i = 1, 2, 3) satisfies the Dini–Lipschitz condition on Qi respectively (but pi does not satisfy Edmunds–R´akosn´ık’s Condition (C), for example, pi has a strict maximum point in − + − Qi ) and p+ 1 (Q1 ) ≤ p2 (Q2 ) ≤ p2 (Q2 ) ≤ p3 (Q3 ). Though Q2 does not admit Lipschitz boundary, since all conditions in Theorem 2.3 are satisfied, C ∞ ( Q ) is dense in W 1,p(x) (Q), and relation (R) is true. Especially, the corresponding integral functional does not allow Lavrentiev phenomenon. www.mn-journal.com

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Combining Theorem 2.1 with Theorem 2.3, we get our main result in this paper as following Theorem 2.11 Let Ω be an open subset of RN and p ∈ L∞ + (Ω). 1◦ . If for any x ∈ Ω, there is an open neighborhood Q ⊂ Ω of x such that (Q, p(x)) satisfies the conditions mentioned in Theorem 2.3, then C ∞ (Ω) ∩ W 1,p(x) (Ω) is dense in W 1,p(x) (Ω). 2◦ . Under the assumptions in 1◦ , if for any x ∈ ∂Ω, there is an open neighborhood U of x such that Q = U ∩Ω satisfies the conditions mentioned in Theorem 2.2, then C ∞ ( Ω ) ∩ W 1,p(x) (Ω) is dense in W 1,p(x) (Ω), C0∞ (Ω) is dense in W01,1 (Ω) ∩ W 1,p(x) (Ω), and relation (R) holds. Sometimes condition (2) in Theorem 2.3 cannot be satisfied on the whole domain. In such situation, by Theorem 2.4 we can consider the local properties for each point. Particularly, we present the following conclusion. Corollary 2.12 Let Qi and Q be the same as stated in Theorem 2.3 (but can be unbounded). For x ∈ Q, set I(x) = {i ∈ {1, 2, . . . , k} : x ∈ Qi }. Denote by |I(x)| the number of the elements in I(x). Let p ∈ L∞ + (Ω) and pi = p|Qi for i = 1, 2, . . . , k. If (1) every pi satisfies the Dini–Lipschitz condition on Qi (consequently we may think pi ∈ C 0 ( Qi )), (2) for every x ∈ Q with |I(x)| ≥ 2, pi (x) = pj (x) for i, j ∈ I(x), i = j, (3) ∂Q and ∂Qi are Lipschitz, then we have 1◦ . C ∞ (Q) ∩ W 1,p(x) (Q) is dense in W 1,p(x) (Q). 2◦ . if condition (2) is satisfied for all point x ∈ Q with |I(x)| ≥ 2, then C ∞ ( Q ) ∩ W 1,p(x) (Q) is dense in W 1,p(x) (Q), and relation (R) is true. P r o o f. Take any x ∈ Q (correspondingly, x ∈ Q ). If |I(x)| = 1, condition (1) shows that x is a p(x)regular point. If |I(x)| ≥ 2, by the continuity of pi on Qi , condition (2) provides an open neighborhood Qx of x small enough (if x ∈ ∂Ω, consider Qx ∩ Ω instead) such that all hypothesis in Theorem 2.2 are satisfied for Qx , and then x is also p(x)-regular, so the conclusion is true. Remark 2.13 In [5] Edmunds and R´akosn´ık have proved the following theorem (see [5, Theorem 1]): Theorem (C) Let Ω ⊂ RN be an open, non-empty set. Let p : Ω → [1, ∞) be a measurable function satisfying the following condition. For every x ∈ Ω there exist numbers 0 < r(x) ≤ 1, h(x) > 0 and a vector ξ(x) ∈ RN \{0} such that h(x) < |ξ(x)| ≤ 1 , B(x, r(x)) + C(x) ⊂ Ω ,

where C(x) = Cξ(x),h(x) = 0
x ∈ Ω,

y ∈ C(x) .

(C)

Then the set C ∞ (Ω) ∩ W k,p(x) (Ω) is dense in W k,p(x) (Ω). This is an important theorem which is often cited in some papers (see, e.g., [11]). Here we point out that the condition (C) in the theorem is misprinted, and the correct form should be the following p(z) ≤ p(z + y) for a.e. z ∈ B(x, r(x)) ,

y ∈ C(x) .

(C∗ )

It is easy to see that in the Zhikov’s example on the non-density as mentioned in the Introduction, the condition (C) is satisfied but the condition (C∗ ) is not satisfied. Remark 2.14 A referee has put forward a very good suggestion, that is, to combine the Dini–Lipschitz condition and Edmunds–R´akosn´ık’s cone-monotonicity condition together to get a more general condition. By this suggestion the following theorem holds: Theorem (DL–C) Under the assumptions of Theorem (C) (i.e., [5, Theorem 1]), if the condition (C∗ ) is replaced by the following condition p(z) ≤ p(z + y) +

L |ln |y||

c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 

for a.e. z ∈ B(x, r(x)) ,

y ∈ C(x) ,

(DL–C) www.mn-journal.com

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where L is a positive constant depending only on x, then the set C ∞ (Ω) ∩ W 1,p(x) (Ω) is dense in W 1,p(x) (Ω). The proof of this theorem is standard. This theorem is useful and powerful. In some cases, as was pointed out by the referee, “this new condition (DL–C) would immediately imply density on Ω without splitting and gluing”. For example, in Example 2.10 the condition (DL–C) is satisfied on Q. However, the condition (DL–C) is not all-powerful. In the following example, the condition (DL–C) is not satisfied but all conditions in Theorem 2.3 are satisfied. 
Example 2.15 Let Q = x ∈ R2 : |x| < 1 , Q1 = {x ∈ Q : x1 > 0, x2 > 0} , Q2 = {x ∈ Q : x1 < 0, x2 > 0} , Q3 = {x ∈ Q : x1 < 0, x2 < 0} , Q4 = {x ∈ Q : x1 > 0, x2 < 0} , and let p(x) = 1, 2, 3, 4 on Q1 , Q2 , Q3 and Q4 , respectively. It is easy to see that all conditions in Theorem 2.3 are satisfied but at the origin the condition (DL–C) is not satisfied. Acknowledgements The authors are very appreciate to the referees for the very careful comments and the valuable suggestions. This research was supported by the NSFC (10371052), NSFC TY (10226033) and NSFC (10205007).

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