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Estimating the gradient of vector fields via div and curl in variable exponent Sobolev spaces
Nonlinear Analysis 192 (2020) 111666
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Estimating the gradient of vector fields via div and curl in variable exponent Sobolev spaces Cholmin Sin a ,∗, Sin-Il Ri b a b
Institute of Mathematics, Academy of Sciences, Pyongyang, Democratic People’s Republic of Korea Department of Mathematics, University of Science, Pyongyang, Democratic People’s Republic of Korea
article
info
Article history: Received 27 June 2019 Accepted 22 September 2019 Communicated by Vicentiu D. Radulescu
abstract We show an estimate of the gradient of vector fields u in terms of div u and curl u in vector-valued variable exponent Sobolev spaces with vanishing normal component on the boundary in simply connected 3D-domains. © 2019 Elsevier Ltd. All rights reserved.
MSC: 35D30 35A23 46E30 46E35 76D03 76A05 Keywords: Div–curl system Calderon–Zygmund operator Variable exponent Sobolev space
1. Introduction In this paper we prove the following. Theorem 1.1. Let p(x) > 1, p ∈ P log (Ω ). Let Ω ⊂ R3 be a C 0,1 bounded and simply connected domain. 1,p(x) Then for all u ∈ Wν (Ω ), there holds ∥∇u∥p(x) ≤ c(∥curl u∥p(x) + ∥div u∥p(x) ), 1,p(x)
where Wν
(Ω ) := {u ∈ W 1,p(x) (Ω ) | u · ν|∂Ω = 0} and ν is the unit outer normal on ∂Ω .
∗ Corresponding author. E-mail address: [email protected] (C. Sin).
https://doi.org/10.1016/j.na.2019.111666 0362-546X/© 2019 Elsevier Ltd. All rights reserved.
(1.1)
C. Sin and S.-I. Ri / Nonlinear Analysis 192 (2020) 111666
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The mathematical study of many problems in classical continuum and electromagnetic field theories often depends on inequalities involving the divergence and rotation of a vector field with either vanishing tangential components or vanishing normal components on the boundary. The inequality (1.1) with p ≡ 2 is proved by Duvaut and Lions [15]. In [13], authors showed (1.1) with p ≡ 2 in any n(≥ 2)-D domains. Wahl [23] proved (1.1) with p = const in 3D simply connected domain and in [1,5,6,8] there have been studied various versions of (1.1) in 3D multi-connected domain. The various mathematical problems with variable exponent have been investigated by many authors in recent years. We refer to the monographs [3,4,12,14,17,18,21] for the advances, the references and the application background of this area and [9,11,20] for recent results . Cekic, et al. [10] showed H¨ older-type norm inequalities in terms of div and curl of the vector-valued functions in variable exponent Lebesgue spaces Lp(x) (Ω ) for p(x) > 3/2. In [2] authors proved that for p(x) > 6/5, (1.2) ∥∇u∥p(x) ≤ c∥curl u∥p(x) ∀u ∈ Wν1,p(x) (Ω ) with div u = 0. in 3D simply connected domain. For the applications of the previous result, we refer to [2,7]. Our result can be applied to some PDEs involving variable exponent, for example, to magnetostatics problems [10] and to p(x)-curl systems arising in electromagnetism [7]. Furthermore, applying the inequality (1.1) yields improvement of the lower bounds of p(x) from not only the inequality (1.2) but also [7, Theorem 1.1] 1,p(x) (Ω ), see [13] and [10, Section 5]. The inequality (1.1) is also used in proving Korn type inequality in Wν for constant p and forthcoming paper [22] for p(x). The main difficulty in the proof of the main result arises from the fact that the norm in Tr W 1,p(x) (Ω ) 1 has not the same form as in usual trace space W 1− p ,p (∂Ω ) for constant p, which prevents us to show (2.5) below for p(x) ̸= const by directly applying an inequality proved in [19, p. 74]. To overcome the difficulty, we perform delicate estimates using an intrinsic norm (see [14, Section 12.1]) in Tr W 1,p(x) (Ω ), which are presented in Step 3 from Section 2. The paper is organized as follows. In the rest of this section, we give preliminaries. In Section 2, we prove Theorem 1.1. Let Ω ⊂ Rn be a domain and p ∈ L∞ (Ω ), p ≥ 1. We introduce the following variable exponent Lebesgue and Sobolev spaces: ˆ { } p(x) Lp(x) (Ω ) := u : Ω → R | u is measurable and ϱp(x) (u) := |u| dx < ∞ , Ω
} u ∥u∥Lp(x) (Ω) := inf λ > 0 | ϱp(x) ( ) ≤ 1 λ ∑ k,p(x) α p(x) W (Ω ) := { u | ∇ u ∈ L (Ω ), ∀ |α| ≤ k}, ∥u∥W k,p(x) (Ω) := ∥∇α u∥p(x) . {
|α|≤k
As usual the term “Tr” means the trace of a function. The trace space Tr W of all functions U ∈ W k,p(x) (Ω ), endowed the quotient norm
k,p(x)
(Ω ) consists of the traces
∥u∥Tr W k,p(x) (Ω) = inf{∥U ∥k,p(x) | U ∈ W k,p(x) (Ω ), Tr U = u}. k,p(x)
The properties of Lp(x) , W k,p(x) and Tr W k,p(x) (Ω ) can be found in books [12,14]. Let us define W0 (Ω ) as the closure of C0∞ (Ω ) in W k,p(x) (Ω ). We use the same notation for functional spaces and norms for both scalar and vector fields. Let us denote p− := essinf p(x). We say that a function p : Ω → R is globally log-H¨older continuous on Ω if there exists a constant Clog > 0 such that |p(x) − p(y)| ≤ and denote p ∈ P log (Ω ).
Clog ln(e + 1/|x − y|)
∀x, y ∈ Ω ,
C. Sin and S.-I. Ri / Nonlinear Analysis 192 (2020) 111666
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2. The proof of main result In this section we prove main result, i.e., Theorem 1.1. The proof is divided in several steps. In the first two steps, we follow the argument from [23, Theorems 2.1 and 3.2] but in part use some knowledge in variable exponent spaces. In the third step we perform delicate estimates using an intrinsic norm in Tr W 1,p(x) (Ω ). By [·, ·] we denote cross product. Note that for all u ∈ C 1,α (Ω ) (ˆ ) ˆ 1 div u(y) (u · ν)(y) u(x) = −grad dy − dσ(y) 4π |x − y| |x − y| (ˆ Ω ) ˆ∂Ω (2.1) 1 curl u(y) [u, ν](y) + curl dy + dσ(y) . 4π Ω |x − y| ∂Ω |x − y| Step 1: At first we assume that u ∈ C 1,α (Ω ) for some α ∈ (0, 1) and u · ν = 0 on ∂Ω . The div–curl system curl v = γ, div v = ϵ in Ω ; −v · ν = 0 on ∂Ω has one and only one classical solution v, see [23, p. 132]. We have set γ = curl u, ϵ = div u. Then of course v = u. From (2.1), the quantity γ ∗ = −[ν, u] satisfies the integral equation ˆ ˆ ] ϵ(y) γ(y) 1 [ ν, grad dy − curl dy (2.2) γ ∗ + Rγ ∗ = 2π Ω |x − y| Ω |x − y| on ∂Ω , where R is the integral operator defined by ˆ [ ˆ [ [ ]] 1 γ ∗ (y) ] 1 1 ∗ ν, curlx ν, gradx Rγ : = dσ(y) = , γ ∗ (y) dσ(y) 2π ∂Ω |x − y| 2π ∂Ω |x − y| ˆ ˆ 1 1 1 1 = ν · γ ∗ (y)gradx dσ(y) − ν · gradx γ ∗ (y)dσ(y), 2π ∂Ω |x − y| 2π ∂Ω |x − y| where the symbols curlx , gradx stand for curl and gradient operations with respect to variable x, respectively. As pointed out in [23], since −1
ν(x) · γ ∗ (y)gradx (|x − y|
) = (ν(x) − ν(y)) · γ ∗ (y)gradx (|x − y|
−1
),
the integral operator R has the property that it can be decomposed into integral operators whose kernels R satisfy the estimates c , x ̸= y, |R(x, y)| ≤ (2.3) |x − y| c|x − z| |R(x, y) − R(z, y)| ≤ x ̸= y, z ̸= y, |x − y|, |z − y| ≤ 1. (2.4) 2 2 , min{|x − y| , |z − y| } In Step 3 below we will prove that ∥Rγ ∗ ∥TrW 1,p(x) (Ω) ≤ c∥γ ∗ ∥Lp(x) (∂Ω) .
(2.5)
As mentioned in [23, p. 133], the null space of I +R consists of the element zero due to simply connectedness of Ω . Thus we can use the trace embedding [16, Corollary 2.4] and Calderon–Zygmund theorem [14, Section 6.3] to get (⏐⏐ ) ˆ ˆ ⏐⏐ ⏐⏐ ⏐⏐ γ(y) ϵ(y) ⏐⏐ ⏐⏐ ⏐⏐ ⏐⏐ dy ⏐⏐ p(x) + ⏐⏐curl dy ⏐⏐ p(x) ∥γ ∗ ∥Lp(x) (∂Ω) ≤ c ⏐⏐grad |x − y| |x − y| L (∂Ω) L (∂Ω) (⏐⏐ ) ˆ Ω ˆΩ ⏐⏐ ⏐⏐ ⏐⏐ ϵ(y) γ(y) ⏐⏐ ⏐⏐ ⏐⏐ ⏐⏐ (2.6) + ⏐⏐curl ≤ ⏐⏐grad dy ⏐⏐ 1,p(x) dy ⏐⏐ 1,p(x) W (Ω) W (Ω) Ω |x − y| Ω |x − y| ( ) ≤ c ∥ϵ∥Lp(x) (Ω) + ∥γ∥Lp(x) (Ω) .
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On the other hand by (2.2) we obtain ∥γ ∗ ∥TrW 1,p(x) (Ω) ≤ ∥Rγ ∗ ∥TrW 1,p(x) (Ω) ) (⏐⏐ ˆ ˆ ⏐⏐ ⏐⏐ ⏐⏐ γ(y) ϵ(y) ⏐⏐ ⏐⏐ ⏐⏐ ⏐⏐ + dy ⏐⏐ dy + ⏐⏐grad ⏐curl ⏐ ⏐ ⏐ |x − y| TrW 1,p(x) (Ω) TrW 1,p(x) (Ω) Ω |x − y| (⏐⏐ ) ˆ Ω ˆ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ϵ(y) γ(y) ⏐⏐ ⏐⏐ ⏐⏐ ⏐⏐ ≤ ⏐⏐grad + ⏐⏐curl dy ⏐⏐ 1,p(x) dy ⏐⏐ 1,p(x) W (Ω) W (Ω) Ω |x − y| Ω |x − y| ( ) ≤ c ∥ϵ∥Lp(x) (Ω) + ∥γ∥Lp(x) (Ω) ,
(2.7)
where we use (2.5), (2.6) and again trace embedding and Calderon–Zygmund theorem. Note that for x ∈ ∂Ω , −u(x) = −u(x) − ν(x) · u(x)ν(x) = −[ν(x), [ν(x), u(x)]] = [ν(x), γ ∗ (x)]. This together with trace theorem yields that there exists a w ∈ W 1,p(x) (Ω ) with w = u on ∂Ω , ∥w∥W 1,p(x) (Ω) ≤ c∥w∥Tr W 1,p(x) (Ω) = c∥u∥Tr W 1,p(x) (Ω) ≤ ∥γ ∗ ∥TrW 1,p(x) (Ω) ) (2.7) ( ≤ c ∥ϵ∥Lp(x) (Ω) + ∥γ∥Lp(x) (Ω) . 1,p(x)
Using (2.1), density of C0∞ (Ω ) in W0 theorem, we obtain that
(Ω ) (for example [14, Theorem 9.1.7]) and again Calderon–Zygmund
∥∇h∥Lp(x) (Ω) ≤ c(∥div h∥Lp(x) (Ω) + ∥curl h∥Lp(x) (Ω) ),
1,p(x)
∀h ∈ W0
(Ω ).
Thus we have ∥∇(u − w)∥Lp(x) (Ω) ≤ c(∥div w∥Lp(x) (Ω) + ∥curl w∥Lp(x) (Ω) + ∥ϵ∥Lp(x) (Ω) + ∥γ∥Lp(x) (Ω) ) ( ) ≤ c ∥ϵ∥Lp(x) (Ω) + ∥γ∥Lp(x) (Ω) , which implies that ( ) ∥∇u∥Lp(x) (Ω) ≤ c ∥ϵ∥Lp(x) (Ω) + ∥γ∥Lp(x) (Ω)
(2.8)
≤ c(∥curl u∥p(x) + ∥div u∥p(x) ). 1,p(x)
Step 2: In this step, our aim is to approximate an element u ∈ Wν (Ω ) by elements uk ∈ C 1,α (Ω ) k with u · ν|∂Ω = 0 satisfying the estimate (2.8). To this end, we choose a sequence {ˆ uk } ⊂ C 2 (Ω ) such that ´ k 1,p(x) k k u ˆ → u in W (Ω ) as k → ∞. Set γ = curl u ˆ , γ = curl u. Note that Ω ϵ dx = 0 for ϵ := divu since k u · ν = 0 on ∂Ω . Let us choose a sequence {ϵ } such that ˆ ϵk ∈ C0∞ (Ω ), ϵk dx = 0, ϵk → ϵ in Lp(x) (Ω ). Ω
Let uk be the solution of the problem curl uk = γ k , div uk = ϵk in Ω ;
−uk · ν = 0 on ∂Ω
Due to the simply connectedness of Ω , this problem has unique solution uk ∈ C α (Ω ) ∩ C 1,α (Ω ) [23, p. 139] and uk can be represented as follows: ˆ ˆ ˆ 1 ϵk (y) 1 γ k (y) 1 [uk , ν](y) (2.9) uk (x) = − grad dy + curl dy + curl dσ(y). 4π 4π 4π Ω |x − y| Ω |x − y| ∂Ω |x − y|
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1,p(x)
On the other hand we have for u ∈ Wν (Ω ) ˆ ˆ ˆ ϵ(y) 1 γ(y) 1 [u, ν](y) 1 dy + curl dy + curl dσ(y). u(x) = − grad 4π |x − y| 4π |x − y| 4π Ω Ω ∂Ω |x − y|
(2.10)
Now we want to show that uk → u in W 1,p(x) (Ω ). k
(2.11)
1,α
According to [23, p. 139], we have u ∈ C (Ω ). By Calderon–Zygmund theorem [14, Section 6.3] and boundedness results for the Riesz potential operator [14, Theorem 6.1.9] we obtain ˆ ˆ ˆ ˆ ϵk (y) ϵ(y) γ k (y) γ(y) grad dy → grad dy curl dy → curl dy in W 1,p(x) (Ω ). (2.12) Ω |x − y| Ω |x − y| Ω |x − y| Ω |x − y| From (2.12) and the representations (2.9) and (2.10), the proof of (2.11) reduces to one of ˆ ˆ [uk , ν](y) [u, ν](y) → curl in W 1,p(x) (Ω ). curl ∂Ω |x − y| ∂Ω |x − y|
(2.13)
Taking into account curl uk = curl u ˆk → curl u and ϵk → ϵ in Lp(x) (Ω ) as k → ∞, we obtain that by (2.7) [uk , ν] → [u, ν] in Tr W 1,p(x) (Ω ). ´ γ(y) ´ ϵ(y) dy, curl Ω |x−y| dy are in Tr W 1,p(x) (Ω ), we can see Furthermore recalling that the traces of grad Ω |x−y| that [u, ν] fulfills the integral equation (2.2) on ∂Ω . Again by Calderon–Zygmund theorem [14, Section 6.3], it follows that the traces of two expressions ´ [uk ,ν](y) ´ 2,p(x) dσ(y), ∂Ω [u,ν](y) (Ω ), and |x−y| dσ(y) are in Tr W ∂Ω |x−y| ˆ ⏐⏐ ˆ [uk , ν](y) ⏐⏐ [u, ν](y) ⏐⏐ ⏐⏐ dσ(y) − dσ(y)⏐⏐ ≤ c∥[uk , ν] − [u, ν]∥Tr W 1,p(x) (Ω) . ⏐⏐ 2,p(x) (Ω) |x − y| |x − y| Tr W ∂Ω ∂Ω and in turn ⏐⏐ ˆ ⏐⏐ ⏐⏐
∂Ω
[uk , ν](y) dσ(y) − |x − y|
ˆ
⏐⏐ [u, ν](y) ⏐⏐ dσ(y)⏐⏐ 2,p(x) ≤ c∥[uk , ν] − [u, ν]∥Tr W 1,p(x) (Ω) , |x − y| W (Ω)
∂Ω
which implies desired convergence relation (2.13). Thus combining (2.12) with (2.13) and using the representations (2.9) and (2.10), we show that there holds (2.11). Step 3: It remains to prove (2.5). By localization method we assume that Ω = R3+ and u ∈ W 1,p(x) (R3+ ) with u · ν = 0 on Bρ (x0 ) and u = 0 in {x3 = 0} \ Bρ+ (x0 ) for a x0 ∈ {x3 = 0} and a radius ρ. Here we denote by Br (x) the ball centered at x with radius r in R2 . Recall the definition of the sharp operator: for a f ∈ L1loc (R2 ) MB♯ r (x) f :=
|f (y) − (f )Br (x) |dy. Br (x)
Then by [14, Theorem 12.1.14] the quotient norm ∥ · ∥Tr W 1,p(x) (R3 ) is equivalent to the norm ∥ · ∥Tr,p(x) for +
f ∈ Tr W 1,p(x) (R3+ ), where ∥f ∥Tr,p(x) It is clear that ∥f ∥Tr,p(x)
ˆ { := inf λ > 0 :
ˆ { ≤ inf λ > 0 :
)p(x) ˆ 1ˆ ( ⏐ f (x) ⏐p(x) } 1 ♯ f ⏐ ⏐ dx + MBr (x) dxdr ≤ 1 . ⏐ ⏐ λ r λ R2 0 R2
)p(x) ˆ 1ˆ ( { } f (x) p(x) } 1 ♯ f | | dx + inf λ > 0 : MBr (x) dxdr ≤ 1 λ r λ R2 0 R2 )p(x) ˆ 1ˆ ( } { 1 ♯ f M dxdr ≤ 1 . = ∥f ∥Lp(x) (R2 ) + inf λ > 0 : r Br (x) λ 0 R2
C. Sin and S.-I. Ri / Nonlinear Analysis 192 (2020) 111666
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By (2.3) and [14, Theorem 6.1.9], we can see that ∥Rγ ∗ ∥Lp(x) (R2 ) ≤ c∥γ ∗ ∥
≤ c∥γ ∗ ∥Lp(x) (R2 )
2+p(x)
L 2p(x) (R2 )
(2.14)
2 since 2+p(x) 2p(x) < 2 and u has a compact support in R . For convenience, from now on we denote Br (x) by Br . By the definition of MB♯ r (x) , it is easy to show that
ˆ
1
0
ˆ R2
)p(x) )p(x) ˆ 1ˆ ( 1 1 ♯ ∗ ∗ ∗ M Rγ dxdr ≤ |Rγ (y) − Rγ (z)|dydz dxdr r Br r Br Br 0 R2 )p1 p(x) ˆ 1ˆ ( p1 1 ∗ ∗ dxdr, = |Rγ (y) − Rγ (z)|dydz r Br Br 0 R2
(
where p1 := inf x∈Bρ (x0 ) p(x). Using Jensen inequality and the fact that
1 2r
≤
1 |y−z|
(2.15)
all y, z ∈ Br , we obtain
) p1 1 ∗ ∗ |Rγ (y) − Rγ (z)|dydz I:= r Br Br 1 p ≤ p |Rγ ∗ (y) − Rγ ∗ (z)| 1 dydz r 1 Br Br ˆ ˆ p |Rγ ∗ (y) − Rγ ∗ (z)| 1 c dydz, ≤ 2+µ 2+p −µ r |y − z| 1 Br Br (
(2.16)
for any µ ∈ (0, 1). It is known (see [19, p. 74]) that by (2.3) and (2.4) ˆ ˆ p |Rγ ∗ (y) − Rγ ∗ (z)| 1 dydz ≤ c∥γ ∗ ∥pL1p1 (Br ) . 2+p1 −µ |y − z| Br Br
(2.17)
From (2.15)–(2.17), it follows that ˆ
1
0
ˆ
(
R2
1 ♯ M Rγ ∗ r Br
ˆ
)p(x)
1
ˆ
(
dxdr ≤ c ˆ
1
ˆ
(
=c R2
0
1 rµ
) p(x) p
∗ p1
r2+µ
R2
0
ˆ
1
1
|γ | dy
dxdr
Br p
|γ ∗ | 1 dy
(2.18)
) p(x) p 1
dxdr.
Br
Now we assume that ∥γ ∗ ∥Lp(x) (R2 ) ≤ 1. From p(x)-Jensen inequality (for example, [14, Theorem 4.2.4]) it follows that for a β ∈ (0, 1), depending only on Clog , (
∗ p1
|γ | dy
β
) p(x) p 1
∗ p(y)
≤
|γ |
Br
( dy + c
Br
Br
1 1 + m (e + |x|) (e + |y|)m
) χ{0<|γ ∗ (y)|≤1} dy,
where m > 2. Since ∥γ ∗ ∥Lp(x) (R2 ) ≤ 1, |γ ∗ |
p(y)
dy ≤
Br
c . r2
It is obvious that (
) (e + |x|)−m + (e + |y|)−m χ{0<|γ ∗ (y)|≤1} dy ≤ c,
Br
where c is independent of r. Thus (
∗ p1
|γ | dy
β Br
) p(x) p 1
≤
c + c. r2
(2.19)
C. Sin and S.-I. Ri / Nonlinear Analysis 192 (2020) 111666
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The inequality (2.18) together with (2.19) implies that ˆ 0
1
ˆ R2
(
1 ♯ M Rγ ∗ r Br
)p(x)
ˆ
ˆ
) p(x) ) p1 ( c + c dxdr dxdr ≤ c r2 0 R2 ˆ 1 ˆ ρ ( ) p(x) ) p1 ( c 1 ≤c + c r′ dr′ dr ≤ c. rµ r2 0 0 1
(
1 rµ
(2.20)
where to perform the last estimate, we use 2+µ p(x) p1 < 3 by choosing ρ sufficiently small. Since the inequality (2.20) holds for γ ∗ with ∥γ ∗ ∥Lp(x) (R2 ) ≤ 1. By the scaling argument we have for all γ ∗ ∈ Lp(x) (R2 ), ˆ { inf λ > 0 :
1
ˆ R2
0
(
1 ♯ Rγ ∗ M r Br λ
)p(x)
} dxdr ≤ 1 ≤ c∥γ ∗ ∥Lp(x) (R2 ) .
Thus we conclude by (2.14) and the previous inequality that ∥Rγ ∗ ∥Tr W 1,p(x) (Ω) ≤ c∥Rγ ∗ ∥Tr,p(x) ≤ c∥γ ∗ ∥Lp(x) (∂Ω) , which is just the estimate (2.5).
□
Remark 2.1. The inequality (1.1) with p ≡ 2 was proved for any n ≥ 2, see [13]. But it seems to be impossible to apply the method from [13] to the problem with p ̸= 2. Remark 2.2. We think that it is possible to prove that the inequality (1.1) continues to hold for all function 1,p(x) v ∈ Wtan (Ω ) := {f ∈ W 1,p(x) (Ω ) | [f, ν] = 0 on ∂Ω }. Acknowledgments The author would like to thank the editor and the anonymous reviewer for their valuable suggestions to improve the quality and impact of this paper. References [1] C. Amrouche, N. Seloula, Lp -Theory for vector potentials and Sobolev’s inequality for vector fields, application to the Stokes equations with pressure boundary conditions, Math. Models Methods Appl. Sci. 23 (2013) 37–92. [2] S. Antontsev, F. Miranda, L. Santos, Blow-up and finite time extinction for p(x, t)-curl systems arising in electromagnetism, J. Math. Anal. Appl. 440 (2016) 300–322. [3] S.N. Antontsev, S. Shmarev, Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions, in: M. Chipot, P. Quittner (Eds.), Handbook of Differential Equations, in: Stationary Partial Differential Equations, vol. 3, Elsevier, 2006, pp. 1–100. [4] S. Antontsev, S. Shmarev, Evolution PDEs with NonstandArd Growth Conditions, Existence, Uniqueness, Localization, Blow-Up, in: Atlantis Studies in Differential Equations, vol. 4, Atlantis Press, 2015. [5] J. Aramaki, Lp Theory for the div–curl system, Int. J. Math. Anal. 8 (6) (2014) 259–271. [6] G. Auchmuty, The main inequality of 3D vector analysis, Math. Models Methods Appl. Sci. 14 (1) (2004) 79–103. [7] A. Bahrouni, D. Repovs, Existence and nonexistence of solutions for p(x)-curl systems arising in electromagnetism, Complex Var. Elliptic Equ. 63 (2) (2018) 292–301. [8] J. Bolik, W. Wahl, Estimating ∇u in terms of div u, curlu, either (ν, u) or ν × u and the topology, Math. Methods Appl. Sci. 20 (1997) 734–744. [9] T.A. Bui, Global W 1,p(·) estimate for renormalized solutions of quasilinear equations with measure data on reifenberg domains, Adv. Nonlinear Anal. 7 (2018) 517–533. [10] B. Cekic, A.V. Kalinin, R.A. Mashiyev, M. Avci, Lp(x) (Ω )-Estimates of vector fields and some applications to magnetostatics problems, J. Math. Anal. Appl. 389 (2012) 838–851. [11] I. Chlebicka, A pocket guide to nonlinear differential equations in musielak-orlicz spaces, Nonlinear Anal. 175 (2018) 1–27. [12] D. Cruz-Uribe, A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, in: Applied and Numerical Harmonic Analysis, Birkhauser, Basel, 2013.
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[13] L. Desvillettes, C. Villani, On a variant of korn’s inequality arising in statistical mechanics, a tribute to j. l. lions, ESAIM Control Optim. Calc. Var. 8 (2002) 603–619. [14] L. Diening, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, in: Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Berlin, 2011. [15] G. Duvaut, J.L. Lions, Inequalities in Mechanics and Physics, Springer, 1976. [16] X. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl. 339 (2008) 1395–1412. [17] V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Non-Standard Function Spaces, 1: Variable Exponent Lebesgue and Amalgam Spaces, in: Operator Theory: Advances and Applications, vol. 248, Birkhauser, 2016. [18] V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Non-Standard Function Spaces, 2: Variable Exponent H¨ older, Morrey–Campanato and Grand Spaces, in: Operator Theory: Advances and Applications, vol. 249, Birkhauser, 2016. [19] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. [20] V. Radulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal. 121 (2015) 336–369. [21] V.D. Radulescu, D.D. Repovs, Partial differential equations with variable exponents, in: Variational Methods and Qualitative Analysis, in: Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015. [22] C. Sin, Existence of weak solutions for steady flows of generalized Newtonian fluid with homogeneous Navier-slip type boundary conditions, in preparation. [23] W. von Wahl, Estimating ∇u by div u and curl u, Math. Methods Appl. Sci. 15 (1992) 123–143.
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Nonlinear Analysis www.elsevier.com/locate/na
Estimating the gradient of vector fields via div and curl in variable exponent Sobolev spaces Cholmin Sin a ,∗, Sin-Il Ri b a b
Institute of Mathematics, Academy of Sciences, Pyongyang, Democratic People’s Republic of Korea Department of Mathematics, University of Science, Pyongyang, Democratic People’s Republic of Korea
article
info
Article history: Received 27 June 2019 Accepted 22 September 2019 Communicated by Vicentiu D. Radulescu
abstract We show an estimate of the gradient of vector fields u in terms of div u and curl u in vector-valued variable exponent Sobolev spaces with vanishing normal component on the boundary in simply connected 3D-domains. © 2019 Elsevier Ltd. All rights reserved.
MSC: 35D30 35A23 46E30 46E35 76D03 76A05 Keywords: Div–curl system Calderon–Zygmund operator Variable exponent Sobolev space
1. Introduction In this paper we prove the following. Theorem 1.1. Let p(x) > 1, p ∈ P log (Ω ). Let Ω ⊂ R3 be a C 0,1 bounded and simply connected domain. 1,p(x) Then for all u ∈ Wν (Ω ), there holds ∥∇u∥p(x) ≤ c(∥curl u∥p(x) + ∥div u∥p(x) ), 1,p(x)
where Wν
(Ω ) := {u ∈ W 1,p(x) (Ω ) | u · ν|∂Ω = 0} and ν is the unit outer normal on ∂Ω .
∗ Corresponding author. E-mail address: [email protected] (C. Sin).
https://doi.org/10.1016/j.na.2019.111666 0362-546X/© 2019 Elsevier Ltd. All rights reserved.
(1.1)
C. Sin and S.-I. Ri / Nonlinear Analysis 192 (2020) 111666
2
The mathematical study of many problems in classical continuum and electromagnetic field theories often depends on inequalities involving the divergence and rotation of a vector field with either vanishing tangential components or vanishing normal components on the boundary. The inequality (1.1) with p ≡ 2 is proved by Duvaut and Lions [15]. In [13], authors showed (1.1) with p ≡ 2 in any n(≥ 2)-D domains. Wahl [23] proved (1.1) with p = const in 3D simply connected domain and in [1,5,6,8] there have been studied various versions of (1.1) in 3D multi-connected domain. The various mathematical problems with variable exponent have been investigated by many authors in recent years. We refer to the monographs [3,4,12,14,17,18,21] for the advances, the references and the application background of this area and [9,11,20] for recent results . Cekic, et al. [10] showed H¨ older-type norm inequalities in terms of div and curl of the vector-valued functions in variable exponent Lebesgue spaces Lp(x) (Ω ) for p(x) > 3/2. In [2] authors proved that for p(x) > 6/5, (1.2) ∥∇u∥p(x) ≤ c∥curl u∥p(x) ∀u ∈ Wν1,p(x) (Ω ) with div u = 0. in 3D simply connected domain. For the applications of the previous result, we refer to [2,7]. Our result can be applied to some PDEs involving variable exponent, for example, to magnetostatics problems [10] and to p(x)-curl systems arising in electromagnetism [7]. Furthermore, applying the inequality (1.1) yields improvement of the lower bounds of p(x) from not only the inequality (1.2) but also [7, Theorem 1.1] 1,p(x) (Ω ), see [13] and [10, Section 5]. The inequality (1.1) is also used in proving Korn type inequality in Wν for constant p and forthcoming paper [22] for p(x). The main difficulty in the proof of the main result arises from the fact that the norm in Tr W 1,p(x) (Ω ) 1 has not the same form as in usual trace space W 1− p ,p (∂Ω ) for constant p, which prevents us to show (2.5) below for p(x) ̸= const by directly applying an inequality proved in [19, p. 74]. To overcome the difficulty, we perform delicate estimates using an intrinsic norm (see [14, Section 12.1]) in Tr W 1,p(x) (Ω ), which are presented in Step 3 from Section 2. The paper is organized as follows. In the rest of this section, we give preliminaries. In Section 2, we prove Theorem 1.1. Let Ω ⊂ Rn be a domain and p ∈ L∞ (Ω ), p ≥ 1. We introduce the following variable exponent Lebesgue and Sobolev spaces: ˆ { } p(x) Lp(x) (Ω ) := u : Ω → R | u is measurable and ϱp(x) (u) := |u| dx < ∞ , Ω
} u ∥u∥Lp(x) (Ω) := inf λ > 0 | ϱp(x) ( ) ≤ 1 λ ∑ k,p(x) α p(x) W (Ω ) := { u | ∇ u ∈ L (Ω ), ∀ |α| ≤ k}, ∥u∥W k,p(x) (Ω) := ∥∇α u∥p(x) . {
|α|≤k
As usual the term “Tr” means the trace of a function. The trace space Tr W of all functions U ∈ W k,p(x) (Ω ), endowed the quotient norm
k,p(x)
(Ω ) consists of the traces
∥u∥Tr W k,p(x) (Ω) = inf{∥U ∥k,p(x) | U ∈ W k,p(x) (Ω ), Tr U = u}. k,p(x)
The properties of Lp(x) , W k,p(x) and Tr W k,p(x) (Ω ) can be found in books [12,14]. Let us define W0 (Ω ) as the closure of C0∞ (Ω ) in W k,p(x) (Ω ). We use the same notation for functional spaces and norms for both scalar and vector fields. Let us denote p− := essinf p(x). We say that a function p : Ω → R is globally log-H¨older continuous on Ω if there exists a constant Clog > 0 such that |p(x) − p(y)| ≤ and denote p ∈ P log (Ω ).
Clog ln(e + 1/|x − y|)
∀x, y ∈ Ω ,
C. Sin and S.-I. Ri / Nonlinear Analysis 192 (2020) 111666
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2. The proof of main result In this section we prove main result, i.e., Theorem 1.1. The proof is divided in several steps. In the first two steps, we follow the argument from [23, Theorems 2.1 and 3.2] but in part use some knowledge in variable exponent spaces. In the third step we perform delicate estimates using an intrinsic norm in Tr W 1,p(x) (Ω ). By [·, ·] we denote cross product. Note that for all u ∈ C 1,α (Ω ) (ˆ ) ˆ 1 div u(y) (u · ν)(y) u(x) = −grad dy − dσ(y) 4π |x − y| |x − y| (ˆ Ω ) ˆ∂Ω (2.1) 1 curl u(y) [u, ν](y) + curl dy + dσ(y) . 4π Ω |x − y| ∂Ω |x − y| Step 1: At first we assume that u ∈ C 1,α (Ω ) for some α ∈ (0, 1) and u · ν = 0 on ∂Ω . The div–curl system curl v = γ, div v = ϵ in Ω ; −v · ν = 0 on ∂Ω has one and only one classical solution v, see [23, p. 132]. We have set γ = curl u, ϵ = div u. Then of course v = u. From (2.1), the quantity γ ∗ = −[ν, u] satisfies the integral equation ˆ ˆ ] ϵ(y) γ(y) 1 [ ν, grad dy − curl dy (2.2) γ ∗ + Rγ ∗ = 2π Ω |x − y| Ω |x − y| on ∂Ω , where R is the integral operator defined by ˆ [ ˆ [ [ ]] 1 γ ∗ (y) ] 1 1 ∗ ν, curlx ν, gradx Rγ : = dσ(y) = , γ ∗ (y) dσ(y) 2π ∂Ω |x − y| 2π ∂Ω |x − y| ˆ ˆ 1 1 1 1 = ν · γ ∗ (y)gradx dσ(y) − ν · gradx γ ∗ (y)dσ(y), 2π ∂Ω |x − y| 2π ∂Ω |x − y| where the symbols curlx , gradx stand for curl and gradient operations with respect to variable x, respectively. As pointed out in [23], since −1
ν(x) · γ ∗ (y)gradx (|x − y|
) = (ν(x) − ν(y)) · γ ∗ (y)gradx (|x − y|
−1
),
the integral operator R has the property that it can be decomposed into integral operators whose kernels R satisfy the estimates c , x ̸= y, |R(x, y)| ≤ (2.3) |x − y| c|x − z| |R(x, y) − R(z, y)| ≤ x ̸= y, z ̸= y, |x − y|, |z − y| ≤ 1. (2.4) 2 2 , min{|x − y| , |z − y| } In Step 3 below we will prove that ∥Rγ ∗ ∥TrW 1,p(x) (Ω) ≤ c∥γ ∗ ∥Lp(x) (∂Ω) .
(2.5)
As mentioned in [23, p. 133], the null space of I +R consists of the element zero due to simply connectedness of Ω . Thus we can use the trace embedding [16, Corollary 2.4] and Calderon–Zygmund theorem [14, Section 6.3] to get (⏐⏐ ) ˆ ˆ ⏐⏐ ⏐⏐ ⏐⏐ γ(y) ϵ(y) ⏐⏐ ⏐⏐ ⏐⏐ ⏐⏐ dy ⏐⏐ p(x) + ⏐⏐curl dy ⏐⏐ p(x) ∥γ ∗ ∥Lp(x) (∂Ω) ≤ c ⏐⏐grad |x − y| |x − y| L (∂Ω) L (∂Ω) (⏐⏐ ) ˆ Ω ˆΩ ⏐⏐ ⏐⏐ ⏐⏐ ϵ(y) γ(y) ⏐⏐ ⏐⏐ ⏐⏐ ⏐⏐ (2.6) + ⏐⏐curl ≤ ⏐⏐grad dy ⏐⏐ 1,p(x) dy ⏐⏐ 1,p(x) W (Ω) W (Ω) Ω |x − y| Ω |x − y| ( ) ≤ c ∥ϵ∥Lp(x) (Ω) + ∥γ∥Lp(x) (Ω) .
C. Sin and S.-I. Ri / Nonlinear Analysis 192 (2020) 111666
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On the other hand by (2.2) we obtain ∥γ ∗ ∥TrW 1,p(x) (Ω) ≤ ∥Rγ ∗ ∥TrW 1,p(x) (Ω) ) (⏐⏐ ˆ ˆ ⏐⏐ ⏐⏐ ⏐⏐ γ(y) ϵ(y) ⏐⏐ ⏐⏐ ⏐⏐ ⏐⏐ + dy ⏐⏐ dy + ⏐⏐grad ⏐curl ⏐ ⏐ ⏐ |x − y| TrW 1,p(x) (Ω) TrW 1,p(x) (Ω) Ω |x − y| (⏐⏐ ) ˆ Ω ˆ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ϵ(y) γ(y) ⏐⏐ ⏐⏐ ⏐⏐ ⏐⏐ ≤ ⏐⏐grad + ⏐⏐curl dy ⏐⏐ 1,p(x) dy ⏐⏐ 1,p(x) W (Ω) W (Ω) Ω |x − y| Ω |x − y| ( ) ≤ c ∥ϵ∥Lp(x) (Ω) + ∥γ∥Lp(x) (Ω) ,
(2.7)
where we use (2.5), (2.6) and again trace embedding and Calderon–Zygmund theorem. Note that for x ∈ ∂Ω , −u(x) = −u(x) − ν(x) · u(x)ν(x) = −[ν(x), [ν(x), u(x)]] = [ν(x), γ ∗ (x)]. This together with trace theorem yields that there exists a w ∈ W 1,p(x) (Ω ) with w = u on ∂Ω , ∥w∥W 1,p(x) (Ω) ≤ c∥w∥Tr W 1,p(x) (Ω) = c∥u∥Tr W 1,p(x) (Ω) ≤ ∥γ ∗ ∥TrW 1,p(x) (Ω) ) (2.7) ( ≤ c ∥ϵ∥Lp(x) (Ω) + ∥γ∥Lp(x) (Ω) . 1,p(x)
Using (2.1), density of C0∞ (Ω ) in W0 theorem, we obtain that
(Ω ) (for example [14, Theorem 9.1.7]) and again Calderon–Zygmund
∥∇h∥Lp(x) (Ω) ≤ c(∥div h∥Lp(x) (Ω) + ∥curl h∥Lp(x) (Ω) ),
1,p(x)
∀h ∈ W0
(Ω ).
Thus we have ∥∇(u − w)∥Lp(x) (Ω) ≤ c(∥div w∥Lp(x) (Ω) + ∥curl w∥Lp(x) (Ω) + ∥ϵ∥Lp(x) (Ω) + ∥γ∥Lp(x) (Ω) ) ( ) ≤ c ∥ϵ∥Lp(x) (Ω) + ∥γ∥Lp(x) (Ω) , which implies that ( ) ∥∇u∥Lp(x) (Ω) ≤ c ∥ϵ∥Lp(x) (Ω) + ∥γ∥Lp(x) (Ω)
(2.8)
≤ c(∥curl u∥p(x) + ∥div u∥p(x) ). 1,p(x)
Step 2: In this step, our aim is to approximate an element u ∈ Wν (Ω ) by elements uk ∈ C 1,α (Ω ) k with u · ν|∂Ω = 0 satisfying the estimate (2.8). To this end, we choose a sequence {ˆ uk } ⊂ C 2 (Ω ) such that ´ k 1,p(x) k k u ˆ → u in W (Ω ) as k → ∞. Set γ = curl u ˆ , γ = curl u. Note that Ω ϵ dx = 0 for ϵ := divu since k u · ν = 0 on ∂Ω . Let us choose a sequence {ϵ } such that ˆ ϵk ∈ C0∞ (Ω ), ϵk dx = 0, ϵk → ϵ in Lp(x) (Ω ). Ω
Let uk be the solution of the problem curl uk = γ k , div uk = ϵk in Ω ;
−uk · ν = 0 on ∂Ω
Due to the simply connectedness of Ω , this problem has unique solution uk ∈ C α (Ω ) ∩ C 1,α (Ω ) [23, p. 139] and uk can be represented as follows: ˆ ˆ ˆ 1 ϵk (y) 1 γ k (y) 1 [uk , ν](y) (2.9) uk (x) = − grad dy + curl dy + curl dσ(y). 4π 4π 4π Ω |x − y| Ω |x − y| ∂Ω |x − y|
C. Sin and S.-I. Ri / Nonlinear Analysis 192 (2020) 111666
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1,p(x)
On the other hand we have for u ∈ Wν (Ω ) ˆ ˆ ˆ ϵ(y) 1 γ(y) 1 [u, ν](y) 1 dy + curl dy + curl dσ(y). u(x) = − grad 4π |x − y| 4π |x − y| 4π Ω Ω ∂Ω |x − y|
(2.10)
Now we want to show that uk → u in W 1,p(x) (Ω ). k
(2.11)
1,α
According to [23, p. 139], we have u ∈ C (Ω ). By Calderon–Zygmund theorem [14, Section 6.3] and boundedness results for the Riesz potential operator [14, Theorem 6.1.9] we obtain ˆ ˆ ˆ ˆ ϵk (y) ϵ(y) γ k (y) γ(y) grad dy → grad dy curl dy → curl dy in W 1,p(x) (Ω ). (2.12) Ω |x − y| Ω |x − y| Ω |x − y| Ω |x − y| From (2.12) and the representations (2.9) and (2.10), the proof of (2.11) reduces to one of ˆ ˆ [uk , ν](y) [u, ν](y) → curl in W 1,p(x) (Ω ). curl ∂Ω |x − y| ∂Ω |x − y|
(2.13)
Taking into account curl uk = curl u ˆk → curl u and ϵk → ϵ in Lp(x) (Ω ) as k → ∞, we obtain that by (2.7) [uk , ν] → [u, ν] in Tr W 1,p(x) (Ω ). ´ γ(y) ´ ϵ(y) dy, curl Ω |x−y| dy are in Tr W 1,p(x) (Ω ), we can see Furthermore recalling that the traces of grad Ω |x−y| that [u, ν] fulfills the integral equation (2.2) on ∂Ω . Again by Calderon–Zygmund theorem [14, Section 6.3], it follows that the traces of two expressions ´ [uk ,ν](y) ´ 2,p(x) dσ(y), ∂Ω [u,ν](y) (Ω ), and |x−y| dσ(y) are in Tr W ∂Ω |x−y| ˆ ⏐⏐ ˆ [uk , ν](y) ⏐⏐ [u, ν](y) ⏐⏐ ⏐⏐ dσ(y) − dσ(y)⏐⏐ ≤ c∥[uk , ν] − [u, ν]∥Tr W 1,p(x) (Ω) . ⏐⏐ 2,p(x) (Ω) |x − y| |x − y| Tr W ∂Ω ∂Ω and in turn ⏐⏐ ˆ ⏐⏐ ⏐⏐
∂Ω
[uk , ν](y) dσ(y) − |x − y|
ˆ
⏐⏐ [u, ν](y) ⏐⏐ dσ(y)⏐⏐ 2,p(x) ≤ c∥[uk , ν] − [u, ν]∥Tr W 1,p(x) (Ω) , |x − y| W (Ω)
∂Ω
which implies desired convergence relation (2.13). Thus combining (2.12) with (2.13) and using the representations (2.9) and (2.10), we show that there holds (2.11). Step 3: It remains to prove (2.5). By localization method we assume that Ω = R3+ and u ∈ W 1,p(x) (R3+ ) with u · ν = 0 on Bρ (x0 ) and u = 0 in {x3 = 0} \ Bρ+ (x0 ) for a x0 ∈ {x3 = 0} and a radius ρ. Here we denote by Br (x) the ball centered at x with radius r in R2 . Recall the definition of the sharp operator: for a f ∈ L1loc (R2 ) MB♯ r (x) f :=
|f (y) − (f )Br (x) |dy. Br (x)
Then by [14, Theorem 12.1.14] the quotient norm ∥ · ∥Tr W 1,p(x) (R3 ) is equivalent to the norm ∥ · ∥Tr,p(x) for +
f ∈ Tr W 1,p(x) (R3+ ), where ∥f ∥Tr,p(x) It is clear that ∥f ∥Tr,p(x)
ˆ { := inf λ > 0 :
ˆ { ≤ inf λ > 0 :
)p(x) ˆ 1ˆ ( ⏐ f (x) ⏐p(x) } 1 ♯ f ⏐ ⏐ dx + MBr (x) dxdr ≤ 1 . ⏐ ⏐ λ r λ R2 0 R2
)p(x) ˆ 1ˆ ( { } f (x) p(x) } 1 ♯ f | | dx + inf λ > 0 : MBr (x) dxdr ≤ 1 λ r λ R2 0 R2 )p(x) ˆ 1ˆ ( } { 1 ♯ f M dxdr ≤ 1 . = ∥f ∥Lp(x) (R2 ) + inf λ > 0 : r Br (x) λ 0 R2
C. Sin and S.-I. Ri / Nonlinear Analysis 192 (2020) 111666
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By (2.3) and [14, Theorem 6.1.9], we can see that ∥Rγ ∗ ∥Lp(x) (R2 ) ≤ c∥γ ∗ ∥
≤ c∥γ ∗ ∥Lp(x) (R2 )
2+p(x)
L 2p(x) (R2 )
(2.14)
2 since 2+p(x) 2p(x) < 2 and u has a compact support in R . For convenience, from now on we denote Br (x) by Br . By the definition of MB♯ r (x) , it is easy to show that
ˆ
1
0
ˆ R2
)p(x) )p(x) ˆ 1ˆ ( 1 1 ♯ ∗ ∗ ∗ M Rγ dxdr ≤ |Rγ (y) − Rγ (z)|dydz dxdr r Br r Br Br 0 R2 )p1 p(x) ˆ 1ˆ ( p1 1 ∗ ∗ dxdr, = |Rγ (y) − Rγ (z)|dydz r Br Br 0 R2
(
where p1 := inf x∈Bρ (x0 ) p(x). Using Jensen inequality and the fact that
1 2r
≤
1 |y−z|
(2.15)
all y, z ∈ Br , we obtain
) p1 1 ∗ ∗ |Rγ (y) − Rγ (z)|dydz I:= r Br Br 1 p ≤ p |Rγ ∗ (y) − Rγ ∗ (z)| 1 dydz r 1 Br Br ˆ ˆ p |Rγ ∗ (y) − Rγ ∗ (z)| 1 c dydz, ≤ 2+µ 2+p −µ r |y − z| 1 Br Br (
(2.16)
for any µ ∈ (0, 1). It is known (see [19, p. 74]) that by (2.3) and (2.4) ˆ ˆ p |Rγ ∗ (y) − Rγ ∗ (z)| 1 dydz ≤ c∥γ ∗ ∥pL1p1 (Br ) . 2+p1 −µ |y − z| Br Br
(2.17)
From (2.15)–(2.17), it follows that ˆ
1
0
ˆ
(
R2
1 ♯ M Rγ ∗ r Br
ˆ
)p(x)
1
ˆ
(
dxdr ≤ c ˆ
1
ˆ
(
=c R2
0
1 rµ
) p(x) p
∗ p1
r2+µ
R2
0
ˆ
1
1
|γ | dy
dxdr
Br p
|γ ∗ | 1 dy
(2.18)
) p(x) p 1
dxdr.
Br
Now we assume that ∥γ ∗ ∥Lp(x) (R2 ) ≤ 1. From p(x)-Jensen inequality (for example, [14, Theorem 4.2.4]) it follows that for a β ∈ (0, 1), depending only on Clog , (
∗ p1
|γ | dy
β
) p(x) p 1
∗ p(y)
≤
|γ |
Br
( dy + c
Br
Br
1 1 + m (e + |x|) (e + |y|)m
) χ{0<|γ ∗ (y)|≤1} dy,
where m > 2. Since ∥γ ∗ ∥Lp(x) (R2 ) ≤ 1, |γ ∗ |
p(y)
dy ≤
Br
c . r2
It is obvious that (
) (e + |x|)−m + (e + |y|)−m χ{0<|γ ∗ (y)|≤1} dy ≤ c,
Br
where c is independent of r. Thus (
∗ p1
|γ | dy
β Br
) p(x) p 1
≤
c + c. r2
(2.19)
C. Sin and S.-I. Ri / Nonlinear Analysis 192 (2020) 111666
7
The inequality (2.18) together with (2.19) implies that ˆ 0
1
ˆ R2
(
1 ♯ M Rγ ∗ r Br
)p(x)
ˆ
ˆ
) p(x) ) p1 ( c + c dxdr dxdr ≤ c r2 0 R2 ˆ 1 ˆ ρ ( ) p(x) ) p1 ( c 1 ≤c + c r′ dr′ dr ≤ c. rµ r2 0 0 1
(
1 rµ
(2.20)
where to perform the last estimate, we use 2+µ p(x) p1 < 3 by choosing ρ sufficiently small. Since the inequality (2.20) holds for γ ∗ with ∥γ ∗ ∥Lp(x) (R2 ) ≤ 1. By the scaling argument we have for all γ ∗ ∈ Lp(x) (R2 ), ˆ { inf λ > 0 :
1
ˆ R2
0
(
1 ♯ Rγ ∗ M r Br λ
)p(x)
} dxdr ≤ 1 ≤ c∥γ ∗ ∥Lp(x) (R2 ) .
Thus we conclude by (2.14) and the previous inequality that ∥Rγ ∗ ∥Tr W 1,p(x) (Ω) ≤ c∥Rγ ∗ ∥Tr,p(x) ≤ c∥γ ∗ ∥Lp(x) (∂Ω) , which is just the estimate (2.5).
□
Remark 2.1. The inequality (1.1) with p ≡ 2 was proved for any n ≥ 2, see [13]. But it seems to be impossible to apply the method from [13] to the problem with p ̸= 2. Remark 2.2. We think that it is possible to prove that the inequality (1.1) continues to hold for all function 1,p(x) v ∈ Wtan (Ω ) := {f ∈ W 1,p(x) (Ω ) | [f, ν] = 0 on ∂Ω }. Acknowledgments The author would like to thank the editor and the anonymous reviewer for their valuable suggestions to improve the quality and impact of this paper. References [1] C. Amrouche, N. Seloula, Lp -Theory for vector potentials and Sobolev’s inequality for vector fields, application to the Stokes equations with pressure boundary conditions, Math. Models Methods Appl. Sci. 23 (2013) 37–92. [2] S. Antontsev, F. Miranda, L. Santos, Blow-up and finite time extinction for p(x, t)-curl systems arising in electromagnetism, J. Math. Anal. Appl. 440 (2016) 300–322. [3] S.N. Antontsev, S. Shmarev, Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions, in: M. Chipot, P. Quittner (Eds.), Handbook of Differential Equations, in: Stationary Partial Differential Equations, vol. 3, Elsevier, 2006, pp. 1–100. [4] S. Antontsev, S. Shmarev, Evolution PDEs with NonstandArd Growth Conditions, Existence, Uniqueness, Localization, Blow-Up, in: Atlantis Studies in Differential Equations, vol. 4, Atlantis Press, 2015. [5] J. Aramaki, Lp Theory for the div–curl system, Int. J. Math. Anal. 8 (6) (2014) 259–271. [6] G. Auchmuty, The main inequality of 3D vector analysis, Math. Models Methods Appl. Sci. 14 (1) (2004) 79–103. [7] A. Bahrouni, D. Repovs, Existence and nonexistence of solutions for p(x)-curl systems arising in electromagnetism, Complex Var. Elliptic Equ. 63 (2) (2018) 292–301. [8] J. Bolik, W. Wahl, Estimating ∇u in terms of div u, curlu, either (ν, u) or ν × u and the topology, Math. Methods Appl. Sci. 20 (1997) 734–744. [9] T.A. Bui, Global W 1,p(·) estimate for renormalized solutions of quasilinear equations with measure data on reifenberg domains, Adv. Nonlinear Anal. 7 (2018) 517–533. [10] B. Cekic, A.V. Kalinin, R.A. Mashiyev, M. Avci, Lp(x) (Ω )-Estimates of vector fields and some applications to magnetostatics problems, J. Math. Anal. Appl. 389 (2012) 838–851. [11] I. Chlebicka, A pocket guide to nonlinear differential equations in musielak-orlicz spaces, Nonlinear Anal. 175 (2018) 1–27. [12] D. Cruz-Uribe, A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, in: Applied and Numerical Harmonic Analysis, Birkhauser, Basel, 2013.
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