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Imbedding theorems in Orlicz–Sobolev space of differential forms
Nonlinear Analysis 96 (2014) 87–95
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Imbedding theorems in Orlicz–Sobolev space of differential forms Shusen Ding a , Yuming Xing b,∗ a
Department of Mathematics, Seattle University, Seattle, WA 98122, USA
b
Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, PR China
article
info
Article history: Received 13 April 2013 Accepted 7 November 2013 Communicated by S. Carl MSC: primary 35J60 secondary 35B45 30C65 47J05 46E35
abstract In this paper, we prove imbedding inequalities with Lϕ -norms in the Orlicz–Sobolev space of the forms and establish Lϕ norm inequalities for the related operators applied to differential forms. We also obtain the global imbedding theorems in Lϕ (m)-averaging domains. © 2013 Elsevier Ltd. All rights reserved.
Keywords: Imbedding inequalities Differential forms Orlicz norms Harmonic equations
1. Introduction The purpose of this paper is to develop the local and global Lϕ imbedding inequalities for the Orlicz–Sobolev space of differential forms satisfying the A-harmonic equation. The imbedding inequalities have been playing a crucial role in the Lp theory of the Sobolev space and partial differential equations. The study and applications of imbedding inequalities are now ubiquitous in different areas, including PDEs and analysis. The investigation of the A-harmonic equation for differential forms has developed rapidly in recent years. The A-harmonic equation is an important extension of the p-harmonic equation div(∇ u|∇ u|p−2 ) = 0 in Rn , p > 1. In the meantime, the p-harmonic equation is a natural generalization of the usual Laplace equation ∆u = 0. Many interesting results concerning the properties of solutions to the A-harmonic equation have been established recently, see [1–5]. As extensions of the functions, differential forms have been widely studied and used in many fields of sciences and engineering, including theoretical physics, general relativity, potential theory and electromagnetism. For instance, differential forms can be used to describe various systems of partial differential equations and to express different geometrical structures on manifolds. Some of them are often utilized in studying deformations of elastic bodies, the related extrema for variational integrals and certain geometric invariance. The norm estimates for functions or differential forms are critical to investigate the properties of the solutions of the partial differential equations, or a system of the partial differential equations. The study of Lp norm inequalities, including Lp imbedding inequalities, for differential forms satisfying some versions of harmonic equations has been well developed during the recent years, see [1–9]. However, the investigation
∗
Corresponding author. Tel.: +86 451 86412607. E-mail addresses: [email protected] (S. Ding), [email protected] (Y. Xing).
0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.11.005
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for Lϕ imbeddings in the Orlicz–Sobolev space of differential forms just started. In this paper, we prove the local and global Lϕ imbedding theorems for the Orlicz–Sobolev space of differential forms. Our main results are presented and proved in Theorems 2.5 and 3.2, respectively. These results enrich the Lp theory of differential forms and can be used to estimate the integrals of the solutions of the related differential system and to study the Lϕ integrability of differential forms. Throughout this paper, we always assume that Ω is a bounded domain in Rn , n ≥ 2, B and σ B are the balls with the n same center and diam(σ B) = σ diam(B). We use |E | to denote the n-dimensional Lebesgue measure of a set E ⊆ R . For 1 a function u, the average of u over B is defined by uB = |B| B udm. All integrals involved in this paper are the Lebesgue integrals. Differential forms are widely used not only in analysis and partial differential equations [1,10], but also in physics [11,12]. Differential forms are extensions of differentiable functions in Rn . For example, the function u(x1 , x2 , . . . , xn ) is n called a 0-form. A differential 1-form u(x) in Rn can be written as u(x) = i=1 ui (x1 , x2 , . . . , xn )dxi , where the coefficient functions ui (x1 , x2 , . . . , xn ), i = 1, 2, . . . , n, are differentiable. Similarly, a differential k-form u(x) can be expressed as u(x) =
uI (x)dxI =
ui1 i2 ···ik (x)dxi1 ∧ dxi2 ∧ · · · ∧ dxik ,
I n n ′ l where I = (i1 , i2 , . . . , ik ), 1 ≤ i1 < i2 < · · · < ik ≤ n. Let ∧l = ∧l (R ) be the set of all l-forms in R , Dp (Ω , ∧ ) be the space p l of all differential l-forms in Ω , and L (Ω , ∧ ) be the l-forms u(x) = I uI (x)dxI in Ω satisfying Ω |uI | < ∞ for all ordered l-tuples I, l = 1, 2, . . . , n. We denote the exterior derivative by d and the Hodge star operator by ⋆. The Hodge codifferential operator d⋆ is given by d⋆ = (−1)nl+1 ⋆ d⋆, l = 1, 2, . . . , n. For u ∈ D′ (Ω , ∧l ) the vector-valued differential form
∇u =
∂u ∂u ,..., ∂ x1 ∂ xn
consists of differential forms ∂∂xu ∈ D′ (Ω , ∧l ), where the partial differentiation is applied to the coefficients of ω. We consider i here the nonlinear partial differential equation d⋆ A(x, du) = B(x, du)
(1.1)
which is called non-homogeneous A-harmonic equation, where A : Ω × ∧l (Rn ) → ∧l (Rn ) and B : Ω × ∧l (Rn ) →
∧l−1 (Rn ) satisfy the conditions: |A(x, ξ )| ≤ a|ξ |p−1 ,
A(x, ξ ) · ξ ≥ |ξ |p
and
|B(x, ξ )| ≤ b|ξ |p−1
(1.2)
for almost every x ∈ Ω and all ξ ∈ ∧l (Rn ). Here a, b > 0 are constants and 1 < p < ∞ is a fixed exponent associated with 1,p (1.1). A solution to (1.1) is an element of the Sobolev space Wloc (Ω , ∧l−1 ) such that
Ω
A(x, du) · dϕ + B(x, du) · ϕ = 0
(1.3)
1,p
for all ϕ ∈ Wloc (Ω , ∧l−1 ) with compact support. If u is a function (0-form) in Rn , the Eq. (1.1) reduces to divA(x, ∇ u) = B(x, ∇ u).
(1.4)
If the operator B = 0, Eq. (1.1) becomes d⋆ A(x, du) = 0
(1.5)
which is called the (homogeneous) A-harmonic equation. Let A : Ω × ∧ (R ) → ∧ (R ) be defined by A(x, ξ ) = ξ |ξ |p−2 with p > 1. Then, A satisfies the required conditions and (1.5) becomes the p-harmonic equation d⋆ (du|du|p−2 ) = 0 for differential forms. See [1–9] for recent results on the A-harmonic equations and related topics. Let D ⊂ Rn be a bounded, convex domain. The following operator Ky with the case y = 0 was first introduced by H. Cartan in [10]. Then, it was extended to the following general version in [13]. For each y ∈ D, there corresponds a linear 1 operator Ky : C ∞ (D, Λl ) → C ∞ (D, Λl−1 ) defined by (Ky ω)(x; ξ1 , . . . , ξl−1 ) = 0 t l−1 ω(tx + y − ty; x − y, ξ1 , . . . , ξl−1 )dt and the decomposition ω = d(Ky ω) + Ky (dω). A homotopy operator T : C ∞ (D, Λl ) → C ∞ (D, Λl−1 ) is defined by averaging Ky over all points y in D l
Tω =
ϕ(y)Ky ωdy ,
n
l
n
(1.6)
D
where ϕ ∈ C0∞ (D) is normalized by
1 0
ϕ(y)dy = 1. For simplicity purpose, we write ξ = (ξ1 , . . . , ξl−1 ). Then, T ω(x; ξ ) = t ϕ(y)ω(tx + y − ty; x − y, ξ )dydt. By substituting z = tx + y − ty and t = s/(1 + s), we have D T ω(x; ξ ) = ω(z , ζ (z , x − z ), ξ )dz , (1.7) l−1
D
D
S. Ding, Y. Xing / Nonlinear Analysis 96 (2014) 87–95
89
∞
where the vector function ζ : D × Rn → Rn is given by ζ (z , h) = h 0 sl−1 (1 + s)n−1 ϕ(z − sh)ds. The integral (1.7) defines a bounded operator T : Ls (D, Λl ) → W 1,s (D, Λl−1 ), l = 1, 2, . . . , n, and the decomposition u = d(Tu) + T (du)
(1.8)
holds for any differential form u. The l-form ωD ∈ D′ (D, Λl ) is defined by
ωD = − ω(y)dy = |D|−1 ω(y)dy, D
l = 0,
and ωD = d(T ω),
l = 1, 2, . . . , n,
(1.9)
D
for all ω ∈ Lp (D, Λl ), 1 ≤ p < ∞. Also, for any differential form u, we have
∥∇(Tu)∥p,D ≤ C |D|∥u∥p,D ,
and
∥Tu∥p,D ≤ C |D|diam(D)∥u∥p,D .
(1.10)
From [14, Page 16], we know that any open subset Ω in R is the union of a sequence of cubes Qk , whose sides are parallel to the axes, whose interiors are mutually disjoint, and whose diameters are approximately proportional to their distances 0 0 from F . Specifically, (i) Ω = ∪∞ k=1 Qk , (ii) Qj ∩ Qk = φ if j ̸= k, (iii) there exist two constants c1 , c2 > 0 (we can take c1 = 1, and c2 = 4), so that c1 diam(Qk ) ≤ distance Qk from F ≤ c2 diam(Qk ). Thus, the definition of the homotopy operator T can be generalized to any domain Ω in Rn : For any x ∈ Ω , x ∈ Qk for some k. Let TQk be the homotopy operator defined on ∞Qk (each cube is bounded and convex). Thus, we can define the homotopy operator TΩ on any domain Ω by TΩ = k=1 TQk χQk (x) . n
2. Local imbedding inequalities The Sobolev imbedding inequalities have been serving as an effective tool in the study of the Sobolev spaces and PDEs. Different versions of the classical Sobolev imbedding inequality for functions (0-forms) have been very well developed and widely used analysis and PDEs in last several decades. In recent years, some versions of the Sobolev imbedding inequality with Lp norms for differential forms or operators applied to differential forms have also been established. See [1, Chapter 5] for recent progress in the study of the Sobolev imbedding inequalities with Lp norms. In this section, we first prove the local Lϕ imbedding inequalities for differential forms satisfying the non-homogeneous A-harmonic equation in a bounded domain. A continuously increasing function ϕ : [0, ∞) → [0, ∞) with ϕ(0 ) = 0, is called an Orlicz function. The Orlicz space Lϕ (Ω ) consists of all measurable functions f on Ω such that
Ω
ϕ
|f | λ
dx < ∞ for some λ = λ(f ) > 0. Lϕ (Ω ) is
equipped with the nonlinear Luxemburg functional
∥f ∥Lϕ (Ω ) = inf
|f | λ>0: ϕ dx ≤ 1 . λ Ω
A convex Orlicz function ϕ is often called a Young function. If ϕ is a Young function, then ∥ · ∥Lϕ (Ω ) defines a norm in Lϕ (Ω ), which is called the Luxemburg norm or Orlicz norm. The space Lϕ (Ω ) is a Banach space when ϕ is a Young function. We ϕ use Wd (Ω , ∧l ) to express the space of l-forms u ∈ Lϕ (Ω , ∧l ) such that du ∈ Lϕ (Ω , ∧l+1 ) and W 1,ϕ (Ω , ∧l ) to denote the ϕ Orlicz–Sobolev space of l-forms which equals Lϕ (Ω , ∧l ) ∩ L1 (Ω , ∧l ) with norm
∥u∥W 1,ϕ (Ω ) = ∥u∥W 1,ϕ (Ω ,∧l ) = diam(Ω )−1 ∥u∥Lϕ (Ω ) + ∥∇ u∥Lϕ (Ω ) .
(2.1)
We should notice that the Orlicz–Sobolev space W 1,ϕ (Ω , ∧l ) of differential forms is an extension of the well known Sobolev space W 1,p (Ω ) of functions. Specifically, if we choose ϕ(t ) = t p , p > 1, the Orlicz–Sobolev space W 1,ϕ (Ω , ∧l ) reduces to the Sobolev space W 1,p (Ω , ∧l ) of differential forms. Also W 1,ϕ (Ω , ∧l ) ⊂ Lϕ (Ω , ∧l ) and W 1,ϕ (Ω , ∧l ) is a Banach space when ϕ is a Young function. Furthermore, we have W 1,p (Ω , ∧0 ) = W 1,p (Ω ). Note that functions are 0-forms. Hence, all theorems proved in this paper for differential forms still hold for functions. Definition 2.1 ([15]). We say a Young function ϕ lies in the class G(p, q, C ), 1 ≤ p < q < ∞, C ≥ 1, if (i) 1/C ≤ ϕ(t 1/p )/g (t ) ≤ C and (ii) 1/C ≤ ϕ(t 1/q )/h(t ) ≤ C for all t > 0, where g is a convex increasing function and h is a concave increasing function on [0, ∞). From [15], each of ϕ, g and h in above definition is doubling in the sense that its values at t and 2t are uniformly comparable for all t > 0, and the consequent fact that C1 t q ≤ h−1 (ϕ(t )) ≤ C2 t q ,
C1 t p ≤ g −1 (ϕ(t )) ≤ C2 t p ,
(2.2)
where C1 and C2 are constants. Also, for all 1 ≤ p1 < p < p2 and α ∈ R, the function ϕ(t ) = t logα+ t belongs to G(p1 , p2 , C ) for some constant C = C (p, α, p1 , p2 ). Here log+ (t ) is defined by log+ (t ) = 1 for t ≤ e; and log+ (t ) = log(t ) for t > e. p
Particularly, if α = 0, we see that ϕ(t ) = t p lies in G(p1 , p2 , C ), 1 ≤ p1 < p < p2 . We will need the following Reverse Hölder inequality.
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Lemma 2.2 ([1]). Let u be a solution of the non-homogeneous A-harmonic equation (1.1) in a domain Ω and 0 < s, t < ∞. Then, there exists a constant C , independent of u, such that
∥du∥s,B ≤ C |B|(t −s)/st ∥du∥t ,σ B
(2.3)
for all balls B with σ B ⊂ Ω for some σ > 1. Theorem 2.3. Let ϕ be a Young function in the class G(p, q, C ), 1 ≤ p < q < ∞, C ≥ 1, Ω be a bounded and convex domain, and T : C ∞ (Ω , ∧l ) → C ∞ (Ω , ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator defined in (1.6). Assume that ϕ(|du|) ∈ L1loc (Ω ) and u is a solution of the non-homogeneous A-harmonic equation (1.1) in Ω . Then, there exists a constant C , independent of u, such that
∥T (du)∥Lϕ (B) ≤ C ∥du∥Lϕ (σ B)
(2.4)
for all balls B with σ B ⊂ Ω . Proof. Applying (1.10), we have
∥T (du)∥q,B ≤ C1 |B|1+1/n ∥du∥q,B
(2.5)
for all balls B with B ⊂ Ω . From Lemma 2.2, for any positive numbers p and q, it follows that
1/q ≤ C2 |B|(p−q)/pq |du|q dx
B
σB
1/p , |du|p dx
(2.6)
where σ is a constant σ > 1. Using Jensen’s inequality for h−1 , (2.2), (2.5), (2.6), (i) in Definition 2.1, and noticing the fact that ϕ and h are doubling, and ϕ is an increasing function, we obtain
ϕ(|T (du)|)dx = h h−1 ϕ (|T (du)|) dx B B −1 h ≤h ϕ (|T (du)|) dx B ≤ h C3 |T (du)|q dx B 1/q ≤ C4 ϕ C3 |T (du)|q dx B 1/q 1+1/n ≤ C4 ϕ C5 |B| |du|q dx B 1/p 1+1/n+(p−q)/pq ≤ C4 ϕ C6 |B| |du|p dx σB 1/p p p(1+1/n)+(p−q)/q |du|p dx ≤ C4 ϕ C6 |B| σB p p(1+1/n)+(p−q)/q ≤ C7 g C6 |B| |du|p dx σB p p(1+1/n)+(p−q)/q = C7 g C6 |B| |du|p dx σ B p ≤ C7 g C6 |B|p(1+1/n)+(p−q)/q |du|p dx σB
≤ C8 Since p ≥ 1, then 1 +
1 n
+
σB
p−q pq
p−q 1 ϕ C6 |B|1+ n + pq |du| dx.
(2.7) 1
> 0. Hence, we have |B|1+ n +
p−q 1 ϕ C6 |B|1+ n + pq |du| ≤ C9 ϕ |du| .
p−q pq
1
≤ |Ω |1+ n +
p−q pq
≤ C5 . Note that ϕ is doubling, we obtain (2.8)
Combining (2.7) and (2.8) yields
ϕ (|T (du)|) dx ≤ C10 B
σB
ϕ (|du|) dx.
(2.9)
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Since each of ϕ, g and h in Definition 2.1 is doubling, from (2.9), we have
ϕ
B
|T (du)| |du| dx ≤ C ϕ dx λ λ σB
(2.10)
for all balls B with σ B ⊂ Ω and any constant λ > 0. From (2.1) and (2.10), we have the following inequality with the Luxemburg norm
∥T (du)∥Lϕ (B) ≤ C ∥du∥Lϕ (σ B) .
(2.11)
The proof of Theorem 2.3 has been completed.
Theorem 2.4. Let ϕ be a Young function in the class G(p, q, C ), 1 ≤ p < q < ∞, C ≥ 1, q(n − p) < np, Ω be a bounded domain and T : C ∞ (Ω , ∧l ) → C ∞ (Ω , ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator defined in (1.6). Assume that u ∈ D′ (Ω , ∧l ) is any differential l-form, ϕ(|u|) ∈ L1loc (Ω ). Then, there exists a constant C , independent of u, such that
∥T (du)∥Lϕ (B) ≤ C ∥du∥Lϕ (B)
(2.12)
for all balls B with B ⊂ Ω . Proof. Note that T (du) = u − uB holds for any differential form u. Using (2.7), we have
ϕ(|T (du)|)dx ≤ C1 ϕ
= C1 ϕ
1/q |T (du)|q dx B
B
1/q |u − uB |q dx .
(2.13)
B
If 1 < p < n, by assumption, we have q < n−p . Using the Poincaré-type inequality for differential forms u np
|u − uB |
np/(n−p)
(n−p)/np
p
≤ C2
dx
1/p
|du| dx
B
,
(2.14)
B
we find that
1/q 1/p |u − uB |q dx ≤ C3 |du|p dx .
B
(2.15)
B
Combining (2.13) and (2.15), we obtain
ϕ(|T (du)|)dx ≤ C1 ϕ C3 B
p
1/p
|du| dx
(2.16)
B
for 1 < p < n. Note that the Lp -norm of |u − uB | increases with p and n−p → ∞ as p → n, it follows that (2.15) still holds when p ≥ n. Since ϕ is increasing, from (2.13) and (2.15), we obtain np
1/p ϕ(|T (du)|)dx ≤ C1 ϕ C3 |u|p dx .
B
(2.17)
B
Applying (2.17), (i) in Definition 2.1, Jensen’s inequality, and noticing that ϕ and g are doubling, we have
1/p ϕ(|T (du)|)dx ≤ C1 ϕ C3 |du|p dx B B ≤ C1 g C4 |du|p dx B
g (|du|p )dx.
≤ C5
(2.18)
B
Using (i) in Definition 2.1 again yields
g (|du|p )dx ≤ C6 B
ϕ(|du|)dx.
(2.19)
B
Combining (2.18) and (2.19), we obtain
ϕ |T (du)| dx ≤ C7 ϕ(|du|)dx. B
B
(2.20)
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From (2.1) and (2.20), we have the following inequality with the Luxemburg norm
∥T (du)∥Lϕ (B) ≤ C ∥du∥Lϕ (B) . The proof of Theorem 2.4 has been completed.
Applying the first inequality in (1.10) to du, we have
∥∇(T (du))∥q,B ≤ C |B|∥du∥q,B (2.21) for all balls B and q > 1. Starting with (2.21) and using the similar method to the proof of Theorem 2.3, we know that the following Lϕ -norm inequality
∥∇(T (du))∥Lϕ (B) ≤ C ∥du∥Lϕ (σ B)
(2.22)
holds if all conditions of Theorem 2.3 are satisfied. Now we are ready to prove the following main theorem, the local Lϕ -norm imbedding inequality in Orlicz–Sobolev space of differential forms. Theorem 2.5. Let ϕ be a Young function in the class G(p, q, C ), 1 ≤ p < q < ∞, C ≥ 1, Ω be a bounded and convex domain. Assume that ϕ(|du|) ∈ L1loc (Ω , m) and u is a solution of the non-homogeneous A-harmonic equation (1.1) in Ω . Then, there exists a constant C , independent of u, such that
∥u − uB ∥W 1,ϕ (B,∧l ) ≤ C ∥du∥Lϕ (σ B)
(2.23)
for all balls B with σ B ⊂ Ω . Proof. Using the definition of the norm in Orlicz–Sobolev space of differential forms, Theorems 2.3 and 2.4, and noticing u − uB = T (du) for any differential form u, we have
∥u − uB ∥W 1,ϕ (B,∧l ) = ∥T (du)∥W 1,ϕ (B,∧l ) = (diam(B))−1 ∥T (du)∥Lϕ (B) + ∥∇(T (du))∥Lϕ (B) ≤ C1 ∥du∥Lϕ (σ1 B) + C2 ∥du∥Lϕ (σ2 B) ≤ C3 ∥du∥Lϕ (σ B)
(2.24)
for any differential form u and all balls B with σ B ⊂ Ω , where σ1 , σ2 > 1 are some constants and σ = max{σ1 , σ2 }. The proof of Theorem 2.5 has been completed. Using (2.1) and (2.23), we have
∥u − uB ∥Lϕ (B) = diam(B) (diam(B))−1 ∥u − uB ∥Lϕ (B) ≤ diam(B) (diam(B))−1 ∥u − uB ∥Lϕ (B) + ∥∇(u − uB )∥Lϕ (B) = diam(B)∥u − uB ∥W 1,ϕ (B,∧l ) ≤ Cdiam(B)∥du∥Lϕ (σ B) ,
(2.25)
that is,
∥u − uB ∥Lϕ (B) ≤ Cdiam(B)∥du∥Lϕ (σ B)
(2.26)
where σ > 1 is a constant. Remark. (i) From above discussion, we find that the local Orlicz–Sobolev imbedding inequality with Lϕ -norm proved in Theorem 2.5 implies directly the local Poincaré inequality (2.26) with Lϕ -norm. (ii) We should also note that Theorems 2.3–2.5 include some existing results as their special cases. For example, selecting ϕ(t ) = t p in Theorem 2.5, we obtain the usual imbedding inequalities with the Lp -norms
∥u − uB ∥W 1,p (B,∧l ) ≤ C ∥du∥p,σ B
(2.27)
(logα+
for all balls B with σ B ⊂ Ω , where σ > 1 is a constant. (iii) The L L)-imbedding inequality is also a special case of Theorem 2.5. Specifically, choosing ϕ(t ) = t p logα+ t in Theorem 2.5, we obtain the following imbedding inequalities with the Lp (logα+ L)-norms. p
Theorem 2.6. Let ϕ(t ) = t p logα+ t, p ≥ 1 and α ∈ R. Assume that ϕ(|du|) ∈ L1loc (Ω , m) and u is a solution of the nonhomogeneous A-harmonic equation (1.1). Then, there exists a constant C , independent of u, such that
∥u − uB ∥W 1,t p logα+ t (B,∧l ) ≤ C ∥du∥Lp (logα+ L)(σ B) .
(2.28)
for all balls B with σ B ⊂ Ω , where σ > 1 is a constant. Similarly, if we choose ϕ(t ) = t p logα+ t or ϕ(t ) = t p in Theorems 2.3 and 2.4, respectively, we will obtain the corresponding special results.
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3. Global imbedding theorem In this section, we extend the local imbedding inequalities into the global Orlicz–Sobolev imbedding inequality with Lϕ -norms in the following Lϕ -averaging domains. Definition 3.1 ([16]). Let ϕ be an increasing convex function on [0, ∞) with ϕ(0) = 0. We call a proper subdomain Ω ⊂ Rn an Lϕ -averaging domain, if |Ω | < ∞ and there exists a constant C such that
Ω
ϕ(τ |u − uB0 |)dx ≤ C sup
B ⊂Ω
ϕ(σ |u − uB |)dx
(3.1)
B
for some ball B0 ⊂ Ω and all u such that ϕ(|u|) ∈ L1loc (Ω ), where τ , σ are constants with 0 < τ < ∞, 0 < σ < ∞ and the supremum is over all balls B ⊂ Ω . From above definition and [1] we see that Ls -averaging domains are special Lϕ -averaging domains when ϕ(t ) = t s in Definition 3.1. Also, uniform domains and John domains are very special Lϕ -averaging domains, see [16,17] for more results about the averaging domains. Now we prove the global Orlicz–Sobolev imbedding inequality with Lϕ -norms in the following theorem. Theorem 3.2. Let ϕ be a Young function in the class G(p, q, C ), 1 ≤ p < q < ∞, C ≥ 1, and Ω be any bounded Lϕ -averaging domain. Assume that ϕ(|du|) ∈ L1 (Ω ) and u ∈ D′ (Ω , ∧0 ) is a solution of the non-homogeneous A-harmonic equation (1.1) in Ω . Then, there exists a constant C , independent of u, such that
∥u − uB0 ∥W 1,ϕ (Ω ) ≤ C ∥du∥Lϕ (Ω )
(3.2)
where B0 ⊂ Ω is some fixed ball. Proof. Using the Poincaré inequality ∥u−uB ∥q,B ≤ C ∥du∥q,B , q > 1, for differential forms (see [1]) and the method developed in the proof of Theorem 2.3, we have the following Poincaré inequality with Lϕ -norm
ϕ (|u − uB |) dx ≤ C1 diam(B)
ϕ (|du|) dx.
(3.3)
B
B
From Definition 3.1 and (3.3), we obtain
∥u − uB0 ∥Lϕ (Ω ) ≤ C2
Ω
ϕ |u − uB0 | dx
ϕ (|u − uB |) dx
≤ C3 sup B ⊂Ω
B
≤ C3 sup C1 diam(B) ϕ (|du|) dx B ⊂Ω
B
≤ C3 sup C1 diam(Ω ) B ⊂Ω
≤ C4 diam(Ω )
Ω
Ω
ϕ (|du|) dx
ϕ (|du|) dx
≤ C5 diam(Ω )∥du∥Lϕ (Ω ) .
(3.4)
From (2.1), (3.4), and noticing that ϕ is doubling and |∇ u| = |du| for any u ∈ ∧0 , it follows that
∥u − uB0 ∥W 1,ϕ (Ω ,∧0 ) = (diam(Ω ))−1 ∥u − uB0 ∥Lϕ (Ω ) + ∥∇ u∥Lϕ (Ω ) ≤ (diam(Ω ))−1 (C5 diam(Ω )∥du∥Lϕ (Ω ) ) + ∥du∥Lϕ (Ω ) ≤ C6 ∥du∥Lϕ (Ω ) We have completed the proof of Theorem 3.2.
(3.5)
Remark. (i) Since the class of Lϕ -averaging domains is a very large class of domains [1], Theorem 3.2, the global Lϕ imbedding theorem, is a far extension of the existing imbedding theorems. For example, we know that the Ls -averaging domains and uniform domains are special cases of the Lϕ -averaging domains. Then, Theorem 3.2 still holds if Ω is an Ls -averaging domain or a uniform domain. (ii) For the case ϕ(t ) = t p , p > 1, inequality (3.2) reduces to the following Lp imbedding inequality
∥u − uB0 ∥W 1,p (Ω ) ≤ C ∥du∥p,Ω .
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S. Ding, Y. Xing / Nonlinear Analysis 96 (2014) 87–95
(iii) Similar to the local case, the global Orlicz–Sobolev Lϕ imbedding inequality (3.2) implies directly the global Poincaré inequality with Lϕ -norm
∥u − uB0 ∥Lϕ (Ω ) ≤ Cdiam(Ω )∥du∥Lϕ (Ω ) ,
(3.6)
where C is a constant. It has been proved that any John domain is a special Lϕ -averaging domain [1]. Hence, we have the following imbedding inequality for John domains. Corollary 3.3. Let ϕ be a Young function in the class G(p, q, C ), 1 ≤ p < q < ∞, C ≥ 1, and Ω be a bounded John domain. Assume that ϕ(|du|) ∈ L1 (Ω ) and u ∈ D′ (Ω , ∧0 ) is a solution of the non-homogeneous A-harmonic equation (1.1) in Ω . Then, there exists a constant C , independent of u, such that
∥u − uB0 ∥W 1,ϕ (Ω ) ≤ C ∥du∥Lϕ (Ω )
(3.7)
where B0 ⊂ Ω is some fixed ball. Choosing ϕ(t ) = t p logα+ t in Theorem 3.2, we obtain the following imbedding inequality with the Lp (logα+ L)-norms. Corollary 3.4. Let ϕ(t ) = t p logα+ t, p ≥ 1, α ∈ R. Assume that ϕ(|du|) ∈ L1 (Ω ) and u ∈ D′ (Ω , ∧0 ) is a solution of the non-homogeneous A-harmonic equation (1.1). Then, there exists a constant C , independent of u, such that
∥u − uB0 ∥W 1,ϕ (Ω ) ≤ C ∥du∥Lϕ (Ω )
(3.8)
for any bounded Lϕ -averaging domain Ω and B0 ⊂ Ω is some fixed ball. 4. Applications As applications of the main theorems proved in last two sections, we will study two examples in this section. If u is a function (0-form) in the homogeneous A-harmonic equation (1.5), then (1.5) reduces to the following A-harmonic equation divA(x, ∇ u) = 0
(4.1)
for functions. If the above operator A : Ω × ∧l (Rn ) → ∧l (Rn ) is defined by A(x, ξ ) = ξ |ξ |p−2 with p > 1. Then, A satisfies the required conditions and (4.1) becomes the usual p-harmonic equation for functions div(∇ u|∇ u|p−2 ) = 0
(4.2)
which is equivalent to
(p − 2)
n n
uxk uxi uxk xi + |∇ u|2 ∆u = 0.
(4.3)
k=1 i=1
If we choose p = 2 in (4.2), we have the Laplace equation ∆u = 0 for functions. Thus, from Theorem 3.2, we have the following example. Example 4.1. Let u be a solution of the usual A-harmonic equation (4.1) or the p-harmonic equation (4.2), ϕ be a Young function in the class G(p, q, C ), 1 ≤ p < q < ∞, C ≥ 1, and Ω be any bounded Lϕ -averaging domain. If ϕ(|du|) ∈ L1 (Ω ), then there exists a constant C , independent of u, such that
∥u − uB0 ∥W 1,ϕ (Ω ) ≤ C ∥du∥Lϕ (Ω ) where B0 ⊂ Ω is some fixed ball. Example 4.2. In some areas of science and engineering, such as astronomy, fluid dynamics, and electromagnetism, we need 1,p to study and use various mappings. Let f : Ω → Rn , f = (f 1 , . . . , f n ), be a mapping of the Sobolev class Wloc (Ω , Rn ), i 1 ≤ p < ∞, whose distributional differential Df = [∂ f /∂ xj ] : Ω → GL(n) is a locally integrable function on Ω with values in the space GL(n) of all n × n-matrices, i, j = 1, 2, . . . , n. A homeomorphism f : Ω → Rn of Sobolev class 1,n Wloc (Ω , Rn ) is said to be K -quasiconformal, 1 ≤ K < ∞, if its differential matrix Df (x) and the Jacobian determinant J = J (x, f ) = detDf (x) satisfy
|Df (x)|n ≤ KJ (x, f ),
(4.4)
where |Df (x)| = max{|Df (x)h| : |h| = 1} denotes the norm of the Jacobi matrix Df (x). It is well known that if the differential matrix Df (x) = [∂ f i /∂ xj ], i, j = 1, 2, . . . , n, of a homeomorphism f (x) = (f 1 , f 2 , . . . , f n ) : Ω → Rn satisfies (4.4), then, each of the functions u = f i (x),
i = 1, 2, . . . , n,
or
u = log |f (x)|,
(4.5)
S. Ding, Y. Xing / Nonlinear Analysis 96 (2014) 87–95
95
is a generalized solution of the quasilinear elliptic equation divA(x, ∇ u) = 0, in Ω − f
−1
(4.6)
(0), where
A = (A1 , A2 , . . . , An ),
∂ A i ( x, ξ ) = ∂ξi
n
n/2 θi,j (x)ξi ξj
i,j=1
and θi,j are some functions, which can be expressed in terms of the differential matrix Df (x) and satisfy C1 (K )|ξ |2 ≤
n
θi,j ξi ξj ≤ C2 (K )|ξ |2
(4.7)
i ,j
for some constants C1 (K ), C2 (K ) > 0. All results proved in Sections 2 and 3 are still true if u is defined in (4.5). Remark. Note that the method developed in this paper can be used to prove both local and global imbedding theorems for operators applied to differential forms, such as the potential operator P in [18], Green’s operator G and the homotopy operator T in [1]. The global imbedding theorems for these operators will also hold in Lϕ -averaging domains. Acknowledgments We would like to thank the referees and the handling editor for their time and effort spent on this paper. We also thank the referees for their precious and thoughtful suggestions which largely improved the presentation of the paper. References [1] R.P. Agarwal, S. Ding, C.A. Nolder, Inequalities for Differential Forms, Springer, 2009. [2] Y. Wang, C. Wu, Sobolev imbedding theorems and Poincaré inequalities for Green’s operator on solutions of the nonhomogeneous A-harmonic equation, Comput. Math. Appl. 47 (2004) 1545–1554. [3] S. Ding, Lipschitz and BMO norm inequalities for operators, Nonlinear Anal. TMA (2009). [4] G. Bao, Ar (λ)-weighted integral inequalities for A-harmonic tensors, J. Math. Anal. Appl. 247 (2000) 466–477. [5] Y. Wang, Two-weight Poincaré type inequalities for differential forms in Ls (µ)-averaging domains, Appl. Math. Lett. 20 (2007) 1161–1166. [6] Y. Xing, Weighted Poincaré-type estimates for conjugate A-harmonic tensors, J. Inequal. Appl. 1 (2005) 1–6. [7] B. Liu, Aλr (Ω )-weighted imbedding inequalities for A-harmonic tensors, J. Math. Anal. Appl. 273 (2) (2002) 667–676. [8] C.A. Nolder, Hardy–Littlewood theorems for A-harmonic tensors, Illinois J. Math. 43 (1999) 613–631. [9] S. Ding, Two-weight Caccioppoli inequalities for solutions of nonhomogeneous A-harmonic equations on Riemannian manifolds, Proc. Amer. Math. Soc. 132 (2004) 2367–2375. [10] H. Cartan, Differential Forms, Houghton Mifflin Co., Boston, 1970. [11] S.K. Sachs, H. Wu, General Relativity for Mathematicians, Springer, New York, 1977. [12] C. Westenholz, Differential Forms in Mathematical Physics, North Holland Publishing, Amesterdam, 1978. [13] T. Iwaniec, A. Lutoborski, Integral estimates for null Lagrangians, Arch. Ration. Mech. Anal. 125 (1993) 25–79. [14] E.M. Stein, Harmonic Analysis, Princeton University Press, Princeton, 1993. [15] S.M. Buckley, P. Koskela, Orlicz–Hardy inequalities, Illinois J. Math. 48 (2004) 787–802. [16] S. Ding, Lϕ (µ)-averaging domains and the quasihyperbolic metric, Comput. Math. Appl. 47 (2004) 1611–1618. [17] S.G. Staples, Averaging domains: from Euclidean spaces to homogeneous spaces, in: Proc. of Conference on Differential & Difference Equations and Applications, Hindawi Publishing, 2006, pp. 1041–1048. [18] H. Bi, Weighted inequalities for potential operators on differential forms, J. Inequal. Appl. (2010) Article ID 713625.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Imbedding theorems in Orlicz–Sobolev space of differential forms Shusen Ding a , Yuming Xing b,∗ a
Department of Mathematics, Seattle University, Seattle, WA 98122, USA
b
Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, PR China
article
info
Article history: Received 13 April 2013 Accepted 7 November 2013 Communicated by S. Carl MSC: primary 35J60 secondary 35B45 30C65 47J05 46E35
abstract In this paper, we prove imbedding inequalities with Lϕ -norms in the Orlicz–Sobolev space of the forms and establish Lϕ norm inequalities for the related operators applied to differential forms. We also obtain the global imbedding theorems in Lϕ (m)-averaging domains. © 2013 Elsevier Ltd. All rights reserved.
Keywords: Imbedding inequalities Differential forms Orlicz norms Harmonic equations
1. Introduction The purpose of this paper is to develop the local and global Lϕ imbedding inequalities for the Orlicz–Sobolev space of differential forms satisfying the A-harmonic equation. The imbedding inequalities have been playing a crucial role in the Lp theory of the Sobolev space and partial differential equations. The study and applications of imbedding inequalities are now ubiquitous in different areas, including PDEs and analysis. The investigation of the A-harmonic equation for differential forms has developed rapidly in recent years. The A-harmonic equation is an important extension of the p-harmonic equation div(∇ u|∇ u|p−2 ) = 0 in Rn , p > 1. In the meantime, the p-harmonic equation is a natural generalization of the usual Laplace equation ∆u = 0. Many interesting results concerning the properties of solutions to the A-harmonic equation have been established recently, see [1–5]. As extensions of the functions, differential forms have been widely studied and used in many fields of sciences and engineering, including theoretical physics, general relativity, potential theory and electromagnetism. For instance, differential forms can be used to describe various systems of partial differential equations and to express different geometrical structures on manifolds. Some of them are often utilized in studying deformations of elastic bodies, the related extrema for variational integrals and certain geometric invariance. The norm estimates for functions or differential forms are critical to investigate the properties of the solutions of the partial differential equations, or a system of the partial differential equations. The study of Lp norm inequalities, including Lp imbedding inequalities, for differential forms satisfying some versions of harmonic equations has been well developed during the recent years, see [1–9]. However, the investigation
∗
Corresponding author. Tel.: +86 451 86412607. E-mail addresses: [email protected] (S. Ding), [email protected] (Y. Xing).
0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.11.005
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for Lϕ imbeddings in the Orlicz–Sobolev space of differential forms just started. In this paper, we prove the local and global Lϕ imbedding theorems for the Orlicz–Sobolev space of differential forms. Our main results are presented and proved in Theorems 2.5 and 3.2, respectively. These results enrich the Lp theory of differential forms and can be used to estimate the integrals of the solutions of the related differential system and to study the Lϕ integrability of differential forms. Throughout this paper, we always assume that Ω is a bounded domain in Rn , n ≥ 2, B and σ B are the balls with the n same center and diam(σ B) = σ diam(B). We use |E | to denote the n-dimensional Lebesgue measure of a set E ⊆ R . For 1 a function u, the average of u over B is defined by uB = |B| B udm. All integrals involved in this paper are the Lebesgue integrals. Differential forms are widely used not only in analysis and partial differential equations [1,10], but also in physics [11,12]. Differential forms are extensions of differentiable functions in Rn . For example, the function u(x1 , x2 , . . . , xn ) is n called a 0-form. A differential 1-form u(x) in Rn can be written as u(x) = i=1 ui (x1 , x2 , . . . , xn )dxi , where the coefficient functions ui (x1 , x2 , . . . , xn ), i = 1, 2, . . . , n, are differentiable. Similarly, a differential k-form u(x) can be expressed as u(x) =
uI (x)dxI =
ui1 i2 ···ik (x)dxi1 ∧ dxi2 ∧ · · · ∧ dxik ,
I n n ′ l where I = (i1 , i2 , . . . , ik ), 1 ≤ i1 < i2 < · · · < ik ≤ n. Let ∧l = ∧l (R ) be the set of all l-forms in R , Dp (Ω , ∧ ) be the space p l of all differential l-forms in Ω , and L (Ω , ∧ ) be the l-forms u(x) = I uI (x)dxI in Ω satisfying Ω |uI | < ∞ for all ordered l-tuples I, l = 1, 2, . . . , n. We denote the exterior derivative by d and the Hodge star operator by ⋆. The Hodge codifferential operator d⋆ is given by d⋆ = (−1)nl+1 ⋆ d⋆, l = 1, 2, . . . , n. For u ∈ D′ (Ω , ∧l ) the vector-valued differential form
∇u =
∂u ∂u ,..., ∂ x1 ∂ xn
consists of differential forms ∂∂xu ∈ D′ (Ω , ∧l ), where the partial differentiation is applied to the coefficients of ω. We consider i here the nonlinear partial differential equation d⋆ A(x, du) = B(x, du)
(1.1)
which is called non-homogeneous A-harmonic equation, where A : Ω × ∧l (Rn ) → ∧l (Rn ) and B : Ω × ∧l (Rn ) →
∧l−1 (Rn ) satisfy the conditions: |A(x, ξ )| ≤ a|ξ |p−1 ,
A(x, ξ ) · ξ ≥ |ξ |p
and
|B(x, ξ )| ≤ b|ξ |p−1
(1.2)
for almost every x ∈ Ω and all ξ ∈ ∧l (Rn ). Here a, b > 0 are constants and 1 < p < ∞ is a fixed exponent associated with 1,p (1.1). A solution to (1.1) is an element of the Sobolev space Wloc (Ω , ∧l−1 ) such that
Ω
A(x, du) · dϕ + B(x, du) · ϕ = 0
(1.3)
1,p
for all ϕ ∈ Wloc (Ω , ∧l−1 ) with compact support. If u is a function (0-form) in Rn , the Eq. (1.1) reduces to divA(x, ∇ u) = B(x, ∇ u).
(1.4)
If the operator B = 0, Eq. (1.1) becomes d⋆ A(x, du) = 0
(1.5)
which is called the (homogeneous) A-harmonic equation. Let A : Ω × ∧ (R ) → ∧ (R ) be defined by A(x, ξ ) = ξ |ξ |p−2 with p > 1. Then, A satisfies the required conditions and (1.5) becomes the p-harmonic equation d⋆ (du|du|p−2 ) = 0 for differential forms. See [1–9] for recent results on the A-harmonic equations and related topics. Let D ⊂ Rn be a bounded, convex domain. The following operator Ky with the case y = 0 was first introduced by H. Cartan in [10]. Then, it was extended to the following general version in [13]. For each y ∈ D, there corresponds a linear 1 operator Ky : C ∞ (D, Λl ) → C ∞ (D, Λl−1 ) defined by (Ky ω)(x; ξ1 , . . . , ξl−1 ) = 0 t l−1 ω(tx + y − ty; x − y, ξ1 , . . . , ξl−1 )dt and the decomposition ω = d(Ky ω) + Ky (dω). A homotopy operator T : C ∞ (D, Λl ) → C ∞ (D, Λl−1 ) is defined by averaging Ky over all points y in D l
Tω =
ϕ(y)Ky ωdy ,
n
l
n
(1.6)
D
where ϕ ∈ C0∞ (D) is normalized by
1 0
ϕ(y)dy = 1. For simplicity purpose, we write ξ = (ξ1 , . . . , ξl−1 ). Then, T ω(x; ξ ) = t ϕ(y)ω(tx + y − ty; x − y, ξ )dydt. By substituting z = tx + y − ty and t = s/(1 + s), we have D T ω(x; ξ ) = ω(z , ζ (z , x − z ), ξ )dz , (1.7) l−1
D
D
S. Ding, Y. Xing / Nonlinear Analysis 96 (2014) 87–95
89
∞
where the vector function ζ : D × Rn → Rn is given by ζ (z , h) = h 0 sl−1 (1 + s)n−1 ϕ(z − sh)ds. The integral (1.7) defines a bounded operator T : Ls (D, Λl ) → W 1,s (D, Λl−1 ), l = 1, 2, . . . , n, and the decomposition u = d(Tu) + T (du)
(1.8)
holds for any differential form u. The l-form ωD ∈ D′ (D, Λl ) is defined by
ωD = − ω(y)dy = |D|−1 ω(y)dy, D
l = 0,
and ωD = d(T ω),
l = 1, 2, . . . , n,
(1.9)
D
for all ω ∈ Lp (D, Λl ), 1 ≤ p < ∞. Also, for any differential form u, we have
∥∇(Tu)∥p,D ≤ C |D|∥u∥p,D ,
and
∥Tu∥p,D ≤ C |D|diam(D)∥u∥p,D .
(1.10)
From [14, Page 16], we know that any open subset Ω in R is the union of a sequence of cubes Qk , whose sides are parallel to the axes, whose interiors are mutually disjoint, and whose diameters are approximately proportional to their distances 0 0 from F . Specifically, (i) Ω = ∪∞ k=1 Qk , (ii) Qj ∩ Qk = φ if j ̸= k, (iii) there exist two constants c1 , c2 > 0 (we can take c1 = 1, and c2 = 4), so that c1 diam(Qk ) ≤ distance Qk from F ≤ c2 diam(Qk ). Thus, the definition of the homotopy operator T can be generalized to any domain Ω in Rn : For any x ∈ Ω , x ∈ Qk for some k. Let TQk be the homotopy operator defined on ∞Qk (each cube is bounded and convex). Thus, we can define the homotopy operator TΩ on any domain Ω by TΩ = k=1 TQk χQk (x) . n
2. Local imbedding inequalities The Sobolev imbedding inequalities have been serving as an effective tool in the study of the Sobolev spaces and PDEs. Different versions of the classical Sobolev imbedding inequality for functions (0-forms) have been very well developed and widely used analysis and PDEs in last several decades. In recent years, some versions of the Sobolev imbedding inequality with Lp norms for differential forms or operators applied to differential forms have also been established. See [1, Chapter 5] for recent progress in the study of the Sobolev imbedding inequalities with Lp norms. In this section, we first prove the local Lϕ imbedding inequalities for differential forms satisfying the non-homogeneous A-harmonic equation in a bounded domain. A continuously increasing function ϕ : [0, ∞) → [0, ∞) with ϕ(0 ) = 0, is called an Orlicz function. The Orlicz space Lϕ (Ω ) consists of all measurable functions f on Ω such that
Ω
ϕ
|f | λ
dx < ∞ for some λ = λ(f ) > 0. Lϕ (Ω ) is
equipped with the nonlinear Luxemburg functional
∥f ∥Lϕ (Ω ) = inf
|f | λ>0: ϕ dx ≤ 1 . λ Ω
A convex Orlicz function ϕ is often called a Young function. If ϕ is a Young function, then ∥ · ∥Lϕ (Ω ) defines a norm in Lϕ (Ω ), which is called the Luxemburg norm or Orlicz norm. The space Lϕ (Ω ) is a Banach space when ϕ is a Young function. We ϕ use Wd (Ω , ∧l ) to express the space of l-forms u ∈ Lϕ (Ω , ∧l ) such that du ∈ Lϕ (Ω , ∧l+1 ) and W 1,ϕ (Ω , ∧l ) to denote the ϕ Orlicz–Sobolev space of l-forms which equals Lϕ (Ω , ∧l ) ∩ L1 (Ω , ∧l ) with norm
∥u∥W 1,ϕ (Ω ) = ∥u∥W 1,ϕ (Ω ,∧l ) = diam(Ω )−1 ∥u∥Lϕ (Ω ) + ∥∇ u∥Lϕ (Ω ) .
(2.1)
We should notice that the Orlicz–Sobolev space W 1,ϕ (Ω , ∧l ) of differential forms is an extension of the well known Sobolev space W 1,p (Ω ) of functions. Specifically, if we choose ϕ(t ) = t p , p > 1, the Orlicz–Sobolev space W 1,ϕ (Ω , ∧l ) reduces to the Sobolev space W 1,p (Ω , ∧l ) of differential forms. Also W 1,ϕ (Ω , ∧l ) ⊂ Lϕ (Ω , ∧l ) and W 1,ϕ (Ω , ∧l ) is a Banach space when ϕ is a Young function. Furthermore, we have W 1,p (Ω , ∧0 ) = W 1,p (Ω ). Note that functions are 0-forms. Hence, all theorems proved in this paper for differential forms still hold for functions. Definition 2.1 ([15]). We say a Young function ϕ lies in the class G(p, q, C ), 1 ≤ p < q < ∞, C ≥ 1, if (i) 1/C ≤ ϕ(t 1/p )/g (t ) ≤ C and (ii) 1/C ≤ ϕ(t 1/q )/h(t ) ≤ C for all t > 0, where g is a convex increasing function and h is a concave increasing function on [0, ∞). From [15], each of ϕ, g and h in above definition is doubling in the sense that its values at t and 2t are uniformly comparable for all t > 0, and the consequent fact that C1 t q ≤ h−1 (ϕ(t )) ≤ C2 t q ,
C1 t p ≤ g −1 (ϕ(t )) ≤ C2 t p ,
(2.2)
where C1 and C2 are constants. Also, for all 1 ≤ p1 < p < p2 and α ∈ R, the function ϕ(t ) = t logα+ t belongs to G(p1 , p2 , C ) for some constant C = C (p, α, p1 , p2 ). Here log+ (t ) is defined by log+ (t ) = 1 for t ≤ e; and log+ (t ) = log(t ) for t > e. p
Particularly, if α = 0, we see that ϕ(t ) = t p lies in G(p1 , p2 , C ), 1 ≤ p1 < p < p2 . We will need the following Reverse Hölder inequality.
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Lemma 2.2 ([1]). Let u be a solution of the non-homogeneous A-harmonic equation (1.1) in a domain Ω and 0 < s, t < ∞. Then, there exists a constant C , independent of u, such that
∥du∥s,B ≤ C |B|(t −s)/st ∥du∥t ,σ B
(2.3)
for all balls B with σ B ⊂ Ω for some σ > 1. Theorem 2.3. Let ϕ be a Young function in the class G(p, q, C ), 1 ≤ p < q < ∞, C ≥ 1, Ω be a bounded and convex domain, and T : C ∞ (Ω , ∧l ) → C ∞ (Ω , ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator defined in (1.6). Assume that ϕ(|du|) ∈ L1loc (Ω ) and u is a solution of the non-homogeneous A-harmonic equation (1.1) in Ω . Then, there exists a constant C , independent of u, such that
∥T (du)∥Lϕ (B) ≤ C ∥du∥Lϕ (σ B)
(2.4)
for all balls B with σ B ⊂ Ω . Proof. Applying (1.10), we have
∥T (du)∥q,B ≤ C1 |B|1+1/n ∥du∥q,B
(2.5)
for all balls B with B ⊂ Ω . From Lemma 2.2, for any positive numbers p and q, it follows that
1/q ≤ C2 |B|(p−q)/pq |du|q dx
B
σB
1/p , |du|p dx
(2.6)
where σ is a constant σ > 1. Using Jensen’s inequality for h−1 , (2.2), (2.5), (2.6), (i) in Definition 2.1, and noticing the fact that ϕ and h are doubling, and ϕ is an increasing function, we obtain
ϕ(|T (du)|)dx = h h−1 ϕ (|T (du)|) dx B B −1 h ≤h ϕ (|T (du)|) dx B ≤ h C3 |T (du)|q dx B 1/q ≤ C4 ϕ C3 |T (du)|q dx B 1/q 1+1/n ≤ C4 ϕ C5 |B| |du|q dx B 1/p 1+1/n+(p−q)/pq ≤ C4 ϕ C6 |B| |du|p dx σB 1/p p p(1+1/n)+(p−q)/q |du|p dx ≤ C4 ϕ C6 |B| σB p p(1+1/n)+(p−q)/q ≤ C7 g C6 |B| |du|p dx σB p p(1+1/n)+(p−q)/q = C7 g C6 |B| |du|p dx σ B p ≤ C7 g C6 |B|p(1+1/n)+(p−q)/q |du|p dx σB
≤ C8 Since p ≥ 1, then 1 +
1 n
+
σB
p−q pq
p−q 1 ϕ C6 |B|1+ n + pq |du| dx.
(2.7) 1
> 0. Hence, we have |B|1+ n +
p−q 1 ϕ C6 |B|1+ n + pq |du| ≤ C9 ϕ |du| .
p−q pq
1
≤ |Ω |1+ n +
p−q pq
≤ C5 . Note that ϕ is doubling, we obtain (2.8)
Combining (2.7) and (2.8) yields
ϕ (|T (du)|) dx ≤ C10 B
σB
ϕ (|du|) dx.
(2.9)
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91
Since each of ϕ, g and h in Definition 2.1 is doubling, from (2.9), we have
ϕ
B
|T (du)| |du| dx ≤ C ϕ dx λ λ σB
(2.10)
for all balls B with σ B ⊂ Ω and any constant λ > 0. From (2.1) and (2.10), we have the following inequality with the Luxemburg norm
∥T (du)∥Lϕ (B) ≤ C ∥du∥Lϕ (σ B) .
(2.11)
The proof of Theorem 2.3 has been completed.
Theorem 2.4. Let ϕ be a Young function in the class G(p, q, C ), 1 ≤ p < q < ∞, C ≥ 1, q(n − p) < np, Ω be a bounded domain and T : C ∞ (Ω , ∧l ) → C ∞ (Ω , ∧l−1 ), l = 1, 2, . . . , n, be the homotopy operator defined in (1.6). Assume that u ∈ D′ (Ω , ∧l ) is any differential l-form, ϕ(|u|) ∈ L1loc (Ω ). Then, there exists a constant C , independent of u, such that
∥T (du)∥Lϕ (B) ≤ C ∥du∥Lϕ (B)
(2.12)
for all balls B with B ⊂ Ω . Proof. Note that T (du) = u − uB holds for any differential form u. Using (2.7), we have
ϕ(|T (du)|)dx ≤ C1 ϕ
= C1 ϕ
1/q |T (du)|q dx B
B
1/q |u − uB |q dx .
(2.13)
B
If 1 < p < n, by assumption, we have q < n−p . Using the Poincaré-type inequality for differential forms u np
|u − uB |
np/(n−p)
(n−p)/np
p
≤ C2
dx
1/p
|du| dx
B
,
(2.14)
B
we find that
1/q 1/p |u − uB |q dx ≤ C3 |du|p dx .
B
(2.15)
B
Combining (2.13) and (2.15), we obtain
ϕ(|T (du)|)dx ≤ C1 ϕ C3 B
p
1/p
|du| dx
(2.16)
B
for 1 < p < n. Note that the Lp -norm of |u − uB | increases with p and n−p → ∞ as p → n, it follows that (2.15) still holds when p ≥ n. Since ϕ is increasing, from (2.13) and (2.15), we obtain np
1/p ϕ(|T (du)|)dx ≤ C1 ϕ C3 |u|p dx .
B
(2.17)
B
Applying (2.17), (i) in Definition 2.1, Jensen’s inequality, and noticing that ϕ and g are doubling, we have
1/p ϕ(|T (du)|)dx ≤ C1 ϕ C3 |du|p dx B B ≤ C1 g C4 |du|p dx B
g (|du|p )dx.
≤ C5
(2.18)
B
Using (i) in Definition 2.1 again yields
g (|du|p )dx ≤ C6 B
ϕ(|du|)dx.
(2.19)
B
Combining (2.18) and (2.19), we obtain
ϕ |T (du)| dx ≤ C7 ϕ(|du|)dx. B
B
(2.20)
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From (2.1) and (2.20), we have the following inequality with the Luxemburg norm
∥T (du)∥Lϕ (B) ≤ C ∥du∥Lϕ (B) . The proof of Theorem 2.4 has been completed.
Applying the first inequality in (1.10) to du, we have
∥∇(T (du))∥q,B ≤ C |B|∥du∥q,B (2.21) for all balls B and q > 1. Starting with (2.21) and using the similar method to the proof of Theorem 2.3, we know that the following Lϕ -norm inequality
∥∇(T (du))∥Lϕ (B) ≤ C ∥du∥Lϕ (σ B)
(2.22)
holds if all conditions of Theorem 2.3 are satisfied. Now we are ready to prove the following main theorem, the local Lϕ -norm imbedding inequality in Orlicz–Sobolev space of differential forms. Theorem 2.5. Let ϕ be a Young function in the class G(p, q, C ), 1 ≤ p < q < ∞, C ≥ 1, Ω be a bounded and convex domain. Assume that ϕ(|du|) ∈ L1loc (Ω , m) and u is a solution of the non-homogeneous A-harmonic equation (1.1) in Ω . Then, there exists a constant C , independent of u, such that
∥u − uB ∥W 1,ϕ (B,∧l ) ≤ C ∥du∥Lϕ (σ B)
(2.23)
for all balls B with σ B ⊂ Ω . Proof. Using the definition of the norm in Orlicz–Sobolev space of differential forms, Theorems 2.3 and 2.4, and noticing u − uB = T (du) for any differential form u, we have
∥u − uB ∥W 1,ϕ (B,∧l ) = ∥T (du)∥W 1,ϕ (B,∧l ) = (diam(B))−1 ∥T (du)∥Lϕ (B) + ∥∇(T (du))∥Lϕ (B) ≤ C1 ∥du∥Lϕ (σ1 B) + C2 ∥du∥Lϕ (σ2 B) ≤ C3 ∥du∥Lϕ (σ B)
(2.24)
for any differential form u and all balls B with σ B ⊂ Ω , where σ1 , σ2 > 1 are some constants and σ = max{σ1 , σ2 }. The proof of Theorem 2.5 has been completed. Using (2.1) and (2.23), we have
∥u − uB ∥Lϕ (B) = diam(B) (diam(B))−1 ∥u − uB ∥Lϕ (B) ≤ diam(B) (diam(B))−1 ∥u − uB ∥Lϕ (B) + ∥∇(u − uB )∥Lϕ (B) = diam(B)∥u − uB ∥W 1,ϕ (B,∧l ) ≤ Cdiam(B)∥du∥Lϕ (σ B) ,
(2.25)
that is,
∥u − uB ∥Lϕ (B) ≤ Cdiam(B)∥du∥Lϕ (σ B)
(2.26)
where σ > 1 is a constant. Remark. (i) From above discussion, we find that the local Orlicz–Sobolev imbedding inequality with Lϕ -norm proved in Theorem 2.5 implies directly the local Poincaré inequality (2.26) with Lϕ -norm. (ii) We should also note that Theorems 2.3–2.5 include some existing results as their special cases. For example, selecting ϕ(t ) = t p in Theorem 2.5, we obtain the usual imbedding inequalities with the Lp -norms
∥u − uB ∥W 1,p (B,∧l ) ≤ C ∥du∥p,σ B
(2.27)
(logα+
for all balls B with σ B ⊂ Ω , where σ > 1 is a constant. (iii) The L L)-imbedding inequality is also a special case of Theorem 2.5. Specifically, choosing ϕ(t ) = t p logα+ t in Theorem 2.5, we obtain the following imbedding inequalities with the Lp (logα+ L)-norms. p
Theorem 2.6. Let ϕ(t ) = t p logα+ t, p ≥ 1 and α ∈ R. Assume that ϕ(|du|) ∈ L1loc (Ω , m) and u is a solution of the nonhomogeneous A-harmonic equation (1.1). Then, there exists a constant C , independent of u, such that
∥u − uB ∥W 1,t p logα+ t (B,∧l ) ≤ C ∥du∥Lp (logα+ L)(σ B) .
(2.28)
for all balls B with σ B ⊂ Ω , where σ > 1 is a constant. Similarly, if we choose ϕ(t ) = t p logα+ t or ϕ(t ) = t p in Theorems 2.3 and 2.4, respectively, we will obtain the corresponding special results.
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3. Global imbedding theorem In this section, we extend the local imbedding inequalities into the global Orlicz–Sobolev imbedding inequality with Lϕ -norms in the following Lϕ -averaging domains. Definition 3.1 ([16]). Let ϕ be an increasing convex function on [0, ∞) with ϕ(0) = 0. We call a proper subdomain Ω ⊂ Rn an Lϕ -averaging domain, if |Ω | < ∞ and there exists a constant C such that
Ω
ϕ(τ |u − uB0 |)dx ≤ C sup
B ⊂Ω
ϕ(σ |u − uB |)dx
(3.1)
B
for some ball B0 ⊂ Ω and all u such that ϕ(|u|) ∈ L1loc (Ω ), where τ , σ are constants with 0 < τ < ∞, 0 < σ < ∞ and the supremum is over all balls B ⊂ Ω . From above definition and [1] we see that Ls -averaging domains are special Lϕ -averaging domains when ϕ(t ) = t s in Definition 3.1. Also, uniform domains and John domains are very special Lϕ -averaging domains, see [16,17] for more results about the averaging domains. Now we prove the global Orlicz–Sobolev imbedding inequality with Lϕ -norms in the following theorem. Theorem 3.2. Let ϕ be a Young function in the class G(p, q, C ), 1 ≤ p < q < ∞, C ≥ 1, and Ω be any bounded Lϕ -averaging domain. Assume that ϕ(|du|) ∈ L1 (Ω ) and u ∈ D′ (Ω , ∧0 ) is a solution of the non-homogeneous A-harmonic equation (1.1) in Ω . Then, there exists a constant C , independent of u, such that
∥u − uB0 ∥W 1,ϕ (Ω ) ≤ C ∥du∥Lϕ (Ω )
(3.2)
where B0 ⊂ Ω is some fixed ball. Proof. Using the Poincaré inequality ∥u−uB ∥q,B ≤ C ∥du∥q,B , q > 1, for differential forms (see [1]) and the method developed in the proof of Theorem 2.3, we have the following Poincaré inequality with Lϕ -norm
ϕ (|u − uB |) dx ≤ C1 diam(B)
ϕ (|du|) dx.
(3.3)
B
B
From Definition 3.1 and (3.3), we obtain
∥u − uB0 ∥Lϕ (Ω ) ≤ C2
Ω
ϕ |u − uB0 | dx
ϕ (|u − uB |) dx
≤ C3 sup B ⊂Ω
B
≤ C3 sup C1 diam(B) ϕ (|du|) dx B ⊂Ω
B
≤ C3 sup C1 diam(Ω ) B ⊂Ω
≤ C4 diam(Ω )
Ω
Ω
ϕ (|du|) dx
ϕ (|du|) dx
≤ C5 diam(Ω )∥du∥Lϕ (Ω ) .
(3.4)
From (2.1), (3.4), and noticing that ϕ is doubling and |∇ u| = |du| for any u ∈ ∧0 , it follows that
∥u − uB0 ∥W 1,ϕ (Ω ,∧0 ) = (diam(Ω ))−1 ∥u − uB0 ∥Lϕ (Ω ) + ∥∇ u∥Lϕ (Ω ) ≤ (diam(Ω ))−1 (C5 diam(Ω )∥du∥Lϕ (Ω ) ) + ∥du∥Lϕ (Ω ) ≤ C6 ∥du∥Lϕ (Ω ) We have completed the proof of Theorem 3.2.
(3.5)
Remark. (i) Since the class of Lϕ -averaging domains is a very large class of domains [1], Theorem 3.2, the global Lϕ imbedding theorem, is a far extension of the existing imbedding theorems. For example, we know that the Ls -averaging domains and uniform domains are special cases of the Lϕ -averaging domains. Then, Theorem 3.2 still holds if Ω is an Ls -averaging domain or a uniform domain. (ii) For the case ϕ(t ) = t p , p > 1, inequality (3.2) reduces to the following Lp imbedding inequality
∥u − uB0 ∥W 1,p (Ω ) ≤ C ∥du∥p,Ω .
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(iii) Similar to the local case, the global Orlicz–Sobolev Lϕ imbedding inequality (3.2) implies directly the global Poincaré inequality with Lϕ -norm
∥u − uB0 ∥Lϕ (Ω ) ≤ Cdiam(Ω )∥du∥Lϕ (Ω ) ,
(3.6)
where C is a constant. It has been proved that any John domain is a special Lϕ -averaging domain [1]. Hence, we have the following imbedding inequality for John domains. Corollary 3.3. Let ϕ be a Young function in the class G(p, q, C ), 1 ≤ p < q < ∞, C ≥ 1, and Ω be a bounded John domain. Assume that ϕ(|du|) ∈ L1 (Ω ) and u ∈ D′ (Ω , ∧0 ) is a solution of the non-homogeneous A-harmonic equation (1.1) in Ω . Then, there exists a constant C , independent of u, such that
∥u − uB0 ∥W 1,ϕ (Ω ) ≤ C ∥du∥Lϕ (Ω )
(3.7)
where B0 ⊂ Ω is some fixed ball. Choosing ϕ(t ) = t p logα+ t in Theorem 3.2, we obtain the following imbedding inequality with the Lp (logα+ L)-norms. Corollary 3.4. Let ϕ(t ) = t p logα+ t, p ≥ 1, α ∈ R. Assume that ϕ(|du|) ∈ L1 (Ω ) and u ∈ D′ (Ω , ∧0 ) is a solution of the non-homogeneous A-harmonic equation (1.1). Then, there exists a constant C , independent of u, such that
∥u − uB0 ∥W 1,ϕ (Ω ) ≤ C ∥du∥Lϕ (Ω )
(3.8)
for any bounded Lϕ -averaging domain Ω and B0 ⊂ Ω is some fixed ball. 4. Applications As applications of the main theorems proved in last two sections, we will study two examples in this section. If u is a function (0-form) in the homogeneous A-harmonic equation (1.5), then (1.5) reduces to the following A-harmonic equation divA(x, ∇ u) = 0
(4.1)
for functions. If the above operator A : Ω × ∧l (Rn ) → ∧l (Rn ) is defined by A(x, ξ ) = ξ |ξ |p−2 with p > 1. Then, A satisfies the required conditions and (4.1) becomes the usual p-harmonic equation for functions div(∇ u|∇ u|p−2 ) = 0
(4.2)
which is equivalent to
(p − 2)
n n
uxk uxi uxk xi + |∇ u|2 ∆u = 0.
(4.3)
k=1 i=1
If we choose p = 2 in (4.2), we have the Laplace equation ∆u = 0 for functions. Thus, from Theorem 3.2, we have the following example. Example 4.1. Let u be a solution of the usual A-harmonic equation (4.1) or the p-harmonic equation (4.2), ϕ be a Young function in the class G(p, q, C ), 1 ≤ p < q < ∞, C ≥ 1, and Ω be any bounded Lϕ -averaging domain. If ϕ(|du|) ∈ L1 (Ω ), then there exists a constant C , independent of u, such that
∥u − uB0 ∥W 1,ϕ (Ω ) ≤ C ∥du∥Lϕ (Ω ) where B0 ⊂ Ω is some fixed ball. Example 4.2. In some areas of science and engineering, such as astronomy, fluid dynamics, and electromagnetism, we need 1,p to study and use various mappings. Let f : Ω → Rn , f = (f 1 , . . . , f n ), be a mapping of the Sobolev class Wloc (Ω , Rn ), i 1 ≤ p < ∞, whose distributional differential Df = [∂ f /∂ xj ] : Ω → GL(n) is a locally integrable function on Ω with values in the space GL(n) of all n × n-matrices, i, j = 1, 2, . . . , n. A homeomorphism f : Ω → Rn of Sobolev class 1,n Wloc (Ω , Rn ) is said to be K -quasiconformal, 1 ≤ K < ∞, if its differential matrix Df (x) and the Jacobian determinant J = J (x, f ) = detDf (x) satisfy
|Df (x)|n ≤ KJ (x, f ),
(4.4)
where |Df (x)| = max{|Df (x)h| : |h| = 1} denotes the norm of the Jacobi matrix Df (x). It is well known that if the differential matrix Df (x) = [∂ f i /∂ xj ], i, j = 1, 2, . . . , n, of a homeomorphism f (x) = (f 1 , f 2 , . . . , f n ) : Ω → Rn satisfies (4.4), then, each of the functions u = f i (x),
i = 1, 2, . . . , n,
or
u = log |f (x)|,
(4.5)
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is a generalized solution of the quasilinear elliptic equation divA(x, ∇ u) = 0, in Ω − f
−1
(4.6)
(0), where
A = (A1 , A2 , . . . , An ),
∂ A i ( x, ξ ) = ∂ξi
n
n/2 θi,j (x)ξi ξj
i,j=1
and θi,j are some functions, which can be expressed in terms of the differential matrix Df (x) and satisfy C1 (K )|ξ |2 ≤
n
θi,j ξi ξj ≤ C2 (K )|ξ |2
(4.7)
i ,j
for some constants C1 (K ), C2 (K ) > 0. All results proved in Sections 2 and 3 are still true if u is defined in (4.5). Remark. Note that the method developed in this paper can be used to prove both local and global imbedding theorems for operators applied to differential forms, such as the potential operator P in [18], Green’s operator G and the homotopy operator T in [1]. The global imbedding theorems for these operators will also hold in Lϕ -averaging domains. Acknowledgments We would like to thank the referees and the handling editor for their time and effort spent on this paper. We also thank the referees for their precious and thoughtful suggestions which largely improved the presentation of the paper. References [1] R.P. Agarwal, S. Ding, C.A. Nolder, Inequalities for Differential Forms, Springer, 2009. [2] Y. Wang, C. Wu, Sobolev imbedding theorems and Poincaré inequalities for Green’s operator on solutions of the nonhomogeneous A-harmonic equation, Comput. Math. Appl. 47 (2004) 1545–1554. [3] S. Ding, Lipschitz and BMO norm inequalities for operators, Nonlinear Anal. TMA (2009). [4] G. Bao, Ar (λ)-weighted integral inequalities for A-harmonic tensors, J. Math. Anal. Appl. 247 (2000) 466–477. [5] Y. Wang, Two-weight Poincaré type inequalities for differential forms in Ls (µ)-averaging domains, Appl. Math. Lett. 20 (2007) 1161–1166. [6] Y. Xing, Weighted Poincaré-type estimates for conjugate A-harmonic tensors, J. Inequal. Appl. 1 (2005) 1–6. [7] B. Liu, Aλr (Ω )-weighted imbedding inequalities for A-harmonic tensors, J. Math. Anal. Appl. 273 (2) (2002) 667–676. [8] C.A. Nolder, Hardy–Littlewood theorems for A-harmonic tensors, Illinois J. Math. 43 (1999) 613–631. [9] S. Ding, Two-weight Caccioppoli inequalities for solutions of nonhomogeneous A-harmonic equations on Riemannian manifolds, Proc. Amer. Math. Soc. 132 (2004) 2367–2375. [10] H. Cartan, Differential Forms, Houghton Mifflin Co., Boston, 1970. [11] S.K. Sachs, H. Wu, General Relativity for Mathematicians, Springer, New York, 1977. [12] C. Westenholz, Differential Forms in Mathematical Physics, North Holland Publishing, Amesterdam, 1978. [13] T. Iwaniec, A. Lutoborski, Integral estimates for null Lagrangians, Arch. Ration. Mech. Anal. 125 (1993) 25–79. [14] E.M. Stein, Harmonic Analysis, Princeton University Press, Princeton, 1993. [15] S.M. Buckley, P. Koskela, Orlicz–Hardy inequalities, Illinois J. Math. 48 (2004) 787–802. [16] S. Ding, Lϕ (µ)-averaging domains and the quasihyperbolic metric, Comput. Math. Appl. 47 (2004) 1611–1618. [17] S.G. Staples, Averaging domains: from Euclidean spaces to homogeneous spaces, in: Proc. of Conference on Differential & Difference Equations and Applications, Hindawi Publishing, 2006, pp. 1041–1048. [18] H. Bi, Weighted inequalities for potential operators on differential forms, J. Inequal. Appl. (2010) Article ID 713625.