# Variable Exponent Spaces of Differential Forms on Riemannian Manifold

Authors: Yongqiang Fu; Lifeng Guo
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We introduce the Lebesgue space and the exterior Sobolev space for differential forms on Riemannian manifoldM which are the Lebesgue space and the Sobolev space of functions on M, respectively, when the degree of differential forms to be zero. After discussing the properties of these spaces, we obtain the existence and uniqueness of weak solution for Dirichlet problems of nonhomogeneous p(m)-harmonic equations with variable growth in W01,p(m)(ΛkM).


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## Body

## 1. Introduction

Gol'dshteĭn et al. introduced spaces of differential forms on Riemannian manifold in [1–3]. The study of spaces for differential forms has been developed rapidly. For example, Lp-Cohomology and Lp,q-Cohomology and applications to some nonlinear PDE were studied in [4–6]; Lp Hodge decomposition theory on the compact and complete Riemannian manifold were discussed in [7, 8]; properties of Riesz transforms of differential forms on complete Riemannian manifold were discussed in [9, 10]; the existence of minima of certain mean-coercive functionals is established in [11]. Many interesting results concerning A-harmonic equations have been established recently (see [12, 13] and the references therein).After Kováčik and Rákosník first discussed theLp(x)(Ω) and W1,p(x)(Ω) spaces in [14], a lot of research has been done concerning these kinds of variable exponent spaces (see [15–19] and the references therein). The existence and uniqueness of solutions for p(x)-Laplacian Dirichlet problems with different types on bounded domains in ℝn have been greatly discussed under various conditions (see [20] for the existence and [21] for the uniqueness). In recent years, the theory on problems with variable exponential growth conditions has important applications in nonlinear elastic mechanics (see [22]), electrorheological fluids (see [23, 24]).The paper is organized as follows. In Section2, we give the necessary definitions and some elementary properties of differential forms on Riemannian manifold. Moreover, we introduce the functional ρp(m),ΛkM on ΛkM and the spaces of differential forms Lp(m)(ΛkM) and W1,p(m)(ΛkM), then discuss some important properties. In Section 3, we show the existence and uniqueness of weak solution for Dirichlet problems of nonhomogeneous p(m)-harmonic equations with variable growth in W01,p(m)(ΛkM).


## 2. Preliminaries

LetM be an arbitrary smooth n-dimensional manifold (Hausdorff and with countable basis). Let T*M=∪m∈MTm*M be the cotangent bundle on M and ΛkT*M (or ΛkM) be the bundles of the exterior k-forms. We will call each fiber u of the bundle ΛkT*M a exterior form of degree k on the manifold M. Here, Λ0M=ℝ and ΛkM={0} in the case k>n or k<0. Given a exterior k-form u(m) and a local chart fα:Uα(⊂M)→ℝn, around m∈Uα, we define the representation of u(m) in this local coordinates system as the exterior k-forms uα on fα(Uα)⊂ℝn given by
(2.1)uα(fα(m))(X1,X2,…,Xk)=((fα-1)*u)(fα(m))(X1,X2,…,Xk)=u(m)(dfα-1(X1),dfα-1(X2),…,dfα-1(Xk)),
for any X1,X2,…,Xk∈ℝn, where dfα-1 is the induced map by fα-1 that takes vectors on Tfα(m)ℝn into vectors on TmM and (fα-1)* is the induced map by fα-1 that takes exterior forms on TmM into exterior forms on Tfα(m)ℝn (see [25]).In this paper we will always assume(M,g) is an n-dimensional smooth orientable complete Riemannian manifold and dμ=det(gij)dx is the Riemannian volume element on (M,g), where the gij are the components of the Riemannian metric g in the chart and dx is the Lebesgue volume element of ℝn. A Riemannian metric g on M induces a scalar product on each fiber of the bundle ΛkM. Hence for any exterior forms u and v of the same degree k, the scalar product 〈u,v〉=〈u(m),v(m)〉 is defined at each point m∈M and the norm of u is given by the formula |u|=〈u,u〉. Let γ:[a,b]→M be a curve of class C1, the length of γ is
(2.2)L(γ)=∫ab‍g(γ(t))((dγdt)(t),(dγdt)(t))dμ.
For m1,m2∈M, let Cm1,m21 be the space of piecewise C1 curves γ:[a,b]→M such that γ(a)=m1 and γ(b)=m2. One can define a distance dg(m1,m2)=infCm1,m21L(γ) on M.The Grassman algebraΛ*M=⊕ΛkM is a graded algebra with respect to the exterior products. We denote by Lloc1(ΛkM) the space of locally integrable exterior forms of degree k (i.e., differential k-forms) on M. The local integrability of an exterior k-form means the local integrability of the components of its coordinate representation in each chart of the Riemannian manifold M. We denote by Cc∞(ΛkM) the vector space of smooth differential forms of degree k with compact support on M.Let(M,g) be is an n-dimensional smooth orientable Riemannian manifold. We define the integral of u, a exterior n-form u with compact support on M (see [26]). Let (Uα,fα) be a local chart of (M,g), we have a partition of unity {πα} subordinate to this cover. Recall that supp(πα)⊆Uα and ∑α‍πα=1. Thus, every παu is an exterior n-form whose support is a subset of Uα and we may write u=∑α‍παu. By definition
(2.3)∫M‍u=∑α‍∫Uα‍παu=∑α‍∫fα(Uα)(fα-1)*(παu)=∑α‍∫fα(Uα)‍(det⁡⁡(gij)παu)∘fα-1dx.We will identify each exterior form of degreek on the n-dimensional Riemannian manifold M with an exterior (n-k)-form on M (see [27]). Using this identification, we can assume that each exterior form u has a weak exterior differential du.Definition 2.1 (see [6]).
We say that an exterior formv∈Lloc1(ΛkM) is the weak exterior differential of a form u∈Lloc1(Λk-1M) and we write du=v if for each φ∈Cc∞(ΛkM), one has
(2.4)∫M‍v∧φ=(-1)k∫M‍u∧dφ.The operator⋆:ΛkM→Λn-kM, also called Hodge star operator (see [27]), has the following properties: for u,v∈ΛkM and φ,ψ∈C∞(M)(a1)
⋆(φu+ψv)=φ⋆u+ψ⋆v,(a2)
⋆⋆u=(-1)k(n-k)u,(a3)
⋆φ=φdμ,(a4)
〈u,v〉=⋆(u∧⋆v)=〈⋆u,⋆v〉,(a5)
u∧⋆v=〈u,v〉dμ.By the operator⋆ and the exterior differentiation d we define the codifferential operator δ by the formula
(2.5)δu=(-1)n(k+1)+1⋆d⋆u∈Lloc⁡1(Λk-1M),
for any differential form u∈Lloc1(ΛkM).The Riemannian measure and the characteristic function of a setA⊆M will be denoted by μ(A) and χA, respectively.Let𝒫(M) be the set of all measurable functions p:M→[1,∞]. For p∈𝒫(M) we put M1=M1p={m∈M:p(m)=1},M∞=M∞p={m∈M:p(m)=∞}, M0=M∖(M1∪M∞), p*=essinfM0p(m) and p*=esssupM0p(m) if μ(M0)>0, p*=p*=1 if μ(M0)=0,cp=∥χM0∥L∞(M)+∥χM1∥L∞(M)+∥χM∞∥L∞(M) and rp=cp+1/p*+1/p*. We always assume that p∈𝒫(M), 𝒫1(M)=𝒫(M)∩L∞(M) and 𝒫2(M)={p∈𝒫1(M):1<essinfMp(m)}. We use the convention 1/∞=0.For a differentialk-form u on M we define the functional ρp(m),ΛkM by
(2.6)ρp(m),ΛkM(u)=∫M∖M∞‍|u|p(m)dμ+esssup⁡⁡M∞|u|.The Lebesgue spaceLp(m)(ΛkM) is the space of differential forms u in Lloc1(ΛkM) such that
(2.7)ρp(m),ΛkM(λu)<∞forsomeλ=λ(u)>0,
with the following norm
(2.8)‖u‖Lp(m)(ΛkM)=inf⁡⁡{λ>0:ρp(m),ΛkM(uλ)≤1}.The exterior Sobolev spaceW1,p(m)(ΛkM) consists of such forms u∈Lp(m)(ΛkM) for which du∈Lp(m)(Λk+1M). The norm is defined by
(2.9)‖u‖W1,p(m)(ΛkM)=‖u‖Lp(m)(ΛkM)+‖du‖Lp(m)(Λk+1M).The spaceW01,p(m)(ΛkM) is defined as the closure of Cc∞(ΛkM) in W1,p(m)(ΛkM).Note thatLp(m)(Λ0M), W1,p(m)(Λ0M) and W01,p(m)(Λ0M) are spaces of functions on M. In this paper we denote them by Lp(m)(M), W1,p(m)(M) and W01,p(m)(M).Givenp∈𝒫(M) we define the conjugate function p′(m)∈𝒫(M) by
(2.10)p′(m)={∞ifm∈M1,1ifm∈M∞,p(m)p(m)-1ifm∈M0.Similar to the proof of properties ofρp(m),Ω and Lp(m)(Ω) for Ω⊂ℝn (see [15, 16, 18]), it is easy to see that ρp(m),ΛkM and Lp(m)(ΛkM) has the following properties:(b1)
ρp(m),ΛkM is convex.(b2)
ρp(m),ΛkM(uχA)≤ρp(m),ΛkM(u) for every subset A⊂M and differential forms u.(b3)
If|u(m)|≥|v(m)| for a.e. m∈M and if ρp(m),ΛkM(u)<∞, then ρp(m),ΛkM(u)≥ρp(m),ΛkM(v), the last inequality is strict if |u|≠|v|.(b4)
If0<ρp(m),ΛkM(u)<∞, then the function λ→ρp(m),ΛkM(u/λ) is continuous and decreasing on the interval [1,∞).(b5)
If0<∥u∥Lp(m)(ΛkM)<∞, then ρp(m),ΛkM(u/∥u∥Lp(m)(ΛkM))≤1.(b6)
Ifp*<∞, then ρp(m),ΛkM(u/∥u∥Lp(m)(ΛkM))=1 for every differential forms u with 0<∥u∥Lp(m)(ΛkM)<∞.(b7)
If∥u∥Lp(m)(ΛkM)≤1, then ρp(m),ΛkM(u)≤∥u∥Lp(m)(ΛkM).(b8)
Ifp∈𝒫1(M) and ∥u∥Lp(m)(ΛkM)>1, then
(2.11)‖u‖Lp(m)(ΛkM)p*≤ρp(m),ΛkM(u)≤‖u‖Lp(m)(ΛkM)p*.(b9)
Ifp∈𝒫1(M) and ∥u∥Lp(m)(ΛkM)<1, then
(2.12)‖u‖Lp(m)(ΛkM)p*≥ρp(m),ΛkM(u)≥‖u‖Lp(m)(ΛkM)p*.Lemma 2.2.
Ifp(m)∈𝒫(M), then the inequality
(2.13)∫M‍|〈u,v〉|dμ≤rp‖u‖Lp(m)(ΛkM)‖v‖Lp′(m)(ΛkM)
holds for every u∈Lp(m)(ΛkM), v∈Lp′(m)(ΛkM).Proof.
Obviously, we can suppose that∥u∥Lp(m)(ΛkM)≠0, ∥v∥Lp′(m)(ΛkM)≠0 and μ(M0)>0. We have
(2.14)1<p(m)<∞,|u(m)|<∞,|v(m)|<∞a.e.m∈M0.
By Young inequality, we have
(2.15)|〈u,v〉|‖u‖Lp(m)(ΛkM)‖v‖Lp′(m)(ΛkM)≤1p(m)(|u|‖u‖Lp(m)(ΛkM))p(m)+1p′(m)(|v|‖v‖Lp′(m)(ΛkM))p′(m).
Integrating over M0 we obtain
(2.16)∫M0‍|〈u,v〉|‖v‖Lp(m)(ΛkM)‖v‖Lp′(m)(ΛkM)dμ≤1p*∫M0‍(|u|‖u‖Lp(m)(ΛkM))p(x)dμ+(1-1p*)∫M0‍(|v|‖v‖Lp′(m)(ΛkM))p′(m)dμ≤1+1p*-1p*.
Then by (b2), we have
(2.17)∫M‍|〈u,v〉|dμ≤(1+1p*-1p*)‖u‖Lp(m)(ΛkM)‖v‖Lp′(m)(ΛkM)‖χM0‖L∞(M)+‖uχM1‖L1(ΛkM)‖vχM1‖L∞(ΛkM)+‖uχM∞‖L∞(ΛkM)‖vχM∞‖L1(ΛkM)≤rp‖u‖Lp(m)(ΛkM)‖v‖Lp′(m)(ΛkM),For differentialk-forms u on M, we define
(2.18)‖|u|‖Lp(m)(ΛkM)=sup⁡ρp′(m),Λn-kM(v)≤1∫M‍u∧v.We denote byΛ(k,n) the set of ordered multi-indices (i1,i2,…,ik) of integers 1≤i1<i2<⋯<ik≤n. Let I=(i1,i2,…,ik) be a multi-index from Λ(k,n). The complement I* of the multi-index I is the multi-index I*=(ik+1,ik+2,…,in) in Λ(n-k,n) where the components il are in {1,…,n}∖{i1,i2,…,ik} for all l=k+1,…,n.Letx1,…,xn be the orientable coordinates on M. Each differential k-form u can be written as the linear combination
(2.19)u=∑1≤i1<⋯<ik≤n‍ui1,…,ikdxi1∧⋯∧dxik=∑I∈Λ(k,n)‍uIdxI.
Here uI are the components of u with respect to natural basis
(2.20)dxI=dxi1∧⋯∧dxik,I=(i1,i2,…,ik)∈Λ(k,n).
For a differential (n-k)-form v=∑L∈Λ(k,n)‍vL*dxL*, we have
(2.21)u∧v=(-1)k(n-k)u∧⋆⋆v=(-1)k(n-k)〈u,⋆v〉dμ=〈⋆u,v〉dμ.
Note that ⋆dxI=det(gij)∑J∈Λ(k,n)‍∏γ=1kgiγjγσ(J)dxJ*, and hence
(2.22)〈⋆u,v〉=det⁡(gij)∑I,J,L∈Λ(k,n)‍∏γ=1kgiγjγ∏β=k+1ngjβlβσ(J)uIvL*on⁡M,
where gij are the components of the inverse matrix of (gij) and σ(J) is the signature of the permutation (j1⋯jn) in the set {1⋯n}.We consider an arbitrary local chartf:V(⊂M)→ℝn on M. Let U be any open set in M, whose closure is compact and is contained in V. Note that the components gij of g in (U,f) satisfy 1/2δij≤gij≤2δij as bilinear forms. Then
(2.23)〈⋆u,v〉=∏l=1ngll∑I∈Λ(k,n)‍σ(I)uIvI*on⁡M.
Thus, if sgnvI*=σ(I)sgnuI, ω=∑I∈Λ(k,n)‍ωI*dxI* with ρp′(m),Λn-kM(ω)≤1 and ωI*=±vI*, we have
(2.24)〈⋆u,ω〉≤〈⋆u,v〉,2-n/2∑I‍|uI||vI*|≤〈⋆u,v〉≤2n/2∑I‍|uI||vI*|on⁡M.
Integrating on K and M, by (2.18) we have
(2.25)0≤‖|uχK|‖Lp(m)(ΛkM)≤‖|u|‖Lp(m)(ΛkM),
for any compact subset K on M. Furthermore, It is easy to see that it is a norm on the class of differential k-forms u with ∥|u|∥Lp(m)(ΛkM)<∞.Lemma 2.3.
Let∥|u|∥Lp(m)(ΛkM)<∞ and ρp′(m),Λn-kM(v)<∞. Then
(2.26)|∫M‍u∧v|≤{‖|u|‖Lp(m)(ΛkM)ifρp′(m),Λn-kM(v)≤1,ρp′(m),Λn-kM(v)‖|u|‖Lp(m)(ΛkM)ifρp′(m),Λn-kM(v)>1.Proof.
The first case follows from (2.18). Assume that ρp′(m),Λn-kM(v)>1, we have
(2.27)ρp′(m),Λn-kM(vρp′(m),Λn-kM(v))≤ρp′(m),Λn-kM(v)ρp′(m),Λn-kM(v)=1,
and so
(2.28)|∫M‍u∧v|=ρp′(m),Λn-kM(v)|∫M‍u∧vρp′(m),Λn-kM(v)|≤ρp′(m),Λn-kM(v)‖|u|‖Lp(m)(ΛkM).Lemma 2.4.
Ifμ(M)=μ(M0),ρp(m),ΛkM(u)<∞ and ∥|u|∥Lp(m)(ΛkM)≤1, then ρp(m),ΛkM(u)≤1.Proof.
If this is not true, we may assume thatρp(m),ΛkM(u)>1, by (b4) there exist λ>1 such that ρp(m),ΛkM(u/λ)=1. Set
(2.29)v=|u|p(m)-2λp(m)-1(⋆u),m∈M,
we have ρp′(m),Λn-kM(v)=ρp(m),ΛkM(u/λ)=1 and so
(2.30)‖|u|‖Lp(m)(ΛkM)≥∫M‍u∧v=λρp(m),ΛkM(uλ)=λ>1,
which is a contradiction.Lemma 2.5.
If∥|u|∥Lp(m)(ΛkM)≤1, then ρp(m),ΛkM(u)≤cp∥|u|∥Lp(m)(ΛkM).Proof.
First, suppose thatρp(m),ΛkM(u)<∞. We have
(2.31)ρp(m),ΛkM(u)=∑j=0,1,∞‍ρp(m),ΛkM(uj),
where uj=uχMj, j=0,1,∞. Set
(2.32)v1={|u|-1(⋆u1)if|u|≠0,0if|u|=0,v0={|u|p(m)-2(⋆u0)if|u|≠0,0if|u|=0.
Then ρp′(m),Λn-kM(v1)=esssupM1|v1|=1 and due to Lemma 2.4,
(2.33)ρp′(m),Λn-kM(v0)=∫M0‍|u0|p(m)dμ≤1.
Hence, Lemma 2.3 yields
(2.34)ρp(m),ΛkM(uj)=∫M∖M∞‍u∧vj≤‖|u|‖Lp(m)(ΛkM),j=0,1.
If μ(M∞)>0, then for every ɛ∈(0,1) there exists a set D⊂M∞ such that 0<μ(D)<∞ and |u(m)|≥esssupM∞|u|ɛ, m∈D. Take
(2.35)v∞={μ(D)-1χD|u|-1(⋆u)if|u|≠0,0if|u|=0,
we have ρp′(m),Λn-kM(v∞)=∫D‍μ(D)-1|u|-1|⋆u|dμ≤1 and so
(2.36)‖|u|‖Lp(m)(ΛkM)≥∫M‍u∧v∞=μ(D)-1∫D‍|u|dμ≥ɛesssup⁡M∞|u|=ɛρp(m),ΛkM(u∞).
Letting ɛ→1 we obtain
(2.37)ρp(m),ΛkM(u∞)≤‖|u|‖Lp(m)(ΛkM).
Hence, (2.31)–(2.37) yield the desired results.
To avoid the assumptionρp(m),ΛkM(u)<∞ we define differential k-forms
(2.38)ut={uχGtif|u|≤t,tuχGt|u|if|u|>t,
where {Gt} is a sequence of compact sets such that Gt⊂Gt+1⊂M, μ(Gt)<∞ for t∈ℕ and M=∪t=1∞Gt. Then for every ut we have ρp(m),ΛkM(ut)<∞,∥|ut∥|Lp(m)(ΛkM)≤∥|u∥|Lp(m)(ΛkM)≤1. By the first part of the proof, ρp(m),ΛkM(ut)≤cp∥|u∥|Lp(m)(ΛkM). It follows let t→∞.Lemma 2.6.
For everyu∈Lp(m)(ΛkM), the following inequalities hold
(2.39)cp-1‖u‖Lp(m)(ΛkM)≤‖|u|‖Lp(m)(ΛkM)≤rp‖u‖Lp(m)(ΛkM).
Furthermore, we have
(2.40)Lp(m)(ΛkM)={u∈Lloc⁡1(ΛkM):‖|u|‖Lp(m)(ΛkM)<∞}.Proof.
Letu∈Lp(m)(ΛkM). If ρp′(m),Λn-kM(v)≤1, then ∥v∥Lp′(m)(Λn-kM)≤1 and Hölder inequality yields
(2.41)∫M‍u∧v≤rp‖u‖Lp(m)(ΛkM)‖v‖Lp′(m)(Λn-kM)≤rp‖u‖Lp(m)(ΛkM).
This gives the second inequality in (2.39) and, consequently, ∥|u∥|Lp(m)(ΛkM)<∞.
Conversely, we can suppose that0<∥|u|∥Lp(m)(ΛkM)<∞. By Lemma 2.5 and following inequalitiy
(2.42)0<‖|u(cp‖|u|‖Lp(m)(ΛkM))|‖Lp(m)(ΛkM)=cp-1≤1,
we get ρp(m),ΛkM(u/(cp∥|u|∥Lp(m)(ΛkM)))≤cpcp-1=1. The first inequality in (2.39) follows and then u∈Lp(m)(ΛkM).We shall say that differentialk-forms ut∈Lp(m)(ΛkM)converge modularly to a differential k-form u∈Lp(m)(ΛkM) if limt→∞ρp(m),ΛkM(ut-u)=0.Next, we consider the relationship between convergence in norm, convergence in modular, and convergence in measure. For the corresponding results for domains inℝn, readers can be referred to [15, 16].Lemma 2.7.
Ifp∈𝒫1(M), then ρp(m),ΛkM(ut)→0 if and only if ∥ut∥Lp(m)(ΛkM)→0.Proof.
According to Lemmas2.5 and 2.6, the norm convergence is stronger than the modular convergence. Suppose that ρp(m),ΛkM(ut)→0, and take ɛ∈(0,1]. For sufficiently large t we have ρp(m),ΛkM(ut)<ɛ≤1 and so
(2.43)ρp(m),ΛkM(ut(ρp(m),ΛkM(ut))1/p*)≤ρp(m),ΛkM(ut)ρp(m),ΛkM(ut)=1,
that is, ∥ut∥Lp(m)(ΛkM)≤(ρp(m),ΛkM(ut))1/p*≤ɛ1/p*. Hence, ∥ut∥Lp(m)(ΛkM)→0.Lemma 2.8.
Ifp∈𝒫1(M) and μ(M)<∞, then ∥ut-u∥Lp(m)(ΛkM)→0 if and only if ut converges to u on M in measure and limt→∞ρp(m),ΛkM(ut)=ρp(m),ΛkM(u).Proof.
If∥ut-u∥Lp(m)(ΛkM)→0, by Lemma 2.7(2.44)lim⁡t→∞∫M‍|ut-u|p(m)dμ=0,
then it is easy to see that ut converges to u on M in measure. Hence by μ(M)<∞, |ut|p(m) converges to |u|p(m) on M in measure and the integrals of the functions |ut-u|p(m) possess absolutely equicontinuity on M. Since
(2.45)|ut|p(m)≤2p*-1(|ut-u|p(m)+|u|p(m)),
the integrals of the |ut|p(m) are also absolutely equicontinuous on M. By Vitali convergence theorem (see [28]), we deduce that limt→∞ρp(m),ΛkM(ut)=ρp(m),ΛkM(u).
Conversely, ifut converges to u on M in measure, we can deduce that |ut-u|p(m) converges to 0 on M in measure. Similar to the above proof, by the inequality
(2.46)|ut-u|p(m)≤2p*-1(|ut|p(m)+|u|p(m)),
and limt→∞ρp(m),ΛkM(ut)=ρp(m),ΛkM(u), we get limt→∞ρp(m),ΛkM(ut-u)=0.Lemma 2.9.
Ifp∈𝒫1(M), then L∞(ΛkM)∩Lp(m)(ΛkM) is dense in Lp(m)(ΛkM).Proof.
Letm0 be some point of M, dg be the distance associated to g and Gt={m∈M:dg(m0,m)<t,t∈ℕ}. Given u∈Lp(m)(ΛkM), we define sequence of differential k-forms by
(2.47)ut={uχGtif|u|≤t,tuχGt|u|if|u|>t.
Then ut∈L∞(ΛkM) and by Lebesgue dominated convergence theorem, we have ρp(m),ΛkM(u-ut)→0. Hence, by Lemma 2.7∥u-ut∥Lp(m)(ΛkM)→0.Lemma 2.10.
Ifp∈𝒫1(M), then Cc∞(ΛkM) is dense in Lp(m)(ΛkM).Proof.
Sincep∈𝒫1(M), we have Cc∞(ΛkM)⊂Lp(m)(ΛkM). By Lemma 2.9, there is a differential k-form ut0∈L∞(ΛkM)∩Lp(m)(ΛkM) such that
(2.48)‖u-ut0‖Lp(m)(ΛkM)≤ɛ.
By Luzin theorem there exists a continuousk-form φ∈C(ΛkM) and an open set D⊂M such that
(2.49)μ(D)<min⁡{1,(ɛ2‖ut0‖L∞(ΛkM))p*},φ=ut0 on M∖D and supM|φ|=supM∖D|ut0|≤∥ut0∥L∞(ΛkM). Thus,
(2.50)ρp(m),ΛkM(ut0-φɛ)≤max⁡{1,(2‖ut0‖L∞(ΛkM)ɛ)p*}μ(D)≤1,
that is,
(2.51)‖ut0-φ‖Lp(m)(ΛkM)≤ɛ.
Sinceφ∈Lp(m)(ΛkM), we have ρp(m),ΛkM(φ)<∞ and there exists a bounded open set G⊂M such that ρp(m),ΛkM(φχM∖G/ɛ)≤1, that is,
(2.52)‖φ-φχG‖Lp(m)(ΛkM)≤ɛ.
Leth be a polynomial differential k-form with supG|φ-h|<ɛmin{1,μ(G)-1}. The polynomial differential k-form means the components of its coordinate representation in each chart of the manifold M are polynomial functions. Then ρp(m),ΛkM((φχG-hχG)/ɛ)≤min{1,μ(G)-1}μ(G)≤1, that is,
(2.53)‖φχG-hχG‖Lp(m)(ΛkM)≤ɛ.
Finally, there exists a compact set K⊂G such that ∥hχG-hχK∥Lp(m)(ΛkM)≤ɛ. Let π∈Cc∞(G) with 0≤π≤1 in G and π=1 on K we obtain the estimate
(2.54)‖hχG-πh‖Lp(m)(ΛkM)≤‖hχG-hχK‖Lp(m)(ΛkM)≤ɛ.
From (2.48)–(2.54), we get
(2.55)‖u-πh‖Lp(m)(ΛkM)≤5ɛ.
Obviously, πh∈Cc∞(ΛkM).Theorem 2.11.
Ifp∈𝒫1(M), then the space Lp(m)(ΛkM) is separable.Proof.
Letu∈Lp(m)(ΛkM),ɛ>0. By the proof of Lemma 2.10, we can fine a continuous k-form φ∈C(ΛkM) and a set Gt0={m∈M:dg(m0,m)<t0} such that
(2.56)‖u-φ‖Lp(m)(ΛkM)≤ɛ,‖φχM∖Gt0‖Lp(m)(ΛkM)≤ɛ,
Let h be a polynomial differential k-form with supGt0|φ-h|<ɛmin{1,μ(Gt0)-1}, v be a polynomial differential k-form with rational coefficients and supGt0|h-v|<ɛmin{1,μ(Gt0)-1}. Then we have
(2.57)‖φχGt0-hχGt0‖Lp(m)(ΛkM)≤ɛ,‖vχGt0-hχGt0‖Lp(m)(ΛkM)≤ɛ.
Thus,
(2.58)‖vχGt0-u‖Lp(m)(ΛkM)≤4ɛ.
Therefore, we conclude that the set of all differential k-forms vχGt is dense in Lp(m)(ΛkM).Theorem 2.12.
Ifp∈𝒫(M), then the space Lp(m)(ΛkM) is complete.Proof.
Let{ut:ut=∑I‍(ut)IdxI} be a Cauchy sequence of differential k-forms in Lp(m)(ΛkM) and ɛ>0. Let {Gl} be a sequence of compact sets such that Gl⊂Gl+1⊂M for l∈ℕ and M=∪l=1∞Gl. There exists t0∈ℕ such that
(2.59)sup⁡ρp′(m),Λn-kM(v)≤1∫Gl‍(ut-uτ)∧v≤ɛ,
for every t,τ≥t0 and l∈ℕ. By (2.24) we have
(2.60)∫Gl‍∑I‍|(ut-uτ)I||vI*|dμ≤2n/2ɛ,
for every v=∑I‍vI*dxI*, ρp′(m),Λn-kM(v)≤1 and sgnvI*=σ(I)sgn(ut-uτ)I. We define vl=φlχGl where |φl|=(1+μ(Gl))-1 for l∈ℕ. Then
(2.61)ρp′(m),Λn-kM(vl)≤∫Gl‍(1+μ(Gl))-p′(m)dμ+(1+μ(Gl))-1≤1,
thus, by (2.60) we get
(2.62)∫Gl‍|ut-uτ|dμ≤2k/2∫Gl‍∑I‍|(ut-uτ)I|dμ≤ɛ2n(1+μ(Gl)),fort,τ≥t0,l∈N.
This means that the sequence {ut} is Cauchy in each L1(ΛkGl). By induction we may find subsequences {ut(l)}t and differential k-forms u(l)∈L1(ΛkGl) such that ut(l)→u(l) a.e.  onGl for  l∈ℕ, and u(l+1)χGl=u(l). Thus, limτ→∞uτ(τ)=limτ→∞u(τ)χGτ=u a.e.  on  M. Replacing uτ by uτ(τ) in (2.60) and using the Fatou lemma we obtain
(2.63)∫Gl‍∑I‍|(ut-u)I||vI*|dμ≤sup⁡⁡τ>t0∫Gl‍∑I|(ut-uτ(τ))I|‍|vI*|dμ≤2n/2ɛ.
Let l→∞, together with (2.24) we have
(2.64)∫M‍(ut-u)∧v≤2nɛ.
Therefore, by (2.18) and (2.24), we obtain ∥|ut-u|∥Lp(m)(ΛkM)≤2nɛ.Theorem 2.13.
Ifp∈𝒫2(M), then the space Lp(m)(ΛkM) is reflexive.Proof.
Let[Lp(m)(ΛkM)]′ denote the dual space to Lp(m)(ΛkM). We will show that [Lp(m)(ΛkM)]′=Lp′(m)(Λn-kM) in steps.
(i) For fixedv∈Lp′(m)(Λn-kM), we define a linear functional Fv on Lp(m)(ΛkM)(2.65)Fv(u)=∫M‍u∧v=∫M‍〈⋆u,v〉dμ.
By Lemma 2.2, we have |Fv(u)|≤rp∥u∥Lp(m)(ΛkM)∥v∥Lp′(m)(Λn-kM), that is,
(2.66)‖Fv‖≤rp‖v‖Lp′(m)(Λn-kM).
Thus, Fv is a bounded linear functional on Lp(m)(ΛkM) and so Fv belongs to [Lp(m)(ΛkM)]′.
(ii) We consider an arbitrary local chartf:V(⊂M)→ℝn on M. Let U be any open set in M, whose closure is compact and contained in V. We define
(2.67)hI(φdxI)=φforI∈Λ(k,n),φ∈Lp(f-1(x))(f(U)).
Since each continuous linear functional f~∈[Lp(f-1(x))(f(U))]′ can be represented uniquely in the form f~(φ)=∫f(U)‍φψf~dx for some ψf~∈Lp′(f-1(x))(f(U)), then for each continuous linear functional f¯∈[Lp(f-1(x))(Λkf(U))]′, we have
(2.68)f¯(ω)=∑I∈Λ(k,n)‍f¯(ωIdxI)=∑I∈Λ(k,n)‍f¯∘hI-1(ωI)=∑I∈Λ(k,n)‍∫f(U)‍ωIψf¯∘hI-1dx=∫f(U)‍ω∧(∑I∈Λ(k,n)‍σ(I)ψf¯∘hI-1dxI*),
that is, f¯ can be represented in the form
(2.69)f¯(ω)=∫f(U)‍ω∧ϖf¯,
where ϖf¯=∑I∈Λ(k,n)‍σ(I)ψf¯∘hI-1dxI*∈Lp′(f-1(x))(f(U)). If ϖ1=∑I‍ϖ1IdxI*,ϖ2=∑I‍ϖ2IdxI* such that
(2.70)f¯(ω)=∫f(U)‍ω∧ϖ1=∫f(U)‍ω∧ϖ2,
for every ω∈Lp(f-1(x))(Λkf(U)). Taking ω=φdxI for I∈Λ(k,n), we have f¯∘hI-1(φ)=f¯(ω)=∫f(U)‍φϖ1Idx=∫f(U)‍φϖ2Idx, then ϖ1I=ϖ2I, that is, ϖ1=ϖ2. Hence ϖf¯ is uniquely determined.
For fixedF∈[Lp(m)(ΛkM)]′ and any u∈Lp(m)(Λn-kM) with compact support we have
(2.71)F(χUu)=F∘f*((f-1)*(χUu))=∫f(U)‍(f-1)*(χUu)∧vF∘f*=∫U‍χUu∧f*(vF∘f*),
where vU=f*(vF∘f*)∈Lp′(m)(Λn-kU) is uniquely determined. For any two sets U1 and U2, the differential forms vU1 and vU2 coincide on U1∩U2 because of the uniqueness of the differential form vU1∩U2. Thus, all the differential forms vU, defined for different U, are compatible with one another, and hence defines a differential form vF on M. The differential form vF locally belongs to the space Lp′(m)(Λn-kU) and satisfies
(2.72)F(u)=∫M‍u∧vF,
for every u∈Lp(m)(ΛkM) with compact support, and is uniquely determined.
Let{Gt} be a sequence of compact sets such that Gt⊂Gt+1⊂M for t∈ℕ and M=∪t=1∞Gt. Then
(2.73)F(u)=F(lim⁡t→∞χGtu)=lim⁡t→∞F(χGtu)=lim⁡t→∞∫M‍χGtu∧vF=∫M‍u∧vF.
If v1,v2 such that
(2.74)F(u)=∫M‍u∧v1=∫M‍u∧v2,
for every u∈Lp(m)(ΛkM). Then for any U, we have F(χUu)=∫M‍χUu∧v1=∫M‍χUu∧v2. Thus χUv1=χUv2 for any U, that is, v1=v2.
Therefore, we conclude that each continuous linear functionalF∈[Lp(x)(ΛkM)]′ can be uniquely represented in the form (2.72).
(iii) We shall show∥vF∥Lp′(m)(Λn-kM)≤C∥F∥ with the constant C dependent only on p(m). We define a differential form u on M(2.75)u(m)={‖vF(m)‖Lp′(m)(Λn-kM)1/(1-p(m))|vF(m)|p′(m)-2(⋆vF(m))if|vF(m)|≠0,0if|vF(m)|=0,
then by (b4) and (b6), we have
(2.76)‖u‖Lp(m)(ΛkM)=inf⁡⁡{λ>0:∫M‍(|vF|λp(m)-1‖vF‖Lp′(m)(Λn-kM))p′(m)dμ≤1}=1.
Moreover
(2.77)|F(u)|=|∫M‍u∧vF|=∫M‍(|vF|‖vF‖Lp′(m)(Λn-kM))p′(m)‖vF‖Lp′(m)(Λn-kM)dμ≥‖vF‖Lp′(m)(Λn-kM)2p*/(p*-1)∫M‍(|vF|(1/2)‖vF‖Lp′(m)(Λn-kM))p′(m)dμ≥‖vF‖Lp′(m)(Λn-kM)2p*/(p*-1).
Hence, we assert that ∥vF∥Lp′(m)(Λn-kM)≤2p*/(p*-1)∥F∥.
Now we reach the conclusion[Lp(m)(ΛkM)]′=Lp′(m)(Λn-kM), and hence Lp(m)(ΛkM) is reflexive.Theorem 2.14.
Ifp∈𝒫2(M), then the exterior Sobolev space W1,p(m)(ΛkM) is a separable, reflexive Banach space.Proof.
We treatW1,p(m)(ΛkM) in a natural way as a subspace of the Cartesian product space Lp(m)(ΛkM)×Lp(m)(Λk+1M). Then we need only to show that W1,p(m)(ΛkM) is a closed subspace of Lp(m)(ΛkM)×Lp(m)(Λk+1M). Let {ut}⊂W1,p(m)(ΛkM) be a convergent sequence. Then {ut} is a convergent sequence in Lp(m)(ΛkM). In view of Theorem 2.12, there exists u∈Lp(m)(ΛkM) such that ut→u in Lp(m)(ΛkM). Similarly there exists u~∈Lp(m)(Λk+1M) such that dut→u~ in Lp(m)(Λk+1M). Then it is easy to see that ut converges to u and dut converges to u~ on M in measure. For any φ∈Cc∞(Λn-k-1M)⊂Lp′(m)(Λn-k-1M), we have
(2.78)∫M‍ut∧dφ=(-1)k+1∫M‍dut∧φ.
Obviously, |ut∧dφ|≤|(ut-u)∧dφ|+|u∧dφ| and |dut∧φ|≤|(dut-u~)∧φ|+|u~∧φ|, then integrals of the functions |ut∧dφ| and |dut∧φ| possess absolutely equicontinuity on M. Hence, by Vitali convergence theorem (see [28]), we get
(2.79)∫M‍u∧dφ=(-1)k+1∫M‍u~∧φ.
Thus, we obtain that du=u~. Then it is immediate that W1,p(m)(ΛkM) is a closed subspace of Lp(m)(ΛkM)×Lp(m)(Λk+1M).Given two Banach spacesX and Y, the symbol X↷Y means that X is continuously embedded in Y.Theorem 2.15.
Let0<μ(M)<∞. If p(m),q(m)∈𝒫(M) and p(m)≤q(m) a.e. m∈M, then
(2.80)Lq(m)(ΛkM)↷Lp(m)(ΛkM).
The norm of the embedding operator (2.80) does not exceed μ(M)+1.Proof.
Sincep(m)≤q(m) a.e. m∈M, then M∞p⊂M∞q. We may assume that u∈Lq(m)(Λkm) with ∥u∥Lq(m)(ΛkM)≤1. Otherwise we can consider u/∥u∥Lq(m)(ΛkM). By (b7) we have ρq(m),ΛkM(u)≤1, in particular, |u(m)|≤1 a.e. m∈M∞q. Then we can write
(2.81)ρp(m),ΛkM(u)≤μ({m∈M∖M∞q:|u|≤1})+∫M∖M∞q‍|u|q(m)dμ+μ(M∞q∖M∞p)+esssup⁡⁡M∞p|u|≤μ(M)+1.
Thus, we have ρp(m),ΛkM(u/(μ(M)+1))≤(μ(M)+1)-1ρp(m),ΛkM(u)≤1. Therefore
(2.82)‖u‖Lp(m)(ΛkM)≤(μ(M)+1)‖u‖Lq(m)(ΛkM).Theorem 2.16.
LetM be a compact Riemannian manifold with a smooth boundary or without boundary and p(m),q(m)∈C(M¯)∩𝒫1(M). Assume that
(2.83)p(m)<n,q(m)<np(m)n-p(m),form∈M¯.
Then
(2.84)W1,p(m)(M)↷Lq(m)(M)
is a continuous and compact embedding.Proof.
We consider an arbitrary local chartf:V(⊂M)→ℝn on M. Let U be any open set in M, whose closure is compact and is contained in V. Choosing a finite subcovering {Uα}α=1,2,…,s of M such that Uα is homeomorphic to the open unit ball B0(1) of ℝn and for any α the components gijα of g in (Uα,fα) satisfy 1/Cδij≤gijα≤Cδij as bilinear forms, where constant C>1 is given. Let {πα} be a smooth partition of unity subordinate to the finite covering {Uα}. It is obvious that if u∈W1,p(m)(M, then παu∈W1,p(m)(Uα) and (fα-1)*(παu)∈W1,p(fα-1(x))(B0(1)). By the definition of integral for differential n-forms on M and Sobolev embedding theorem in [16], we have the following continuous and compact embedding:
(2.85)W1,p(m)(Uα)↷Lq(m)(Uα),foreachα=1,2,…,s.
Since u=∑α=1s‍παu, we can assert that W1,p(m)(M)⊂Lq(m)(M), and the embedding is continuous and compact.Letu∈Lp(m)(ΛkM), we say that u is absolutely continuous with respect to the norm ∥·∥Lp(m)(ΛkM), if G⊂M be a measurable subset, we have
(2.86)lim⁡μ(G)→0‖uχG‖Lp(m)(ΛkM)=0.Theorem 2.17.
Ifp∈𝒫1(M), u∈Lp(m)(ΛkM) is absolutely continuous with respect to the norm ∥·∥Lp(m)(ΛkM).Proof.
By Lemma2.9, there is a differential k-form ut0∈L∞(ΛkM)∩Lp(m)(ΛkM) such that
(2.87)‖u-ut0‖Lp(m)(ΛkM)<ɛ2.
Since ut0 is bounded, we can find ɛ0>0 such that when μ(G)<ɛ0, the following inequalities hold
(2.88)‖ut0χG‖Lp(m)(ΛkM)<ɛ2.
Hence, we get
(2.89)‖uχG‖Lp(m)(ΛkM)≤‖(u-ut0)χG‖Lp(m)(ΛkM)+‖ut0χG‖Lp(m)(ΛkM)≤‖u-ut0‖Lp(m)(ΛkM)+‖ut0χG‖Lp(m)(ΛkM)<ɛ.


## 3. Applications

In this section, we shall show some applications of the exterior Sobolev space to Dirichlet problems with variable growth on Riemannian manifold. We shall assume thatΩ⊂M is a bounded domain with smooth boundary and p(m)∈𝒫2(Ω).The nonhomogeneousp(m)-harmonic equation for differential forms with variable growth on Ω belong to the nonlinear elliptic equations which take the form
(3.1)δ(du|du|p(m)-2)+u|u|p(m)-2=f(m).Definition 3.1.
A differential formω is a weak solution for the following Dirichlet problems
(3.2)δ(du|du|p(m)-2)+u|u|p(m)-2=f(m),inΩ,u=0,on⁡∂Ω,
where f(m)∈Lp′(m)(Λk-1Ω), if ω∈W01,p(m)(Λk-1Ω) satisfies
(3.3)∫Ω‍〈dω|dω|p(m)-2,dv〉+〈ω|ω|p(m)-2,v〉dμ=∫Ω‍〈f(m),v〉dμ,
for every v∈W01,p(m)(Λk-1Ω).We are now ready to show an application of exterior Sobolev spacesW01,p(m)(Λk-1Ω) to Dirichlet problems (3.2).LetX=W01,p(m)(Λk-1Ω), X′ be the dual space to X and (·,·) denote a dual between X and X′. Consider the following functional:
(3.4)I(u)=∫Ω‍1p(m)(|du|p(m)+|u|p(m))dμ,u∈X.
We denote J=I′:X→X′, then
(3.5)(J(u),v)=∫Ω‍〈du|du|p(m)-2,dv〉dμ+∫Ω‍〈u|u|p(m)-2,v〉dμ:=(J1(u),v)+(J2(u),v),
where u,v∈X. Here,
(3.6)(J1(u),v)=∫Ω‍〈du|du|p(m)-2,dv〉dμ,(J2(u),v)=∫Ω‍〈u|u|p(m)-2,v〉dμ.Lemma 3.2.
J
=
I
′
:
X
→
X
′ is a continuous, bounded, and strictly monotone operator.Proof.
It is obvious thatJ is continuous and bounded. For any y,z∈ℝN, we have the following inequalities (see [29]) from which we can get the strictly monotonicity of J:(h1)
(|z|p-2z-|y|p-2y)·(z-y)≥(1/2)p|z-y|p,p∈[2,∞),(h2)
[(|z|p-2z-|y|p-2y)·(z-y)](|z|p+|yp|)(2-p)/p≥(p-1)2|z-y|2,p∈(1,2).Lemma 3.3.
J
=
I
′
:
X
→
X
′ is a mapping of type (S+), that is, if ut⇀u weakly in X and limsupt→∞(J(ut)-J(u),ut-u)≤0, then ut→u strongly in X.Proof.
By Lemma3.2, if ut⇀u weakly in X and limsupt→∞(J(ut)-J(u),ut-u)≤0, we have limt→∞(J(ut)-J(u),ut-u)=0. In view of (h1) and (h2), limt→∞(Ji(ut)-Ji(u),ut-u)=0(i=1,2). Let Ω1={m∈Ω:p(m)<2},Ω2={m∈Ω:p(m)≥2} and vt=〈|ut|p(m)-2ut-|u|p(m)-2u,ut-u〉. Then there is a constant C>0 such that
(3.7)∫Ω2‍|ut-u|p(m)dμ≤C∫Ω2‍vtdμ⟶0,∫Ω1‍|ut-u|p(m)dμ≤C∫Ω1‍vtp(m)/2(|ut|p(m)+|up(m)|)(2-p(m))/2dμ≤C‖vtp(m)/2χΩ1‖L2/p(m)(Ω)‖(|ut|p(m)+|up(m)|)(2-p(m))/2χΩ1‖L2/(2-p(m))(Ω)⟶0.
Therefore, by (3.7)
(3.8)lim⁡t→∞∫Ω‍|ut-u|p(m)dμ=0.
Similar to the proof above, we can obtain
(3.9)lim⁡t→∞∫Ω‍|dut-du|p(m)dμ=0.
From Lemma 2.8, we have ut→u strongly in X, that is, J is a mapping of type (S+).Lemma 3.4.
The mappingJ is coercive, that is,
(3.10)(J(u),u)‖u‖X⟶∞as‖u‖X⟶∞.Proof.
Takingɛ0=(1/2)∥u∥Lp(m)(Λk-1Ω), we have
(3.11)∫Ω‍|u|p(m)dμ‖u‖Lp(m)(Λk-1Ω)=∫Ω‍(|u|‖u‖Lp(m)(Λk-1Ω)-ɛ0)p(m)(‖u‖Lp(m)(Λk-1Ω)-ɛ0)p(m)‖u‖Lp(m)(Λk-1Ω)dμ≥(‖u‖Lp(m)(Λk-1Ω)-ɛ0)p*‖u‖Lp(m)(Λk-1Ω)≥‖u‖Lp(m)(Λk-1Ω)p*2p*‖u‖Lp(m)(Λk-1Ω)⟶∞,
as ∥u∥Lp(m)(Λk-1,Ω)→∞. Similarly, we also obtain
(3.12)∫Ω‍|du|p(m)dμ‖du‖Lp(m)(ΛkΩ)⟶∞as‖du‖Lp(m)(Λk,Ω)⟶∞.
Thus, for fixed constant K>0, there exists N=N(K) such that
(3.13)∫Ω‍|u|p(m)dμ‖u‖Lp(m)(Λk-1Ω)>2K,if‖u‖Lp(m)(Λk-1,Ω)>N,∫Ω‍|du|p(m)dμ‖du‖Lp(m)(ΛkΩ)>2K,if‖du‖Lp(m)(Λk,Ω)>N.
We take N0=2N, if ∥u∥X>N0 and ∥du∥Lp(m)(Λk,Ω)≥∥u∥Lp(m)(Λk-1,Ω), then
(3.14)(J(u),u)‖u‖X=∫Ω‍|du|p(m)dμ+∫Ω‍|u|p(m)dμ‖du‖Lp(m)(Λk,Ω)+‖u‖Lp(m)(Λk-1,Ω)≥∫Ω‍|du|p(m)dμ2‖du‖Lp(m)(Λk,Ω)>K,
if ∥u∥X>N0 and ∥u∥Lp(m)(Λk-1,Ω)>∥du∥Lp(m)(Λk,Ω), then
(3.15)(J(u),u)‖u‖X≥∫Ω‍|u|p(m)dμ2‖u‖Lp(m)(Λk-1,Ω)>K.
Hence, (J(u),u)/∥u∥X→∞as∥u∥X→∞, that is, the mapping J is coercive.Lemma 3.5.
J
:
X
→
X
′ is a homeomorphism.Proof.
By Lemmas3.2 and 3.4 and the theorem of Minty-Browder (see [30]), J is a bijection. Hence J has an inverse mapping J-1:X′→X. Therefore, the continuity of J-1 is sufficient to ensure J to be a homeomorphism.
Ifvt,v∈X′ and vt→v strongly in X′, let ut=J-1(vt), u=J-1(v), then J(ut)=vt and J(u)=v. As J is coercive, we have {ut} is bounded in X. Without loss of generality, we can assume that ut⇀u¯ weakly in X. Since vt→v strongly in X′, then
(3.16)lim⁡t→∞(J(ut)-J(u¯),ut-u¯)=lim⁡t→∞(J(ut),ut-u¯)=lim⁡t→∞(J(ut)-J(u),ut-u¯)=0.
Since J is a mapping of type (S+), ut→u¯ strongly in X. By Lemma 3.2, we conclude that ut→u strongly in X, so J-1 is continuous.It is immediate to obtain the following conclusion from the above lemmas.Theorem 3.6.
Iff(m)∈[W01,p(m)(Λk-1Ω)]′, then Dirichlet problems (3.2) has a unique weak solution in W01,p(m)(Λk-1Ω).Ifk=1, that is, u is a function on Ω, let ∇ be the gradient operator on M. One has the following corollary.Corollary 3.7.
Iff(m)∈[W01,p(m)(Ω)]′, then Dirichlet problems
(3.17)-div(∇u|∇u|p(m)-2)+u|u|p(m)-2=f(m),inΩ,u=0,on∂Ω,
has a unique weak solution in W01,p(m)(Ω).



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*Source: 575819-2012-08-17.xml*