We introduce the Lebesgue space and the exterior Sobolev space for
differential forms on Riemannian manifold which are the Lebesgue space
and the Sobolev space of functions on , 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 harmonic equations with variable growth in .
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, -Cohomology and -Cohomology and applications to some nonlinear PDE were studied in [4–6]; 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 . Many interesting results concerning -harmonic equations have been established recently (see [12, 13] and the references therein).
After Kováčik and Rákosník first discussed the and spaces in , 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 -Laplacian Dirichlet problems with different types on bounded domains in have been greatly discussed under various conditions (see  for the existence and  for the uniqueness). In recent years, the theory on problems with variable exponential growth conditions has important applications in nonlinear elastic mechanics (see ), electrorheological fluids (see [23, 24]).
The paper is organized as follows. In Section 2, we give the necessary definitions and some elementary properties of differential forms on Riemannian manifold. Moreover, we introduce the functional on and the spaces of differential forms and , then discuss some important properties. In Section 3, we show the existence and uniqueness of weak solution for Dirichlet problems of nonhomogeneous -harmonic equations with variable growth in .
Let be an arbitrary smooth -dimensional manifold (Hausdorff and with countable basis). Let be the cotangent bundle on M and (or ) be the bundles of the exterior -forms. We will call each fiber of the bundle a exterior form of degree on the manifold . Here, and in the case or . Given a exterior -form and a local chart , around , we define the representation of in this local coordinates system as the exterior -forms on given by
for any , where is the induced map by that takes vectors on into vectors on and is the induced map by that takes exterior forms on into exterior forms on (see ).
In this paper we will always assume is an -dimensional smooth orientable complete Riemannian manifold and is the Riemannian volume element on , where the are the components of the Riemannian metric in the chart and is the Lebesgue volume element of . A Riemannian metric on induces a scalar product on each fiber of the bundle . Hence for any exterior forms and of the same degree , the scalar product is defined at each point and the norm of is given by the formula . Let be a curve of class , the length of is
For , let be the space of piecewise curves such that and . One can define a distance on .
The Grassman algebra is a graded algebra with respect to the exterior products. We denote by the space of locally integrable exterior forms of degree (i.e., differential -forms) on . The local integrability of an exterior -form means the local integrability of the components of its coordinate representation in each chart of the Riemannian manifold . We denote by the vector space of smooth differential forms of degree with compact support on .
Let be is an -dimensional smooth orientable Riemannian manifold. We define the integral of , a exterior -form with compact support on (see ). Let be a local chart of , we have a partition of unity subordinate to this cover. Recall that and . Thus, every is an exterior -form whose support is a subset of and we may write . By definition
We will identify each exterior form of degree on the -dimensional Riemannian manifold with an exterior -form on (see ). Using this identification, we can assume that each exterior form has a weak exterior differential .
Definition 2.1 (see ). We say that an exterior form is the weak exterior differential of a form and we write if for each , one has
The operator , also called Hodge star operator (see ), has the following properties: for and ),(),(),(),().
By the operator and the exterior differentiation we define the codifferential operator by the formula
for any differential form .
The Riemannian measure and the characteristic function of a set will be denoted by and , respectively.
Let be the set of all measurable functions . For we put , , and if , if and . We always assume that , and . We use the convention .
For a differential -form on we define the functional by
The Lebesgue space is the space of differential forms in such that
with the following norm
The exterior Sobolev space consists of such forms for which . The norm is defined by
The space is defined as the closure of in .
Note that , and are spaces of functions on . In this paper we denote them by , and .
Given we define the conjugate function by
Similar to the proof of properties of and for (see [15, 16, 18]), it is easy to see that and has the following properties:() is convex.() for every subset and differential forms .() If for a.e. and if , then , the last inequality is strict if .() If , then the function is continuous and decreasing on the interval .() If , then .() If , then for every differential forms with .() If , then .() If and , then
() If and , then
Lemma 2.2. If , then the inequality
holds for every , .
Proof. Obviously, we can suppose that , and . We have
By Young inequality, we have
Integrating over we obtain
Then by (), we have
For differential -forms on , we define
We denote by the set of ordered multi-indices of integers . Let be a multi-index from . The complement of the multi-index is the multi-index in where the components are in for all .
Let be the orientable coordinates on . Each differential -form can be written as the linear combination
Here are the components of with respect to natural basis
For a differential -form , we have
Note that , and hence
where are the components of the inverse matrix of and is the signature of the permutation in the set .
We consider an arbitrary local chart on . Let be any open set in , whose closure is compact and is contained in . Note that the components of in satisfy as bilinear forms. Then
Thus, if , with and , we have
Integrating on and , by (2.18) we have
for any compact subset on . Furthermore, It is easy to see that it is a norm on the class of differential -forms with .
Lemma 2.3. Let and . Then
Proof. The first case follows from (2.18). Assume that , we have
Lemma 2.4. If and , then .
Proof. If this is not true, we may assume that , by () there exist such that . Set
we have and so
which is a contradiction.
Lemma 2.5. If , then .
Proof. First, suppose that . We have
where , . Set
Then and due to Lemma 2.4,
Hence, Lemma 2.3 yields
If , then for every there exists a set such that and , . Take
we have and so
Letting we obtain
Hence, (2.31)–(2.37) yield the desired results. To avoid the assumption we define differential -forms
where is a sequence of compact sets such that , for and . Then for every we have . By the first part of the proof, . It follows let .
Lemma 2.6. For every , the following inequalities hold
Furthermore, we have
Proof. Let . If , then and Hölder inequality yields
This gives the second inequality in (2.39) and, consequently, . Conversely, we can suppose that . By Lemma 2.5 and following inequalitiy
we get . The first inequality in (2.39) follows and then .
We shall say that differential -forms converge modularly to a differential -form if .
Next, we consider the relationship between convergence in norm, convergence in modular, and convergence in measure. For the corresponding results for domains in , readers can be referred to [15, 16].
Lemma 2.7. If , then if and only if .
Proof. According to Lemmas 2.5 and 2.6, the norm convergence is stronger than the modular convergence. Suppose that , and take . For sufficiently large we have and so
that is, . Hence, .
Lemma 2.8. If and , then if and only if converges to on in measure and .
Proof. If , by Lemma 2.7
then it is easy to see that converges to on in measure. Hence by , converges to on in measure and the integrals of the functions possess absolutely equicontinuity on . Since
the integrals of the are also absolutely equicontinuous on . By Vitali convergence theorem (see ), we deduce that . Conversely, if converges to on in measure, we can deduce that converges to on in measure. Similar to the above proof, by the inequality
and , we get .
Lemma 2.9. If , then is dense in .
Proof. Let be some point of , be the distance associated to and . Given , we define sequence of differential -forms by
Then and by Lebesgue dominated convergence theorem, we have . Hence, by Lemma 2.7.
Lemma 2.10. If , then is dense in .
Proof. Since , we have . By Lemma 2.9, there is a differential -form such that
By Luzin theorem there exists a continuous -form and an open set such that
on and . Thus,
Since , we have and there exists a bounded open set such that , that is,
Let be a polynomial differential -form with . The polynomial differential -form means the components of its coordinate representation in each chart of the manifold are polynomial functions. Then , that is,
Finally, there exists a compact set such that . Let with in and on we obtain the estimate
From (2.48)–(2.54), we get
Theorem 2.11. If , then the space is separable.
Proof. Let . By the proof of Lemma 2.10, we can fine a continuous -form and a set such that
Let be a polynomial differential -form with , be a polynomial differential -form with rational coefficients and . Then we have
Therefore, we conclude that the set of all differential -forms is dense in .
Theorem 2.12. If , then the space is complete.
Proof. Let be a Cauchy sequence of differential -forms in and . Let be a sequence of compact sets such that for and . There exists such that
for every and . By (2.24) we have
for every , and . We define where for . Then
thus, by (2.60) we get
This means that the sequence is Cauchy in each . By induction we may find subsequences and differential -forms such that a.e. on for , and . Thus, a.e. on . Replacing by in (2.60) and using the Fatou lemma we obtain
Let , together with (2.24) we have
Therefore, by (2.18) and (2.24), we obtain .
Theorem 2.13. If , then the space is reflexive.
Proof. Let denote the dual space to . We will show that in steps. (i) For fixed , we define a linear functional on
By Lemma 2.2, we have , that is,
Thus, is a bounded linear functional on and so belongs to . (ii) We consider an arbitrary local chart on . Let be any open set in , whose closure is compact and contained in . We define
Since each continuous linear functional can be represented uniquely in the form for some , then for each continuous linear functional , we have
that is, can be represented in the form
where . If such that
for every . Taking for , we have , then , that is, . Hence is uniquely determined. For fixed and any with compact support we have
where is uniquely determined. For any two sets and , the differential forms and coincide on because of the uniqueness of the differential form . Thus, all the differential forms , defined for different , are compatible with one another, and hence defines a differential form on . The differential form locally belongs to the space and satisfies
for every with compact support, and is uniquely determined. Let be a sequence of compact sets such that for and . Then
If such that
for every . Then for any , we have . Thus for any , that is, . Therefore, we conclude that each continuous linear functional can be uniquely represented in the form (2.72). (iii) We shall show with the constant dependent only on . We define a differential form on
then by () and (), we have