Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 575819, 22 pages

doi:10.1155/2012/575819

Research Article

Variable Exponent Spaces of Differential Forms on Riemannian Manifold

Yongqiang Fu and Lifeng Guo

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 30 May 2012; Accepted 22 July 2012

Academic Editor: Alberto Fiorenza

Copyright © 2012 Yongqiang Fu and Lifeng Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 .

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, -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 [11]. 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 [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 -Laplacian Dirichlet problems with different types on bounded domains in 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 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 .

2. Preliminaries

     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 [25]).

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 [26]). 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 [27]). Using this identification, we can assume that each exterior form has a weak exterior differential .

Definition 2.1 (see [6]). 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 [27]), 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

and so

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 [28]), 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,

that is,

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

Obviously, .

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

Thus,

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

Moreover

Hence, we assert that .

Now we reach the conclusion , and hence is reflexive.

Theorem 2.14. If , then the exterior Sobolev space is a separable, reflexive Banach space.

Proof. We treat in a natural way as a subspace of the Cartesian product space . Then we need only to show that is a closed subspace of . Let be a convergent sequence. Then is a convergent sequence in . In view of Theorem 2.12, there exists such that in . Similarly there exists such that in . Then it is easy to see that converges to and converges to on in measure. For any , we have

Obviously, and , then integrals of the functions and possess absolutely equicontinuity on . Hence, by Vitali convergence theorem (see [28]), we get

Thus, we obtain that . Then it is immediate that is a closed subspace of .

Given two Banach spaces and , the symbol means that is continuously embedded in .

Theorem 2.15. Let . If and a.e. , then

The norm of the embedding operator (2.80) does not exceed .

Proof. Since a.e. , then . We may assume that with . Otherwise we can consider . By () we have , in particular, a.e. . Then we can write

Thus, we have . Therefore

Theorem 2.16. Let be a compact Riemannian manifold with a smooth boundary or without boundary and . Assume that

Then

is a continuous and compact embedding.

Proof. We consider an arbitrary local chart on . Let be any open set in , whose closure is compact and is contained in . Choosing a finite subcovering of such that is homeomorphic to the open unit ball of and for any the components of in satisfy as bilinear forms, where constant is given. Let be a smooth partition of unity subordinate to the finite covering . It is obvious that if , then and . By the definition of integral for differential -forms on and Sobolev embedding theorem in [16], we have the following continuous and compact embedding:

Since , we can assert that , and the embedding is continuous and compact.

Let , we say that is absolutely continuous with respect to the norm , if be a measurable subset, we have

Theorem 2.17. If , is absolutely continuous with respect to the norm .

Proof. By Lemma 2.9, there is a differential -form such that

Since is bounded, we can find such that when , the following inequalities hold

Hence, we get

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 that is a bounded domain with smooth boundary and .

The nonhomogeneous -harmonic equation for differential forms with variable growth on belong to the nonlinear elliptic equations which take the form

Definition 3.1. A differential form is a weak solution for the following Dirichlet problems

where , if satisfies

for every .

We are now ready to show an application of exterior Sobolev spaces to Dirichlet problems (3.2).

Let , be the dual space to and denote a dual between and . Consider the following functional:

We denote , then

where . Here,

Lemma 3.2. is a continuous, bounded, and strictly monotone operator.

Proof. It is obvious that is continuous and bounded. For any , we have the following inequalities (see [29]) from which we can get the strictly monotonicity of :

(), ().

Lemma 3.3. is a mapping of type , that is, if weakly in and , then strongly in .

Proof. By Lemma 3.2, if weakly in and , we have . In view of () and (), . Let and . Then there is a constant such that

Therefore, by (3.7)

Similar to the proof above, we can obtain

From Lemma 2.8, we have strongly in , that is, is a mapping of type .

Lemma 3.4. The mapping is coercive, that is,

Proof. Taking , we have

as . Similarly, we also obtain

Thus, for fixed constant , there exists such that

We take , if and , then

if and , then

Hence, , that is, the mapping is coercive.

Lemma 3.5. is a homeomorphism.

Proof. By Lemmas 3.2 and 3.4 and the theorem of Minty-Browder (see [30]), is a bijection. Hence has an inverse mapping . Therefore, the continuity of is sufficient to ensure to be a homeomorphism.

If and strongly in , let , , then and . As is coercive, we have is bounded in . Without loss of generality, we can assume that weakly in . Since strongly in , then

Since is a mapping of type , strongly in . By Lemma 3.2, we conclude that strongly in , so is continuous.

It is immediate to obtain the following conclusion from the above lemmas.

Theorem 3.6. If , then Dirichlet problems (3.2) has a unique weak solution in .

If , that is, is a function on , let be the gradient operator on . One has the following corollary.

Corollary 3.7. If , then Dirichlet problems

has a unique weak solution in .

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