- Zhenhua Cao
^{1, 2}, - Gejun Bao
^{2}Email author and - Haijing Zhu
^{3}

**2010**:767150

https://doi.org/10.1155/2010/767150

© Zhenhua Cao et al. 2010

**Received: **9 December 2009

**Accepted: **31 March 2010

**Published: **7 April 2010

## Abstract

Firstly, we define an order for differential forms. Secondly, we also define the supersolution and subsolution of the -harmonic equation and the obstacle problems for differential forms which satisfy the -harmonic equation, and we obtain the relations between the solutions to -harmonic equation and the solution to the obstacle problem of the -harmonic equation. Finally, as an application of the obstacle problem, we prove the existence and uniqueness of the solution to the -harmonic equation on a bounded domain with a smooth boundary , where the -harmonic equation satisfies where is any given differential form which belongs to .

## Keywords

## 1. Introduction

Recently, a large amount of work about the -harmonic equation for the differential forms has been done. In 1999 Nolder gave some properties for the solution to the -harmonic equation in [1], and different versions of these properties had been established in [2–4]. The properties of the nonhomogeneous -harmonic equation have been discussed in [5–10]. In the above papers, we can think that the boundary values were zero. In this paper, we mainly discuss the existence and uniqueness of the solution to -harmonic equation with boundary values on a bounded domain .

Now let us see some notions and definitions about the -harmonic equation .

with the summation over all and all integers . And the norm of is given by .

## 2. The Obstacle Problem

In this section, we introduce the main work of this paper, which defining the supersolution and subsolution of the -harmonic equation and the obstacle problems for differential forms which satisfy the -harmonic equation, and the proof for the uniqueness of the solution to the obstacle problem of the -harmonic equations for differential forms. We can see this work about functions in [11, Chapter 3 and Appendix ] in detail. We use the similar methods in [11] to do the main work for differential forms.

We firstly give the comparison about differential forms according to the comparison's definition about functions in .

Definition 2.1.

Suppose that and belong to , we say that if for any given , we have for all ordered -tuples , .

Remark 2.2.

The above definition involves the order for differential forms which we have been trying to avoid giving. We know that many differential forms can not be compared based on the above definition since there are so many inequalities to be satisfied. However, at the moment, we can not replace this definition by another one and we are working on it now. We just started our research on the obstacle problem for differential forms satisfying the -harmonic equation and we hope that our work will stimulate further research in this direction.

Definition 2.3.

then we say that is a supersolution (subsolution) to (2.1).

According to the above definition, we can get the following theorem.

Theorem 2.4.

A differential form is a solution to (2.1) if and only if is both supersolution and subsolution to (2.1).

Proof.

Therefore is a solution to (2.1).

Next we will introduce the obstacle problem to -harmonic equation, whose definition is according to the same definition as the obstacle problem of quasilinear elliptic equation. For the obstacle problem of quasilinear elliptic equation we can see [11] for details.

Definition 2.5.

A differential form is called a solution to the obstacle problem of -harmonic equation (2.1) with obstacle and boundary values or a solution to the obstacle problem of -harmonic equation (2.1) in if satisfies (2.11) for any .

If , then we denote that We have some relations between the solution to quasilinear elliptic equation and the solution to obstacle problem in PDE. As to differential forms, we also have some relations between the solution to -harmonic equation and the solution to obstacle problem of -harmonic equation. We have the following two theorems.

Theorem 2.6.

If a differential form is a supersolution to (2.1), then is a solution to the obstacle problem of (2.1) in . For any , if is a solution to the obstacle problem of (2.1) in , then is a supersolution to (2.1) in .

Proof.

So is a solution to the obstacle problem of (2.1) in .

Theorem 2.7.

A differential form is a solution to (2.1) if and only if is a solution to the obstacle problem of (2.1) in with satisfying .

Proof.

Thus is a solution to (2.1) in .

So the theorem is proved.

The following we will discuss the existence and uniqueness of the solution to the obstacle problem of (2.1) in and the solution to (2.1). First we introduce a definition and two lemmas.

Definition 2.8 (see [11]).

whenever is a sequence in with .

By the definition of in [12], we can easily get the following lemma.

Lemma 2.9.

Lemma 2.10 (see [11]).

Using the same methods in [11, Appendix ], we can prove the existence and uniqueness of the solution to the obstacle problem of (2.1).

Theorem 2.11.

If is nonempty, then there exists a unique solution to the obstacle problem of (2.1) in .

Proof.

Then we only prove that is a closed convex subset of and is monotone, coercive, and weakly continuous on .

Thus is weakly continuous on .

for any . Then the theorem is proved.

By Theorem 2.7, we can see that the solution to the obstacle problem of (2.1) in is a solution of (2.1) in . Then by theorem, we can get the existence and uniqueness of the solution to -harmonic equation.

Corollary 2.12.

Proof.

So is solution to -harmonic equation in with a boundary value .

## Declarations

### Acknowledgment

This work is supported by the NSF of P.R. China (no. 10771044).

## Authors’ Affiliations

## References

- Nolder CA: Hardy-Littlewood theorems for -harmonic tensors.
*Illinois Journal of Mathematics*1999, 43(4):613–632.MathSciNetMATHGoogle Scholar - Ding S: Weighted Caccioppoli-type estimates and weak reverse Hölder inequalities for -harmonic tensors.
*Proceedings of the American Mathematical Society*1999, 127(9):2657–2664. 10.1090/S0002-9939-99-05285-5MathSciNetView ArticleMATHGoogle Scholar - Bao G: -weighted integral inequalities for -harmonic tensors.
*Journal of Mathematical Analysis and Applications*2000, 247(2):466–477. 10.1006/jmaa.2000.6851MathSciNetView ArticleMATHGoogle Scholar - Yuming X: Weighted integral inequalities for solutions of the -harmonic equation.
*Journal of Mathematical Analysis and Applications*2003, 279(1):350–363. 10.1016/S0022-247X(03)00036-2MathSciNetView ArticleMATHGoogle Scholar - Ding S: Two-weight Caccioppoli inequalities for solutions of nonhomogeneous -harmonic equations on Riemannian manifolds.
*Proceedings of the American Mathematical Society*2004, 132(8):2367–2395. 10.1090/S0002-9939-04-07347-2MathSciNetView ArticleMATHGoogle Scholar - D'Onofrio L, Iwaniec T: The -harmonic transform beyond its natural domain of definition.
*Indiana University Mathematics Journal*2004, 53(3):683–718. 10.1512/iumj.2004.53.2462MathSciNetView ArticleMATHGoogle Scholar - Ding S: Local and global norm comparison theorems for solutions to the nonhomogeneous -harmonic equation.
*Journal of Mathematical Analysis and Applications*2007, 335(2):1274–1293. 10.1016/j.jmaa.2007.02.048MathSciNetView ArticleMATHGoogle Scholar - Cao Z, Bao G, Li R, Zhu H: The reverse Hölder inequality for the solution to -harmonic type system.
*Journal of Inequalities and Applications*2008, 2008:-15.Google Scholar - Bao G, Cao Z, Li R: The Caccioppoli estimates for the solutions to -harmonic type equation.
*Dynamics of Continuous, Discrete & Impulsive Systems. Series A*2009, 16(S1):104–108.MathSciNetMATHGoogle Scholar - Cao Z, Bao G, Xing Y, Li R: Some Caccioppoli estimates for differential forms.
*Journal of Inequalities and Applications*2009, 2009:-11.Google Scholar - Heinonen J, Kilpeläìnen T, Martio O:
*Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs*. Oxford University Press, New York, NY, USA; 1993:vi+363.Google Scholar - Iwaniec T, Lutoborski A: Integral estimates for null Lagrangians.
*Archive for Rational Mechanics and Analysis*1993, 125(1):25–79. 10.1007/BF00411477MathSciNetView ArticleMATHGoogle Scholar

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