- Research Article
- Open Access
The Obstacle Problem for the -Harmonic Equation
© Zhenhua Cao et al. 2010
- Received: 9 December 2009
- Accepted: 31 March 2010
- Published: 7 April 2010
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 .
- Convex Subset
- Differential Form
- Nonempty Closed Convex Subset
- Obstacle Problem
- Main Work
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 , 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 .
where and So we have
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  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 .
Suppose that and belong to , we say that if for any given , we have for all ordered -tuples , .
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.
then we say that is a supersolution (subsolution) to (2.1).
According to the above definition, we can get the following theorem.
A differential form is a solution to (2.1) if and only if is both supersolution and subsolution to (2.1).
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  for details.
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.
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 .
So is a solution to the obstacle problem of (2.1) in .
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 .
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 ).
whenever is a sequence in with .
By the definition of in , we can easily get the following lemma.
For any , we have and .
Lemma 2.10 (see ).
Using the same methods in [11, Appendix ], we can prove the existence and uniqueness of the solution to the obstacle problem of (2.1).
If is nonempty, then there exists a unique solution to the obstacle problem of (2.1) in .
Then we only prove that is a closed convex subset of and is monotone, coercive, and weakly continuous on .
So is convex.
Thus , so is closed in .
Thus is monotone.
Therefore is coercive 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.
for any .
So is solution to -harmonic equation in with a boundary value .
This work is supported by the NSF of P.R. China (no. 10771044).
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