The Obstacle Problem for the -Harmonic Equation

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 .

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 .

Let denote the standard orthogonal basis of . For we denote by the linear space of all -vectors, spanned by the exterior product corresponding to all ordered -tuples , . The Grassmann algebra is a graded algebra with respect to the exterior products of and , then its inner product is obtained by

(1.1)

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

The Hodge star operator : is defined by the rule if , then

(1.2)

where and So we have

Throughout this paper, is an open subset, for any constant , denotes a cube such that , where denotes the cube whose center is as same as and . We say that is a differential -form on if every coefficient of is Schwartz distribution on . The space spanned by differential -form on is denoted by . We write for the -form on with for all ordered -tuple . Thus is a Banach space with the norm

(1.3)

Similarly denotes those -forms on with all coefficients in . We denote the exterior derivative by

(1.4)

and its formal adjoint operator (the Hodge codifferential operator)

(1.5)

The operators and are given by the formulas

(1.6)

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.

By the some definitions as the solution, supersolution (or subsolution) to quasilinear elliptic equation, we can give the definitions of the solution, supersolution (or subsolution) to -harmonic equation

(2.1)

Definition 2.3.

If a differential form satisfies

(2.2)

for any , then we say that is a solution to (2.1). If for any , we have

(2.3)

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

We can see that if is a subsolution to (2.1), then for , we have

(2.4)

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.

The sufficiency is obvious, we only prove the necessity. For any , we suppose that ,

(2.5)

by Definition 2.3, it holds that

(2.6)

So

(2.7)

Using in place of , we also can get

(2.8)

Thus

(2.9)

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.

Suppose that is a bounded domain. that is any differential form in which satisfies any that is function in with values in the extended reals , and . Let

(2.10)

The problem is to find a differential form in such that for any , we have

(2.11)

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.

If is a solution to the obstacle problem of (2.1) in , then for any , we have , so it holds that

(2.12)

Thus is a supersolution to (2.1) in . Conversely, if is a supersolution to (2.1) in , then for any , we have

(2.13)

Thus let , then we have

(2.14)

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.

If is a solution to the obstacle problem of (2.1) in , then for any , we have . So we can obtain

(2.15)

By using in place of , we have

(2.16)

So

(2.17)

Thus is a solution to (2.1) in .

Conversely, if is a solution to (2.1) in , then for any , we have Now let , then we have

(2.18)

Thus

(2.19)

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]).

Suppose that is a reflexive Banach space in with dual space , and let denote a pairing between and . If is a closed convex set, then a mapping is called monotone if

(2.20)

for all in . Further, is called coercive on if there exists such that

(2.21)

whenever is a sequence in with .

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

Lemma 2.9.

For any , we have and .

Lemma 2.10 (see [11]).

Let be a nonempty closed convex subset of and let be monotone, coercive, and weakly continuous on . Then there exists an element in such that

(2.22)

whenever .

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.

Let , then . Let

(2.23)

where and . Denote that

(2.24)

We define a mapping such that for any , we have . So for any , we have

(2.25)

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

() is convex. For any , we have such that

(2.26)

So for any , we have

(2.27)

Since

(2.28)

thus

(2.29)

So is convex.

() is closed in . Suppose that is a sequence converging to in . Then by the real functions' Poincaré inequality and Lemma 2.9, we have

(2.30)

Thus is a bounded sequence in . Because is a closed and convex subset of , we denote that and . Then for any in tuples, according to Theorems and in [11], we have a function such that

(2.31)

According to Lemma 2.9 and the uniqueness of a limit of a convergence sequence, we only let

(2.32)

Thus , so is closed in .

() is monotone. Since operator satisfies

(2.33)

so for all , it holds that

(2.34)

Thus is monotone.

() is coercive on . For any fixed , we have

(2.35)

So

(2.36)

When and , we can obtain

(2.37)

Therefore is coercive on .

(5) is weakly continuous on . Suppose that is a sequence that converge to on . Pick a subsequence such that a.e. in . Since the mapping is continuous for a.e. , we have

(2.38)

a.e. . Because -norms of are uniformly bounded, we have that

(2.39)

weakly in . Because the weak limit is independent of the choice of the subsequence, it follows that

(2.40)

weakly in . Thus for any , we have

(2.41)

Thus is weakly continuous on .

By Lemma 2.10, we can find an element in such that

(2.42)

for any , that is to say, there exists such that and

(2.43)

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.

Suppose that is a bounded domain with a smooth boundary and . There is a differential form such that

(2.44)

weakly in , that is to say,

(2.45)

for any .

Proof.

Let and be a solution to the obstacle problem of (2.1) in . For any , we have both and belong to . Then

(2.46)

Thus

(2.47)

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