- Research Article
- Open access
- Published:
The Obstacle Problem for the
-Harmonic Equation
Journal of Inequalities and Applications volume 2010, Article number: 767150 (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
.
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ3_HTML.gif)
Similarly denotes those
-forms on
with all coefficients in
. We denote the exterior derivative by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ4_HTML.gif)
and its formal adjoint operator (the Hodge codifferential operator)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ5_HTML.gif)
The operators and
are given by the formulas
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ7_HTML.gif)
Definition 2.3.
If a differential form satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ8_HTML.gif)
for any , then we say that
is a solution to (2.1). If for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ10_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ11_HTML.gif)
by Definition 2.3, it holds that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ12_HTML.gif)
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ13_HTML.gif)
Using in place of
, we also can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ14_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ16_HTML.gif)
The problem is to find a differential form in such that for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ17_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ18_HTML.gif)
Thus is a supersolution to (2.1) in
. Conversely, if
is a supersolution to (2.1) in
, then for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ19_HTML.gif)
Thus let , then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ20_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ21_HTML.gif)
By using in place of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ22_HTML.gif)
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ24_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ25_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ26_HTML.gif)
for all in
. Further,
is called coercive on
if there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ27_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ28_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ29_HTML.gif)
where and
. Denote that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ30_HTML.gif)
We define a mapping such that for any
, we have
. So for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ31_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ32_HTML.gif)
So for any , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ33_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ34_HTML.gif)
thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ36_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ37_HTML.gif)
According to Lemma 2.9 and the uniqueness of a limit of a convergence sequence, we only let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ38_HTML.gif)
Thus , so
is closed in
.
()
is monotone. Since operator
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ39_HTML.gif)
so for all , it holds that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ40_HTML.gif)
Thus is monotone.
()
is coercive on
. For any fixed
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ41_HTML.gif)
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ42_HTML.gif)
When and
, we can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ43_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ44_HTML.gif)
a.e. . Because
-norms of
are uniformly bounded, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ45_HTML.gif)
weakly in . Because the weak limit is independent of the choice of the subsequence, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ46_HTML.gif)
weakly in . Thus for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ47_HTML.gif)
Thus is weakly continuous on
.
By Lemma 2.10, we can find an element in
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ48_HTML.gif)
for any , that is to say, there exists
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ49_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ50_HTML.gif)
weakly in , that is to say,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ51_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ52_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F767150/MediaObjects/13660_2009_Article_2244_Equ53_HTML.gif)
So is solution to
-harmonic equation
in
with a boundary value
.
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Acknowledgment
This work is supported by the NSF of P.R. China (no. 10771044).
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Cao, Z., Bao, G. & Zhu, H. The Obstacle Problem for the -Harmonic Equation.
J Inequal Appl 2010, 767150 (2010). https://doi.org/10.1155/2010/767150
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DOI: https://doi.org/10.1155/2010/767150