The Obstacle Problem for the Harmonic Equation

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Journal of Inequalities and Applications Volume 2010, Article ID 767150,11pages doi:10.1155/2010/767150

Research Article

The Obstacle Problem for the

A

-Harmonic Equation

Zhenhua Cao,

1, 2

_{Gejun Bao,}

2

_{and Haijing Zhu}

3

1_{Department of Mathematics, Jiangxi Normal University, Nanchang 330022, China} 2_{Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China}

3_{College of Mathematics and Physics, Shandong Institute of Light Industry, Jinan 250353, China}

Correspondence should be addressed to Gejun Bao,[email protected]

Received 9 December 2009; Revised 26 March 2010; Accepted 31 March 2010

Academic Editor: Shusen Ding

Copyrightq2010 Zhenhua Cao et al. 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.

Firstly, we define an order for differential forms. Secondly, we also define the supersolution and subsolution of theA-harmonic equation and the obstacle problems for differential forms which satisfy theA-harmonic equation, and we obtain the relations between the solutions toA-harmonic equation and the solution to the obstacle problem of the A-harmonic equation. Finally, as an application of the obstacle problem, we prove the existence and uniqueness of the solution to theA-harmonic equation on a bounded domainΩwith a smooth boundary∂Ω, where theA -harmonic equation satisfiesd_A_{x, du} ₀_{, x}_∈_Ω_;_u_{ρ, x}_∈_∂_Ω_,_where_ρ_{is any given di}_ff_erential

form which belongs toW1,p_Ω_,_Λl−1_.

1. Introduction

Recently, a large amount of work about theA-harmonic equation for the diﬀerential forms has been done. In 1999 Nolder gave some properties for the solution to theA-harmonic equation in1 , and diﬀerent versions of these properties had been established in2–4 . The properties of the nonhomogeneousA-harmonic equation have been discussed in5–10 . 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 toA-harmonic equation with boundary values on a bounded domainΩ.

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

d_A_{x, du} _0.

Let e1, e2, . . . , en denote the standard orthogonal basis ofRn. For l 0,1, . . . , n,we denote byΛl _Λl_Rn_{the linear space of all}_l_{-vectors, spanned by the exterior product}_e

I

ei1∧ei2∧· · ·∧eilcorresponding to all orderedl-tuplesI i1, i2, . . . , il, 1≤i1< i2 <· · ·< il≤n.

The Grassmann algebraΛ ⊕Λl_{is a graded algebra with respect to the exterior products of}

ααIeI∈ΛandββIeI ∈Λ, then its inner product is obtained by

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with the summation over allI i1, i2, . . . , iland all integersl0,1, . . . , n. And the norm of

ααIeI∈Λis given by|α|α, α1/2.

The Hodge star operator :Λl _→ _Λn−l _{is defined by the rule if} _ω _ω IdxI

ωi1,i2,...,ildxi1∧dxi2· · · ∧dxil, then

ω −1IωIdxJ, 1.2

whereI ll1/2l_k₁ikandJ 1,2, . . . , n−I.So we have ω −1ln−lω. Throughout this paper,Ω⊂ Rn _{is an open subset, for any constant}_{σ >} _1,_Q_denotes a cube such thatQ ⊂ σQ ⊂ Ω, where σQdenotes the cube whose center is as same asQ

and diamσQ σdiamQ. We say thatααIeI ∈Λis a differentiall-form onΩif every coefficientαI of αis Schwartz distribution onΩ. The space spanned by differentiall-form on Ω is denoted by DΩ,Λl_{. We write} _Lp_Ω_,_Λl _{for the} _l_-form _α _α

IdxI onΩ with

αI ∈LpΩfor all orderedl-tupleI. ThusLpΩ,Λlis a Banach space with the norm

α_p,_Ω

Ω|α|

p_dx1/p

⎛

⎝

Ω

I

|αI|2

p/2

dx

⎞ ⎠

1/p

. 1.3

SimilarlyWk,pΩ,Λldenotes thosel-forms onΩwith all coeﬃcients inWk,pΩ. We denote the exterior derivative by

d:DΩ,Λl−→DΩ,Λl1 forl0,1,2, . . . , n 1.4

and its formal adjoint operatorthe Hodge codiﬀerential operator

d:DΩ,Λl−→DΩ,Λl−1. 1.5

The operatorsdandd_{are given by the formulas}

dα

I

dαI∧dxI, d −1nl1 d . _1.6

2. The Obstacle Problem

In this section, we introduce the main work of this paper, which defining the supersolution and subsolution of theA-harmonic equation and the obstacle problems for diﬀerential forms which satisfy theA-harmonic equation, and the proof for the uniqueness of the solution to the obstacle problem of theA-harmonic equations for diﬀerential forms. We can see this work about functions in11, Chapter 3 and Appendix I in detail. We use the similar methods in

11 to do the main work for diﬀerential forms.

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Definition 2.1. Suppose thatα_IαIxdxI andβ IβIxdxI belong toΛl, we say that

α ≥ β if for any given x, we haveαIx ≥ βIxfor all ordered l-tuplesI i1, i2, . . . , il, 1≤i1 < i2<· · ·< il≤n.

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 theA-harmonic equation and we hope that our work will stimulate further research in this direction.

By the some definitions as the solution, supersolutionor subsolutionto quasilinear elliptic equation, we can give the definitions of the solution, supersolutionor subsolution toA-harmonic equation

dAx, du 0. 2.1

Definition 2.3. If a diﬀerential formu∈W_loc1,pΩ,Λl−1_satisfies

Ω

Ax, du, dϕdx0, 2.2

for any ϕ ∈ W₀1,pΩ,Λl−1_{, then we say that} _u _{is a solution to} _2.1_{. If for any 0} _≤ _ϕ _∈

W_loc1,pΩ,Λl−1, we have

ΩAx, du, dϕdx≥0≤0, 2.3

then we say thatuis a supersolutionsubsolutionto2.1.

We can see that ifuis a subsolution to2.1, then for 0≥ϕ∈W₀1,pΩ,Λl−1_{, we have}

ΩAx, du, dϕdx≥0. 2.4

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

Theorem 2.4. A diﬀerential formu ∈ W_loc1,pΩ,Λl−1_{is a solution to}_2.1_{if and only if}_u_{is both}

supersolution and subsolution to2.1.

Proof. The suﬃciency is obvious, we only prove the necessity. For anyϕ∈W₀1,pΩ,Λl−1_{, we} suppose thatϕ_IϕIdxI,

ϕ1

I

ϕ_IdxI ≥0, ϕ2

I

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byDefinition 2.3, it holds that

ΩAx, du, dϕ1dx≥0,

ΩAx, du, dϕ2dx≥0. 2.6

So

0≤

ΩAx, du, dϕ1dx

ΩAx, du, dϕ2dx

ΩAx, du, dϕ1dϕ2dx

ΩAx, du, dϕdx.

2.7

Using−ϕin place ofϕ, we also can get

ΩAx, du, dϕdx≤0. 2.8

Thus

ΩAx, du, dϕdx0. 2.9

Thereforeuis a solution to2.1.

Next we will introduce the obstacle problem toA-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 see11 for details.

Suppose thatΩis a bounded domain. thatψ _IψIdxI is any diﬀerential form inΩ which satisfies anyψI that is function in Ωwith values in the extended reals−∞,∞ , and

ρ∈W1,p_Ω_,_Λl−1_{. Let}

Kψ,ρ

Ω,Λl−1v∈W1,pΩ,Λl−1:v≥ψ a.e., v−ρ∈W₀1,pΩ,Λl−1. 2.10

The problem is to find a diﬀerential form in Kψ,ρΩ,Λl−1 such that for any v ∈

Kψ,ρΩ,Λl−1, we have

ΩAx, dudv−u ≥0. 2.11

Definition 2.5. A diﬀerential formu∈ Kψ,ρΩ,Λl−1is called a solution to the obstacle problem of A-harmonic equation2.1with obstacle ψ and boundary valuesρ or a solution to the obstacle problem of A-harmonic equation2.1inKψ,ρΩ,Λl−1if usatisfies2.11for any

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Ifψ ρ, then we denote that Kψ,ψΩ,Λl−1 KψΩ,Λl−1.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 A -harmonic equation and the solution to obstacle problem ofA-harmonic equation. We have the following two theorems.

Theorem 2.6. If a diﬀerential formuis a supersolution to2.1, thenuis a solution to the obstacle problem of2.1inKψ,uΩ,Λl−1. For anyKψ,ρΩ,Λl−1, ifuis a solution to the obstacle problem of

2.1inKψ,ρΩ,Λl−1, thenuis a supersolution to2.1inΩ.

Proof. Ifuis a solution to the obstacle problem of2.1inKψ,ρΩ,Λl−1, then for any 0≤ϕ∈

W₀1,pΩ,Λl−1_{, we have}_v_u_ϕ_{∈ K}

ψ,ρΩ,Λl−1, so it holds that

ΩAx, du, dϕdx

ΩAx, du, dv−dudx≥0. 2.12

Thusuis a supersolution to2.1inΩ. Conversely, ifuis a supersolution to2.1inΩ, then for anyv∈ KuΩ,Λl−1, we have

v−u≥0, v−u∈W₀1,pΩ,Λl−1. 2.13

Thus letϕv−u, then we have

0≤

ΩAx, du, dϕdx

ΩAx, du, dv−dudx. 2.14

Souis a solution to the obstacle problem of2.1inKψ,uΩ,Λl−1.

Theorem 2.7. A diﬀerential formuis a solution to2.1if and only ifuis a solution to the obstacle problem of2.1inKψ,ρΩ,Λl−1withρsatisfyingu−ρ∈W01,pΩ,Λl−1.

Proof. If is a solution to the obstacle problem of 2.1 in Kψ,ρΩ,Λl−1, then for any ϕ ∈

W₀1,pΩ,Λl−1_{, we have}_v_u_ϕ_u₋_ρ_ρ_ϕ_{∈ K}

−∞,ρΩ,Λl−1. So we can obtain

ΩAx, du, dϕdx

ΩAx, du, dv−dudx≥0. 2.15

By using−ϕin place ofϕ, we have

ΩAx, du, d

−ϕdx

ΩAx, du, dv−dudx≥0. 2.16

So

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Conversely, ifuis a solution to2.1inΩ, then for anyv ∈ K−∞,ρΩ,Λl−1, we have

v−u∈W₀1,pΩ,Λl−1_._{Now let}_ϕ_v₋_u_{, then we have}

0

ΩAx, du, dϕdx

Thus

0≤

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 Kψ,ρΩ,Λl−1 and the solution to 2.1. First we introduce a definition and two lemmas.

Definition 2.8 see11 . Suppose that X is a reflexive Banach space in Ω with dual space

X, and let·,·denote a pairing betweenXandX. IfK ⊂ X is a closed convex set, then a mapping £ :K → Xis called monotone if

£u−£v, u−v≥0, 2.20

for alluvinK. Further, £ is called coercive onKif there existsϕ∈Ksuch that

£uj−£ϕ, uj−ϕ

uj−ϕ

−→ ∞, 2.21

wheneverujis a sequence inKwithuj → ∞.

By the definition of∇uin12 , we can easily get the following lemma.

Lemma 2.9. For anyu∈W1,pΩ,Λl, we have|du| ≤ |∇u|and|∇|u|| ≤ |∇u|.

Lemma 2.10see11 . LetKbe a nonempty closed convex subset ofX and let £ : K → Xbe monotone, coercive, and weakly continuous onK. Then there exists an elementuinKsuch that

£u, u−v≥0, 2.22

wheneverv∈K.

Using the same methods in 11, Appendix I , we can prove the existence and uniqueness of the solution to the obstacle problem of2.1.

Theorem 2.11. If Kψ,ρΩ,Λl−1 is nonempty, then there exists a unique solution to the obstacle

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Proof. LetXLp_Ω_,_Λl_{, then}_X_Lp/p−1_Ω_,_Λl_{. Let}

f, g

Ωf, gdx, 2.23

wheref∈Lp_Ω_,_Λl_and_g _∈_Lp/p−1_Ω_,_Λl_{. Denote that}

Kdv:v∈ Kψ,ρ

Ω,Λl−1_. _2.24

We define a mapping £ :K → Xsuch that for anyv∈K, we have £vAx, v. So for any

u∈Lp_Ω_,_Λl_{, we have}

£v, u

ΩAx, v, udx. 2.25

Then we only prove that Kis a closed convex subset ofX and £ : K → X is monotone, coercive, and weakly continuous onK.

1Kis convex. For anyx1, x2∈K, we havev1, v2∈ Kψ,ρΩ,Λl−1such that

x1dv1, x2dv2. 2.26

So for anyt∈0,1, we have

tx1 1−tx2tdv1 1−tdv2dtv1 1−tv2. 2.27

Since

tv1 1−tv2−ρt

v1−ρ

1−tv2−ρ

∈ Kψ,ρ

Ω,Λl−1_, _2.28

thus

tx1 1−tx2∈K. 2.29

SoKis convex.

2Kis closed inX. Suppose thatdvi∈Kis a sequence converging tovinX. Then by the real functions’ Poincar´e inequality andLemma 2.9, we have

Ω

vi−ρpdx≤cdiamΩp

Ω

_∇vi−ρpdx

≤cdiamΩp

Ω

_∇_v_i_{− ∇}_ρp

dx≤M <∞.

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Thusviis a bounded sequence inW1,pΩ,Λl−1. BecauseKψ,ρΩ,Λl−1is a closed and convex subset ofW1,p_Ω_,_Λl−1_{, we denote that}_v

i IvIidxI andρ

IρIdxI. Then for anyI in

l−1 tuples, according to Theorems 1.30 and 1.31 in11 , we have a functionvI_{such that}

vI_i −→vI weakly, vI−ρI ∈W₀1,pΩ, ∇vI_i −→ ∇vI

∂vI

∂x1, . . . ,

∂vI

∂xn

weakly.

2.31

According toLemma 2.9and the uniqueness of a limit of a convergence sequence, we only let v I n

i1

∂vI

∂xi

dxi∧dxI. 2.32

Thusv∈K, soKis closed inX.

3£ is monotone. Since operatorAsatisfies

Ax, ξ1−Ax, ξ2, ξ1−ξ2 ≥0, 2.33

so for allu, v∈K, it holds that

£u−£v, u−v

ΩAx, u−Ax, v, u−vdx≥0. 2.34

Thus £ is monotone.

4£ is coercive onK. For any fixedϕ∈K, we have

£u−£ϕ, u−ϕ

Ω

Ax, u−Ax, ϕ, u−ϕdx

ΩAx, u, udx

Ω

Ax, ϕ, ϕdx−

Ω

Ax, u, ϕdx

−

Ω

Ax, ϕ, udx

≥K−1

Ω|u|

p_dx_K−1

Ω _ϕp

dx−K

Ω|u|

p−1_ϕ_dx₋

Ω _ϕp−1_|

u|dx

≥K−1upϕp−Kup−1ϕuϕp−1

≥K−12−pu−ϕu−ϕp−1−K2p−1ϕϕp−1u−ϕp−1

−Kϕp−1ϕu−ϕ.

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So

_u_j₋_ϕ ≥K−12−puj−ϕp−1−K2p−1ϕ

ϕp−1

_u_j₋_ϕuj−ϕp−2

−Kϕp−1 ϕ

uj−ϕ

1

.

2.36

Whenuj → ∞anduj−ϕ → ∞, we can obtain

uj−ϕ

−→ ∞. 2.37

Therefore £ is coercive onK.

5£ is weakly continuous onK. Suppose thatui ∈Kis a sequence that converge to

u∈KonX. Pick a subsequenceuijsuch thatuij → ua.e. inΩ. Since the mappingξ → Ax, ξ

is continuous for a.e.x, we have

Ax, uij

−→Ax, u, 2.38

a.e.x∈Ω. BecauseLp/p−1_Ω_,_Λl_{-norms of}_A_{x, u}

ijare uniformly bounded, we have that

Ax, uij

−→Ax, u 2.39

weakly in Lp/p−1_Ω_,_Λl_{. Because the weak limit is independent of the choice of the} subsequence, it follows that

Ax, ui−→Ax, u 2.40

weakly inLp/p−1_Ω_,_Λl_{. Thus for any}_v_∈_Lp_Ω_,_Λl_{, we have}

£ui, v

Ω£ui, vdx−→

Ω£u, vdx £u, v. 2.41

Thus £ is weakly continuous onK.

ByLemma 2.10, we can find an elementuinKsuch that

£u, v−u≥0, 2.42

for anyv∈K, that is to say, there existsu∈ Kψ,ρΩ,Λl−1such thatduuand

ΩAx, du, dv−dudx £du, dv−du≥0, 2.43

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By Theorem 2.7, we can see that the solution u to the obstacle problem of 2.1 in

K−∞,ρΩ,Λl−1is a solution of 2.1 inΩ. Then by theorem, we can get the existence and uniqueness of the solution toA-harmonic equation.

Corollary 2.12. Suppose that Ω is a bounded domain with a smooth boundary ∂Ω and ρ ∈ W1,p_Ω_,_Λl−1_{. There is a di}_ﬀ_{erential form}_u_∈_W1,p_Ω_,_Λl−1_{such that}

d_A_{x, du} ₀_, _x_∈_Ω_,

uρ, x∈∂Ω 2.44

weakly inΩ, that is to say,

Ω

Ax, du, dϕdx0, 2.45

for anyϕ∈W₀1,pΩ,Λl−1_.

Proof. Letψ −∞andube a solution to the obstacle problem of2.1inKψ,ρΩ,Λl−1. For anyϕ∈W₀1,pΩ,Λl−1, we have bothuϕandu−ϕbelong toKψ,ρΩ,Λl−1. Then

ΩAx, du, dϕdx≥0, −

ΩAx, du, dϕdx≥0. 2.46

Thus

Souis solution toA-harmonic equationd_A_{x, du} _{0 in}_Ω_{with a boundary value}_ρ_.

Acknowledgment

This work is supported by the NSF of P.R. Chinano. 10771044.

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