Stability of weak solutions to obstacle problem in Clifford analysis

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R E S E A R C H

Open Access

Stability of weak solutions to obstacle

problem in Clifford analysis

Yueming Lu

*

_{and Gejun Bao}

*_{Correspondence:}

[email protected]

Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, P.R. China

Abstract

This paper is concerned with the regularity properties of weak solutions to the obstacle problem for Cliﬀord-valued functions. Our main results are a global reverse Hölder inequality and stability of the weak solutions to the obstacle problem.

Keywords: Cliﬀord-valued function; obstacle problem; higher integrability; stability

1 Introduction

Lete,e, . . . ,en be the standard basis ofRnwith the relationeiej+ejei= –δij. Fork=

, , . . . ,n, we denote by Ck_n=Ck_n(Rn_{) the linear space of all} _k_{-multivectors, spanned}

by the reduced products eI = eiei· · ·eik, corresponding to all ordered k-tuples I = (i,i, . . . ,ik), ≤i <i<· · ·<ik≤n. Thus, Cliﬀord algebraCn=kCkn is a graded

algebra, especiallyC_n=RandC_n=Rn.R⊂C⊂H⊂C_n⊂ · · ·is an increasing chain, whereHis the Hamilton’s algebra of quaternions. Foru∈Cn,ucan be written as

u=

I uIeI=

≤i<···<ik≤n

ui,...,ikei· · ·eik,

where ≤k≤n.

The norm of u ∈Cn is given by |u| = (

IuI)/. Cliﬀord conjugation eα· · ·eαk = (–)keαk· · ·eα. Foru=

IuIeI∈Cn,v=

JvJeJ∈Cn,

u,v=

I

uIeI,

J vJeJ

=

I uIvI

deﬁnes the corresponding inner product onCn. Denote by Sc(u) the scalar part ofu, the

coeﬃcient of the elemente, and we also haveu,v= Sc(uv). The Dirac operator used in this paper is given by

D=

n

j=i ej

∂

∂xj. ()

Throughout this paper, we writeD(,Cn) for the space of Cliﬀord-valued functions

whose coefficients are Schwartz distributions on. Forp> , we writeLp(,Ck_n) for the space of Clifford-valued functionsu, whose coefficients belong to the usualLp₍_{) space.}

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It is a reﬂexive Banach space endowed with the norm

up= ˆ

|u|pdx 

p .

Also,W,p₍_,_Ck

n) is the Sobolev space of Cliﬀord-valued functionsu, whose coeﬃcients

belong toW,p₍_{). It is a reﬂexive Banach space endowed with the norm}

u,p=up+up,

whereu= (_∂∂_xu

, . . . ,

∂u

∂xn), ∂u

∂xi=

I

∂uI ∂xieI.

Letbe a bounded domain. This paper is concerned with theA-Dirac equation

DAp(x,Du) = , ()

where  <p<∞,Ap:×Cn→Cnpreserves the grading of the Cliﬀord algebra and

satisﬁes the following conditions for some constants  <a≤b<∞:

(i) The mappingx→Ap(x,ξ)is measurable for allξ∈Ck_n. (ii) The mappingξ→Ap(x,ξ)is continuous for a.e.x∈. (iii) |Ap(x,ξ)| ≤b|ξ|p–.

(iv) Ap(x,ξ) –Ap(x,η),ξ–η ≥a|ξ–η|p_{for all}_x_∈_with_ξ₌_η_.

Deﬁnition .[] A Cliﬀord-valued functionu∈W,p(,Ck_n) fork= , , . . . ,n, is a weak solution to equation () if for allϕ∈W,p₍_,_Ck

n) with compact support, we have ˆ

Ap(x,Du)Dϕ= . ()

In [, ], Nolder explained how quasi-linear elliptic equations

–divAp(x,∇u) =  ()

arise as components of Dirac systems () and discussed some properties of the weak so-lutions to the scalar parts of equations, such as the Caccioppoli estimate and the remov-ability theorem. In [], Heinonenet al.studied the quasi-linear elliptic equations () by means of potential theory systematically. Many other mathematicians also work on prop-erties of solutions to equations (), such as the regularity, stability, convergence and so on, see [–].

This paper is organized as follows. In Section , some preliminary results about Clifford-valued functions are presented. In Section , higher integrability of weak solutions to ob-stacle problem for Clifford-valued functions are obtained. Section  is concerned with the stability of the weak solutions to obstacle problem. For other works about Clifford analysis and Dirac equations, see [, , –].

2 Preliminary results

Lemma . C_∞(,Ck

n)is dense in W

,p

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Proof Let u=_IuI(x)eI ∈W_,p(,Ck

Here, we denote the Grassmann algebra=_kk,dis the exterior derivative opera-tor, andd∗is the formal adjoint operator. For more details about diﬀerential form, see []. From [], we know that the linear mapping

λ:ei∧ · · · ∧eik→ei· · ·eik (k≥)

n), there exists a constantC, such that

max du , d∗u ≤C|Du|. ()

In [], Iwaniec gave the following Poincaré inequality for diﬀerential forms.

Theorem . For each u_∈_C∞ (,

k

),there exits a constant C(n,p),  <p<∞such that _∇u_p≤C(n,p)du_p+d∗u_p. ()

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Proposition . For each u∈W_,p(,Ck

Similar to the process of the Ponicaré inequality for diﬀerential forms in [], the fol-lowing inequalities for Cliﬀord-valued function can be obtained.

Corollary . Let u∈W_,s(,Ck_n),  <s<∞,then,there exists a constant C such that

In this section, we will prove the higher integrability of the weak solutions to the obstacle problemKf_ψ,p(,Ck_n). Here, we assume that the complement ofsatisﬁes the measure density condition, it means that there exists a positive constantC>  such that

c∩B ≥C|B|, ()

whereBis a ball inRn_{. In order to prove the higher integrability, we need the following}

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Lemma . Let Bbe a ball inRn,and let g,h∈Lp(B)be a nonnegative function

satisfy-ing

B

gpdx≤c

B g dx

p

+ B

hqdx

()

for all balls B⊂B⊂B,  <p<∞.Then,for each  <σ < ,p<s<p+ _n–pp–_n_kp, we

have

σB gsdx



s

≤ n

σns( –_σ) n p

B gpdx



p +

B hsdx



s

. ()

Deﬁnition . For Cliﬀord-valued functionsa=_IaI(x)eI andb=_IbI(x)eI, we say thata≥binifaI(x)≥bI(x) for a.e.x∈and all ordered tupleI.

Let ψ = _IψIeI be a Cliﬀord-valued function in W,p(,Ckn), where ψI : →

[–∞, +∞). f ∈W,p₍_,_Ck

n) is a function, which gives the boundary. We consider the

obstacle problem

Kf,p

ψ

,Ck_n=v∈W,p,Ck_n:v–f ∈W_,p,Ck_n,v≥ψa.e. ()

for Cliﬀord-valued functions. To avoid trivialities, we always assume that the setKfψ,p(,

Ck

n) is not empty.

Deﬁnition . We say that a functionu∈W_loc,p(,Ck_n) is a weak solution to the obstacle problemKfψ,p(,Ckn), if

ˆ

Ap(x,Du),Dv–Du

dx≥, ()

wheneverv∈K_ψf,p(,Ck_n).

Remark . Ifuis a weak solution to obstacle problem (), thenuholds for the scalar part of (),i.e.,

ˆ

ScA(x,Du)Dϕdx= . ()

Remark . We assumef ≥ψin. Indeed, denotef =_IfIeI,ψ=

IψIeI. Then, since u–f ∈W,p(,Ckn),

≤(ψI–fI)+≤(u–fI)+.

Thus, we have (ψI–fI)+∈W,p() andfI= (ψI–fI)++fI≥ψIin. Letf =

IfIeI, then f ≥ψin.

Theorem . Suppose that the complement ofsatisﬁes the measure density condition

(),and let u∈W_loc,p(,Ck

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W,p+δ₍_,_Ck

a.e.. Thus,vis an admissible test function, and we have

≤

SinceAsatisﬁes (iii), (iv), combining the Hölder inequality and the Young inequality, it follows from () that

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whereθ=n+p_n–> . By means of condition (), we have|B∩| ≤( –C)|B|, then ()

Then, the following reverse Hölder estimate

B

Sinceis a bounded domain,can be covered by a ﬁnite number of balls such that the

previous inequality holds, then the estimates () follows immediately.

4 Stability

In this section, we will show that the weak solutions to obstacle problem are stable under some suitable assumptions.

Suppose that {pi} is a sequence such thatpi→pand for eachi= , , . . . , there is an

operatorApisatisfying (i)-(iv) forp=piand for a.e.x∈,

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uniformly on compact subset ofRn_{. We consider the obstacle problem}

Kf,pi ψ

,Ck_n=v∈W,pi_,_Ck

n

:v–f∈W,pi 

,Ck_n,v≥ψa.e.. () Assume thatApipreserves the grading of the Cliﬀord algebra and converges toAp(x,ξ) uniformly on compact subset ofRn_{. Suppose that}_u

i∈W,pi(,Ckn) is the weak solution

to the followingKf,pi

ψ (,Ckn)-obstacle problem ˆ

Api(x,Dui),Dv–Dui

dx≥ ()

for eachv∈Kf,pi

ψ (,Ckn) andu∈W,p(,Ckn) is the weak solution to ˆ

Ap(x,Du),Dv–Du

dx≥ ()

for allv∈Kfψ,p(,Ckn).

Deﬁnition . We call a sequenceuj∈Lp₍_,_C

n) converges weakly inLp(,Cn) touif ˆ

vujdx→ ˆ

vu dx, ()

wheneverv∈L

p p–₍_,_C

n). Denote it byuju.

Remark . Suppose thatuiweakly converges touinLp₍_,_Ck n), then ˆ

v,uidx→ ˆ

v,udx, ()

wheneverv∈W,p₍_,_Ck n).

Now, we start with the main result of this section.

Theorem . Suppose that the complement ofsatisﬁes the measure density condition

().Let pi,Api(x,ξ),Ap(x,ξ)be deﬁned as described above.Let the boundary value

func-tion f be in W,p+τ₍_,_Ck

n)for someτ> .Then,there is a small numberδsuch that the

sequence{ui}of weak solutions to the obstacle problem()has a subsequence,which con-verges to the weak solution uof()in W,p+δ(,Ckn)for anyδ<δ.

Before proving Theorem ., we need the following lemmas.

Lemma . [] Suppose that a bounded open set satisﬁes ().Let pi>  and ui ∈ W,pi₍_), _i_{= , , . . . ,}_{and suppose that pi}→_{p and ui}→_u

a.e in. Ifθ ∈W,p+(), > and if ui–θ∈W,pi

 ()with

ˆ

_∇(ui–θ) pi

dx≤M ()

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Lemma . Let uibe described in Theorem..There exists a constantγ> such that

for eachγ∈[,γ],

ˆ

|Dui|p+ γ

 _dx≤_C_, _()

where i is suﬃciently large and C is independent of i.

Proof By virtue of Remark ., we can usef as a test function in (), so that

ˆ

Api(x,Dui),Df –Dui

dx≥.

From the structure conditions onAiand the Hölder inequality, it follows

a ˆ

|Dui|pi_dx≤_b

ˆ

|Dui|pi_dx

pi–

pi ˆ

|Df|pi_dx



pi .

Then we have

ˆ

|Dui|pi_dx≤_C

ˆ

|Df|pi_dx_. _()

By the Hölder inequality and Young’s inequality, () becomes

ˆ

|Dui|pi_dx≤_C

||+

ˆ

|Df|p+σdx

, ()

wheniis suﬃciently large andσ> . According to Theorem ., there exists a constant

γ>  such that for eachγ ∈[,γ],

ˆ

|Dui|pi+γdx pi

pi+γ

≤C ˆ

|Dui|pi_dx₊

ˆ

|Df|pi+γdx pi

pi+γ

. ()

Sincepi→p, chooseisuﬃciently large such thatp–γ_ ≤pi≤p+γ_. Thenp+γ_ ≤p+γ. So, there exists a constant C, such that

ˆ

|Dui|p+γ _dx≤_C_.

Proof of Theorem. Letκ=_nn_–. Sinceui–f ∈W_,p(,Ck

n), we chooseiso large that p+γ_ ≤κpiandpi≤p+γ_, using Lemma ., we get

ui–fp+γ_ ≤Cui–fκpi

≤Cui–fnpi n–_pi

≤CDui–Dfpi

≤CDuip+_γ +Dfp+γ_

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Implied by the Minkowski inequality that

ui_p₊γ

 ≤ ui–fp+γ +fp+γ

≤CDui_p₊γ

 +Dfp+γ +fp+γ

.

Then it follows from Lemma . that

ui W,p+

γ

_(,_Ckn)≤C. ()

Writeδ=γ_,ui=_Iui_IeI, according to (), we haveui∈W,p+δ₍_,_Ck

n). So, we can

ex-tract a subsequence, still denote byui

I, such that

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

ui_IuI, inW,p+δ(),

ui

I→uI, inLp+δ(), ui

I→uI, pointwise a.e. in.

()

Then∇ui

I→ ∇uI. Letu=

IuIeI, thenui→uinLp+δ();ui→upointwise a.e. in.

Since∇ui

I∇uI, for eachj= , , . . . ,n, we have ˆ

yα ∂ui

I

∂xjdx→ ˆ

yα ∂uI

∂xj dx,

wheneveryα∈W,p(). Then,

´

yDuidx→

´

yDu dxfor eachy∈W,p(,Cn). This yieldsDuiDuinLp₍_,_C

n).

The next stage is to extract a further subsequence, so thatDui→Dupointwise almost everywhere in.

Because ofui–f ∈W,pi

 (,Ckn) andui–f →u–f, we getu–f ∈W

,pi

 (,Ckn) a.e. in

, that is to say,u∈Kf,pi

ψ (,Ckn). So, ˆ

Api(x,Dui),Du–Dui

dx≥. ()

Moreover, by the convergence assumption, we obtain

Api(x,Du)→Ap(x,Du) almost everywhere in.

From this, if follows thatui→uinW,p+δ₍_,_Ck

n) for allδ<δ.

At last, we will show thatu=u. Using Lemma (.), we getu–f ∈W,p(,Ckn), then u∈K_ψf,p(,Ck_n). This yields

ˆ

Ap(x,Du),Du–Du

dx≥. ()

On the other hand,

ˆ

Api(x,Dui),Du–Dui

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Leti→ ∞, sinceui→uinW,p₍_,_Ck

n), we get ˆ

Ap(x,Du),Du–Du

dx≥. ()

Combining () and (), we have

≤

ˆ

Ap(x,Du) –Ap(x,Du),Du–Du

dx≤.

It follows thatu=u. This proof is completed.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally in this paper. They read and approved the ﬁnal manuscript.

Acknowledgements

The authors would like to thank the anonymous referees for their time and thoughtful suggestions. The research is supported by the National Science Foundation of China (#11071048).

Received: 24 May 2013 Accepted: 6 August 2013 Published: 20 August 2013

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