R E S E A R C H
Open Access
Convergence of very weak solutions to
A
-Dirac equations in Clifford analysis
Yueming Lu
*and Hui Bi
*Correspondence:
ymlu.hit@gmail.com Department of Applied Mathematics, Harbin University of Science and Technology, Xue Fu Road, Harbin, 150080, P.R. China
Abstract
This paper is concerned with the very weak solutions toA-Dirac equations DA(x,Du) = 0 with Dirichlet boundary data. By means of the decomposition in a Clifford-valued function space, convergence of the very weak solutions to A-Dirac equations is obtained in Clifford analysis.
MSC: Primary 35J25; secondary 31A35
Keywords: Dirac equations; very weak solutions; decomposition; convergence
1 Introduction
In this paper, we shall consider a nonlinear mappingA:×Cn→Cnsuch that (N) x→A(x,ξ)is measurable for allξ∈Cn,
(N) ξ→A(x,ξ)is continuous for a.e.x∈,
(N) |A(x,ξ) –A(x,ζ)| ≤b|ξ–ζ|p–,
(N) (A(x,ξ) –A(x,ζ),ξ–ζ)≥a|ξ–ζ|(|ξ|+|ζ|)p–, where <a≤b<∞.
The exponentp> will determine the natural Sobolev class, denoted byW,p(,C n),
in which to consider theA-Dirac equations
DA(x,Du) = . ()
We callu∈Wloc,p(,Cn) a weak solution to () if
A(x,Du)Dϕdx= ()
for eachϕ∈Wloc,p(,Cn) with compact support.
Definition . Fors>max{,p– }, a Clifford-valued functionu∈Wloc,s(,Cn) is called a
very weak solution of equation () if it satisfies () for allϕ∈W,s–ps+(,C
n) with compact
support.
Remark . It is clear that ifs=p, the very weak solution is identity to the weak solution to equation ().
It is well known that theA-harmonic equations
–divA(x,∇u) = ()
arise in the study of nonlinear elastic mechanics. More exactly, by means of qualitative the-ory of solutions to (), we can study the above physics problems at equilibrium. Moreover, basic theories of () of degenerate condition have been studied by Iwaniec, Heinonenet al.systematically in [–]. While the regularity is not good enough, the existence of weak solutions to elliptic equations maybe not obtained in the corresponding function space, then the concept of ‘very weak solution’ is produced in order to study the solutions to elliptic equations in a wider space. Also, there are many researchers’ works on the prop-erties of the very weak solutions to various versions ofA-harmonic equations, see [–]. In , Gürlebeck and Sprößig studied the quaternionic analysis and elliptic boundary value problems in []; for more about Clifford analysis and its applications, see [–]. In , Nolder introduced theA-Dirac equations () and explained how the quasi-linear elliptic equations () arise as components of Dirac equations (). After that, Fu, Zhang, Bisciet al.studied this problem on the weighted variable exponent spaces, see [–]. Wang and Chen studied the relation betweenA-harmonic operator andA-Dirac system in []. In [], Lianet al.studied the weak solutions toA-Dirac equations in whole. For other works in this new field, we refer readers to [–].
This paper is concerned with the very weak solutions to a nonlinearA-Dirac equation with Dirichlet bound data
⎧ ⎨ ⎩
DA(x,Du) = ,
u–u∈W,s(,Cn).
()
We study the convergence of the very weak solutions to equation () without the homo-geneityA(x,λξ) =|λ|p–λA(x,ξ).
2 Preliminary results
Let e,e, . . . ,en be the standard basis ofRn with the relationeiej+ejei= –δij. For l=
, , . . . ,n, we denote byCk
n=Ckn(Rn) the linear space of allk-vectors, spanned by the
reduced productseI=eiei· · ·eik, corresponding to all orderedk-tuplesI= (i,i, . . . ,ik), ≤i<i<· · ·<ik≤n. Thus, Clifford algebraCn=⊕Cknis a graded algebra andCn=R
andCn=Rn.R⊂C⊂H⊂Cn⊂ · · ·is an increasing chain. 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|= (
IuI)/. Clifford conjugation eα· · ·eαk = (–)ke
αk· · ·eα. For eachI= (i,i, . . . ,ik), we have
Foru=IuIeI∈Cn,v=
JvJeJ∈Cn,
u,v=
I
uIeI,
J
vJeJ =
I
uIvI
defines the corresponding inner product onCn. Foru∈Cn,Sc(u) denotes the scalar
part ofu, that is, the coefficient of the elementu. Also we haveu,v=Sc(uv). The Dirac operator is given by
D=
n
j=
ej
∂ ∂xj
.
uis called a monogenic function ifDu= . AlsoD= –, whereis the Laplace operator which operates only on coefficients.
Throughout the paper, is a bounded domain.C∞(,Cn) is the space of
Clifford-valued functions inwhose coefficients belong toC∞(). Fors> , denote byLs(,C n)
the space of Clifford-valued functions inwhose coefficients belong to the usualLs()
space. Denoted by∇u= (∂x∂u
,
∂u ∂x, . . . ,
∂u
∂xn), thenW
,s(,C
n) is the space of Clifford-valued
functions in whose coefficients as well as their first distributional derivatives are in
Ls(). We similarly writeW,s
loc(,C) andW
,s
(,Cn).
LetG(x) = ω n
x
|x|n, the Teodorescu operator here is given by
Tf =
G(x–y)f(y) dy.
Next, we introduce the Borel-Prompieu result for a Clifford-valued function.
Theorem .([]) Ifis a domain inRn,then for each f ∈C∞
(,Cn),we have
f(z) =
∂
G(x–z) dσ(x)f(x) –
G(x–z)Df(x) dx, ()
where z∈.
According to Theorem ., Lian proved the following theorem.
Theorem .([]) (Poincaré inequality) For every u∈W,s(,Cn), <s<∞,there
ex-ists a constant c such that
|u|sdx≤c||n
|Du|sdx. ()
Fors> , we can find thatf =TDf whenf ∈W,s(,Cn). Also, we havef =DTf when
f ∈Ls(,C
n), see [].
Lemma .([]) Suppose Clifford-valued function u∈C∞(,Cn), <p<∞,then there
exists a constant c such that
∇(Tu)pdx≤c(n,p,)
Then we can easily get the following lemma.
Lemma . Let u be a Clifford-valued function in W,s(,Cn), <s<∞.Then there
exists a constant c such that
|∇u|sdx=
|∇TDu|sdx≤c
|Du|sdx. ()
Remark . From Theorem . and Lemma ., we know that, for Clifford-valued func-tionu∈W,p(,Cn), we have
|u|p+|∇u|pdx≤c(n,p,)
|Du|pdx.
3 Decomposition in Clifford-valued function space
In this section, we mainly discuss the properties of decomposition of Clifford-valued func-tions, these properties play an important role in studying the solutions ofA-Dirac equa-tions.
In [], Kähler gave the following decomposition for Clifford-valued function space
Ls(,C n):
Ls(,Cn) =
kerD∩Ls(,Cn)
⊕DW,s(,Cn). ()
This means that forω∈Ls(,C
n), there exist the uniquenessα∈Ls(,Cn)∩kerD,β∈
W,s(,Cn) such thatω=α+Dβ.
Let (X,μ) be a measure space and letEbe a separable complex Hilbert space. Consider a bounded linear operatorF:Ls(X,E)→Ls(X,E) for allr∈[s
,s], where ≤s<s≤ ∞. Denote its norm byFs.
Lemma .([]) Suppose that s
s ≤ +ε≤
s s.Then
F|f|εf
s
+ε≤K|ε|f
+ε
r ()
for each f ∈Ls(X,E)∩G,where
K= s(s–s) (s–s)(s–s)
Fs+Fs
.
Remark . For Clifford-valued functionω=α+Dβ∈W,s(,Cn), letFω=α, then we
haveF(Dω) = .
Proof From (), forDω∈Ls(,Cn), there exist
α∈kerD∩Ls(,Cn), β∈DW,s(,Cn)
such thatDω=α+Dβ, thenF(Dω) =α. So we haveDDω=Dα+DDβ, this means that
⎧ ⎨ ⎩
(ω–β) = , ω–β∈W,s(,Cn).
()
Lemma . For eachω∈W,s(,C
n),max{,p– } ≤s<p,there existμ∈W
,+sε
(,
Cn),π∈L
s
+ε(,Cn)such that
|Dω|εDω=Dμ+π, ()
and also
π s +ε =F
|Dω|εDω s
+ε ≤k|ε|Dω
+ε s ,
Dμ s
+ε =|Dω|
ε
Dω–π s
+ε ≤CDω
+ε s .
()
Proof We can get () from () immediately, so it is only needed to prove (). It follows by the definition of the operatorFthatF(Dω) = and
F:W,s(,Cn)→W,s(,Cn)
is a bounded linear mapping. And according to Lemma ., we have
π s +ε =F
|Dω|εDω s
+ε ≤k|ε|Dω
+ε s .
Then by Minkowski’s inequality, we get
Dμ s
+ε =|Dω|
εDω–π
s +ε
≤|Dω|εDω s +ε +
π s
+ε
≤cDω+s ε,
this completes the proof.
4 Main results
In this section, we will show the convergence of very weak solutions of (). Suppose that {u,j}, j= , , . . . , is the sequence converging to u in W,s(,Cn), and let uj ∈
Wloc,s(,Cn),j= , , . . . , be the very weak solutions of the boundary value problem
⎧ ⎨ ⎩
DA(x,Duj) = ,
uj–u,j∈W,s(,Cn),
()
wheremax{,p– } ≤s<p.
The main result of this section is the following theorem.
Theorem . Under the hypotheses above, for max{,p– } ≤s≤p, there exists u∈ W,s(,C
n)such that ujconverges to u in Wloc,s(,Cn)and u is a very weak solution to
equation()satisfying u–u∈W,s(,Cn).
We first discuss the non-homogeneousA-Dirac equations
DA(x,g+Dω) = , ()
Definition . The Clifford-valued functionω∈W,s(,C
n) with compact support.
Theorem . Let ω∈W,s(,C
n)be a very weak solution to (),then there exists a
constant c such that
Proof From Lemma ., we have the following decomposition:
|Dω|sDω=Dϕ+f, ()
Combining with (), it follows
Then, by Hölder, it yields
According to Lemma ., there is
And then, combining with Young’s inequality, we get
a
This proof is completed.
Corollary . Suppose that u∈W,s(,Cn),max{,p– } ≤s<p,u∈W,s(,Cn)is a
By means of Minkowski’s inequality, we obtain
This completes the proof.
whenjis sufficiently large. Using Lemma ., we have quence, still denoted by{uI
j}anduI∈W,s(), such that
The next stage is to extract a further subsequence, so thatDuj→Dupointwise a.e. in.
Through
Then it follows from the structure condition (N) that
that is to say Duj → Du a.e. in . Since ξ →A(x,ξ) is continuous, for each ϕ ∈
Next, we show that
n) with compact support.
At last, we show that u–u∈W,s(,Cn). Let u,j=
the theorem follows.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to express their gratitude to the editors and anonymous reviewers for their valuable suggestions which improved the presentation of this paper.
Received: 2 May 2015 Accepted: 24 June 2015
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