R E S E A R C H
Open Access
A
-Harmonic operator in the Dirac system
Fengfeng Sun and Shuhong Chen
**Correspondence:
[email protected] School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, Fujian 363000, China
Abstract
In this paper, we show how anA-harmonic operator arises fromA-Dirac systems under natural growth condition. By the method of removable singularities for solutions toA-Dirac systems with natural growth conditions, we establish the fact that
A-harmonic equations are the real part of the correspondingA-Dirac systems.
Keywords: A-harmonic operator;A-Dirac system; Caccioppoli estimate; natural growth condition
1 Introduction
In this paper, we derive the relation between anA-harmonic operator andA-Dirac systems
under natural growth condition. The equation defined by anA-harmonic operator in the
current paper is
–divA(x,∇u) =f(x,∇u), (.)
where
A(x,ξ) :×Rn→Rn, (.)
x→A(x,ξ) is measurable for allξ, andξ→A(x,ξ) is continuous for a.e.x∈. Further assume thatA(x,ξ) satisfies the following structure growth conditions for some constants
a> andλ> , with exponentp> :
A(x,ξ),ξ≥λ|ξ|p, A(x,ξ)≤a|ξ|p–, (.)
andf(x,ξ) satisfies the following natural growth condition for certain constantsCandC:
f(x,∇u)≤C|∇u|p+C, |u|<M< +∞andλ>CM. (.)
The exponentprepresents an exponent throughout the paper.
Definition . We call the functionu∈Wloc,p()∩L∞() a weak solution to (.) under structure growth conditions (.) and (.) if the equality
A(x,∇u),∇φ= –
f(x,∇u),φ (.)
holds for allφ∈W,p() with compact support.
TheA-Dirac systems in the paper are of the form
–DA˜(x,Du) =f(x,Du). (.)
The main purpose of this paper is to show that under natural growth condition, an
equa-tion defined by anA-harmonic operator is the real part of the corresponding A-Dirac
system. The properties of the Dirac operator are stated in Section . Further discussion on nonlinear Dirac equations can be found in [–] and references therein.
In order to obtain the desired results, we prove that under natural growth condition, a
result concerning removable singularities for equations defined by anA-harmonic
opera-tor satisfying the Lipschitz condition or of bounded mean oscillation extends to Clifford-valued solutions to corresponding Dirac equations.
The method of removability theorems was introduced in [] for monogenic functions with modulus of continuityω(r), in which the sets ofrnω(r)-Hausdorff measure are remov-able. The results were extended to Hölder continuous analytic functions by Kaufman and Wu in []. Then, the sets satisfying a certain geometric condition related to Minkowski
dimension were shown to be removable forA-harmonic functions in Hölder and bounded
mean oscillation classes in []. In the case of Hölder continuity, this was sharpened in [] to a precise condition for removable sets forA-harmonic functions in terms of Hausdorff dimension. In [], the sets satisfying a generalized Minkowski-type inequality were remov-able for solutions to theA-Dirac equation which satisfy a certain oscillation condition.
In this paper, we show that the above results even hold for inhomogeneity equations
defined by anA-harmonic operator andA-Dirac systems, and we establish that under
natural growth condition, the removable theorem holds for solutions to the corresponding
A-Dirac equations and the main result can be stated as follows.
Theorem . Let E be a relatively closed subset of.Suppose that u∈Lploc()has dis-tributional first derivatives in,u is a solution to the scalar part of A-Dirac system(.)
under natural growth condition in\E,and u is of p,k-oscillation in\E.If for each compact subset K of E,
K()\Kd(x,K)
p(k–)–k<∞, (.)
then u extends to a solution of the A-Dirac equation in.
2 A-Dirac operator and correspondence definition
In this section, we introduce anA-Dirac system. In order to definite theA-Dirac operator, we present the definitions and notations of Clifford algebra at first [].
We writeUnfor the real universal Clifford algebra overRn. The Clifford algebra is gen-erated overRby the basis of reduced products
{e,e, . . . ,ee, . . . ,e· · ·en}, (.)
where{e,e, . . . ,en}is an orthonormal basis ofRnwith the relationeiej+ejei= –δij. We writeefor the identity. The dimension ofUnisR
n
C⊂H⊂U⊂ · · ·. The Clifford algebraUnis graded algebra asUn=
lUnl, whereUnl are
those elements whose reduced Clifford products have lengthl.
ForA∈Un,Sc(A) denotes the scalar part ofA, that is, the coefficient of the elemente.
Throughout,⊂Ris a connected and open set with boundary∂. A Clifford-valued
functionu:→Uncan be written asu=
αuαeα, where eachuαis real-valued andeα
are reduced products. The norm used here is given by|αuαeα|= (
αuα)
. This norm
is sub-multiplicative,|AB| ≤C|A||B|. The Dirac operator used here is
D= n
j=
ej ∂ ∂xj
. (.)
Also,D= –. Hereis the Laplace operator which operates only on coefficients. A
func-tion is monogenic whenDu= .
Throughout, Qis a cube in with volume|Q|. We writeσQfor the cube with the
same center asQand with side lengthσ times that ofQ. Forq> , we writeLq(,Un) for
the space of Clifford-valued functions inwhose coefficients belong to the usualLq()
space. Also,W,p(,Un) is the space of Clifford-valued functions inwhose coefficients as well as their first distributional derivatives are inLq(). We also writeLq
loc(,Un) for
Lq(,U
n), where the intersection is over allcompactly contained in . We
simi-larly writeWloc,p(,Un). Moreover, we writeM={u:→Un|Du= }for the space of
monogenic functions in.
Furthermore, we define the Dirac-Sobolev space
WD,p() =
u∈Un
|u|p+
|Du|p<∞
. (.)
The local spaceWlocD,pis similarly defined. Notice that ifuis monogenic, thenu∈Lp() if and only ifu∈WD,p(). Also, it is immediate thatW,p()⊂WD,p().
Under such definitions and notations, we can introduce the operator ofA-Dirac. Define
the linear isomorphismθ:Rn→Unby
θ(ω, . . . ,ωn) = n
i=
ωiei. (.)
Forx,y∈Rn, withDudefined byθ(∇φ) =Dφfor a real-valued functionφ, we have
–Reθ(x)θ(y)=x,y, (.)
θ(x)=|x|. (.)
Here,A˜(x,ξ) :×U→Uis defined by
˜
A(x,ξ) =θAx,θ–ξ, (.)
which means that (.) is equivalent to
ReθA(x,∇u)θ(∇φ)dx=
ReA˜(x,Du)Dφdx=
For the Clifford conjugation (ej· · ·ejl) = (–)lejl· · ·ej, we define a Clifford-valued inner
product asαβ¯ . Moreover, the scalar part of this Clifford inner productRe(αβ¯ ) is the usual inner product inRn,α,β, whenαandβare identified as vectors.
For convenience, we replace A˜ withA and recast the structure equations above and
define the operators:
A(x,ξ) :×Un→Un, (.)
whereApreserves the grading of the Clifford algebra,x→A(x,ξ) is measurable for all ξ, and ξ →A(x,ξ) is continuous for a.e. x∈. Furthermore, here A(x,ξ) satisfies the structure conditions withp> :
ReA(x,ξ)ξ≥λ|ξ|p, (.)
A(x,ξ)≤a|ξ|p– (.)
for some constanta> . Hence we can definite the weak solution of equation (.) as
follows.
Definition . A Clifford-valued functionu∈WlocD·p(,Unk), fork= , , . . . ,n, is a weak solution to equation (.) under structure conditions (.) and (.), and further assume that the inhomogeneity termf(x,Du) satisfies the natural growth condition
f(x,Du)≤a|Du|p+a, |u|<M< +∞andλ>aM (.)
for certain constantsaandaandλ> . Then for allφ∈W,p(,Unk) with compact sup-port, we have
A(x,Du)Dφdx=
f(x,Du)φdx. (.)
Notice that whenAis the identity, then the homogeneity part of (.),
A(x,Du)Dφdx= , (.)
is the Clifford Laplacian. Moreover, these equations generalize the important case of the
P-Dirac equation
D|Du|n–Du= . (.)
Here,A(x,ξ) =|ξ|p–ξ.
3 The proof of main results
In this section, we establish the main results. At first, a suitable Caccioppoli estimate for solutions to (.) is necessary. Just as which appears in [, ].
Theorem . Let u be a solution to the scalar part of(.)defined by Definition.,and Q is a cube in.Then,forσ> ,we have
Q
|Du|pdx≤C|Q|–pn
σQ
|u–uσQ|pdx+C|Q| n(p–)+p
n(p–) . (.)
Proof Let the cut-off functionη∈C∞(),η> ,η= inQand|Dη| ≤C|Q|–/n. Choose φ= (u–uσQ)ηpas a test function in (.). Then
Dφ=pηp–(Dη)(u–uσQ) +ηpDu. The solution definition (.) yields
σQ
A(x,Du)pηp–(Dη)(u–uσQ)dx+
σQ
A(x,Du)ηpDu dx
=
σQ
f(x,Du)(u–uσQ)ηpdx. (.)
Using the structure condition (.), we have
λ
σQ
|Du|pηpdx
≤
σQ
f(x,Du)(u–uσQ)ηpdx+
σQ
A(x,Du)pηp–(Dη)(u–uσQ)dx
=I+I. (.)
By natural growth condition (.) and then using Young’s inequality, we have
I≤
σQ
a|Du|p(u–uσQ)ηpdx+
σQ
aηp(u–uσQ)dx ≤aM
σQ|
Du|pηpdx+
σQ
u–uσQ |Q|n
a|Q|
nηpdx
≤aM
σQ
|Du|pηpdx+ε
σQ
(u–uσQ)p |Q|pn
dx+C(ε)
σQ
a|Q|
nηp
p
p–dx. (.)
Notice that
σQ
a|Q|
nηp
p
p–dx≤C
|Q|
n(p–)+p
n(p–) . (.)
Then, using (.), Hölder’s and then Young’s inequalities, we have
|I| ≤pa
σQ
|u–uσQ||∇η||Du|p–|η|p–dx
≤C
σQ|
u–uσQ|p|∇η|pdx
p
σQ|
Du|pηpdx
≤C(ε)
σQ
|u–uσQ|p|∇η|pdx
p·p
+ε
σQ
|Du|pηpdx
≤C
σQ
|u–uσQ|p|Q|–pndx+ε
σQ
|Du|pηpdx. (.)
Combining (.), (.) with (.) in inequality (.), we obtain
(λ–aM–ε)
|Du|p|η|pdx
≤C+C(ε)
|Q|–pn
σQ
|u–uσQ|pdx+εC|Q|
n(p–)+p n(p–) .
Choosingεsmall enough such thatλ–aM–ε> , we have
Q|
Du|pdx≤C|Q|–pn
σQ|
u–uσQ|pdx+C|Q| n(p–)+p
n(p–) .
This completes the proof of Theorem ..
In order to remove singularity of solutions to the A-Dirac system, we also need the
fact that real-valued functions satisfy various regularity properties. Thus we have Defi-nition . [].
Definition . Assume thatu∈L
loc(,Un),q> and that –∞<k< . We say thatuis of
q,k-oscillation inwhen
sup
Q⊂
|Q|–(qk+n)/qn inf uQ∈MQ
Q
|u–uQ|q
/q
<∞. (.)
The infimum over monogenic functions is natural since they are trivially solutions to anA-Dirac equation just as constants are solutions to anA-harmonic equation. Ifuis a
function andq= , then (.) is equivalent to the usual definition of the bounded mean
oscillation whenk= and (.) is equivalent to the usual local Lipschitz condition when <k≤ []. Moreover, at least whenuis a solution to anA-harmonic equation, (.) is equivalent to a local order of growth condition when –∞<k< [, ]. In these cases, the supremum is finite if we chooseuQto be the average value of the functionuover the cubeQ. It is easy to see that in condition (.) the expansion factor ‘’ can be replaced by any factor greater than ‘’.
If the coefficients of anA-Dirac solutionuare of bounded mean oscillation, local Hölder continuous, or of a certain local order of growth, thenuis in an appropriate oscillation class [].
Notice that monogenic functions satisfy (.) just as the space\of constants is a subspace of the bounded mean oscillation and Lipschitz spaces of real-valued functions.
We remark that it follows from Hölder’s inequality that if s≤q and if u is of q,k -oscillation, thenuis ofs,k-oscillation.
Lemma . Suppose that F∈L
loc(,R),F> a.e.,r∈Randσ,σ> .If
sup
σQ⊂Q
|Q|r
Q
F dx<∞, (.)
then
sup
σQ⊂Q
|Q|r
Q
F dx<∞. (.)
Then we proceed to prove the main result, Theorem ..
Proof of Theorem. LetQbe a cube in the Whitney decomposition of\E. The de-composition consists of closed dyadic cubes with disjoint interiors which satisfy
(a) \E=Q∈WQ,
(b) |Q|/n≤d(Q,∂)≤|Q|/n,
(c) (/)|Q|/n≤ |Q|/n≤|Q|/nwhenQ∩Qis not empty.
Here,d(Q,∂) is the Euclidean distance betweenQand the boundary of[]. IfA⊂Rnandr> , then we define ther-inflation ofAas
A(r) =B(x,r). (.)
Using the Caccioppoli estimate and thep,k-oscillation condition, we have
Q
|Du|pdx≤C|Q|–pn
σQ
|u–uσQ|pdx+C|Q| n(p–)+p
n(p–)
≤C inf uQ∈MσQ
|Q|–pn
σQ
|u–uσQ|pdx+C|Q| n(p–)+p
n(p–)
≤C|Q|–pn· |Q| pk+n
n +C|Q| n(p–)+p
n(p–) ≤C|Q|a. (.)
Here,a= (n+pk–p)/nand note that –∞<k≤. Since the problem is local (use a parti-tion of unity), we show that (.) holds wheneverφ∈W,p(B(x,r)) withx∈Eandr>
sufficiently small. Chooser= (/√n)min{,d(x,∂)}, and letK=E∩ ¯B(x, r). ThenKis
a compact subset ofE. Also, letWbe those cubes in the Whitney decomposition of\E
which meetB=B(x,r). Notice that each cubeQ∈Wlies inK()\K. Letγ=p(k– ) –k.
First, sinceγ ≥–, from [] we havem(K) =m(E) = . Also, sincea–n≥γ, using (.) and (.), we obtain
B(x,r)
|Du|pdx≤C
Q∈W
|Q|a/n≤C
Q∈W
d(Q,K)a
≤C
Q∈W
Q
d(x,K)a–ndx≤C
K()\Kd(x,K) a–ndx
≤C
K()\Kd(x,K)
γdx<∞. (.)
Henceu∈WlocD,p().
Next, letB=B(x,r) and assume thatψ∈C∞(B). Also, letWj,j= , , . . . , be those cubes
Consider the scalar functions
φj=max
–j–d(x,K)j, . (.)
Thus eachφj,j= , , . . . , is Lipschitz, equal to onKand as suchψ( –φj)∈W,p(B\E) with compact support. Hence
B
A(x,Du)Dψ–f(x,Du)ψdx
=
B\E
A(x,Du)Dψ( –φj)–f(x,Du)ψ( –φj)dx
+
B
A(x,Du)D(ψ φj) –f(x,Du)ψ φj
dx. (.)
Let
J=
B\E
A(x,Du)Dψ( –φj)
–f(x,Du)ψ( –φj)
dx,
J=
B
A(x,Du)D(ψ φj) –f(x,Du)ψ φj
dx.
Sinceuis a solution inB\E,J= .
Also, we have
J=
B
A(x,Du)ψDφjdx+
B
φjA(x,Du)Dψdx–
B
f(x,Du)ψ φjdx
=J+J+J. (.)
Now there exists a constantCsuch that|ψ| ≤C<∞. Hence, using Hölder’s inequality,
J≤C
Q∈Wj
Q
A(X,Du)|Dφj|dx
≤C
Q∈Wj
Q
|Du|p–|Dφj|dx
≤C
Q∈Wj
Q
|Du|pdx
(p–)/p
Q
|Dφj|pdx
/p
. (.)
Next, using the estimate (.), the above becomes J≤C
Q∈Wj
|Q|(p(k–)+n)(p–)/npj|Q|/p. (.)
Now, forx∈Q∈Wj,d(x,K) is bounded above and below by a multiple of|Q|/nand for
Q∈Wj,|Q|/n≤–j. Hence J≤C
Q∈Wj
|Q|–/n+/p+(p(k–)+n)(p–)/np≤C
Wj
d(x,K)p(k–)–kdx. (.)
SinceWj⊂K()\Kand|
Next, again using Hölder’s inequality, we obtain
J≤sup B
|Dψ|
Wj
|Du|pdx
(p–)/p Wj
/p
≤C
K\K()|
Du|pdx
(p–)/p Wj
/p
. (.)
Sinceu∈Wloc,D() and|Wj| → asj→ ∞, we have thatJ→ asj→ ∞. Hence
J→.
In order to estimateJ, using (.), we have
J=
B
f(x,Du)ψ φjdx≤C
B
|Du|pdx+C
B
dx=J+J. (.)
Similar as the estimate of (.), using the Caccioppoli inequality and inequality (.), we get
J ≤ C
Q∈Wj
Q|
Du|pdx≤C
Q∈Wj
|Q|(n+pk–p)/n
≤ C
Wj
d(x,K)n+pk–pdx≤C
Wj
d(x,K)p(k–)–kdx. → (j→ ∞)
and
J ≤ C
Q∈Wj
Q
dx=C
Q∈Wj
|Q|
≤ C
Wj
d(x,K)ndx≤C
Wj
d(x,K)n+pk–pdx
≤ C
Wj
d(x,K)p(k–)–kdx → (j→ ∞).
HenceJ→.
Combining the estimatesJandJin equation (.), we prove Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
FS participated in design of the study and drafted the manuscript. SC participated in conceived of the study and the amendment of the paper. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 11201415), Natural Science Foundation of Fujian Province (2012J01027) and Training Programme Foundation for Excellent Youth Researching Talents of Fujian’s Universities (JA12205).
References
1. Abreu-Blaya, R, Bory-Reyes, J, Peña-Peña, D: Jump problem and removable singularities for monogenic functions. J. Geom. Anal.17(1), 1-13 (2007)
2. Nolder, CA, Ryan, J:p-Dirac operators. Adv. Appl. Clifford Algebras19(2), 391-402 (2009)
3. Nolder, CA:A-Harmonic equations and the Dirac operator. J. Inequal. Appl.2010, Article ID 124018 (2010) 4. Nolder, CA: NonlinearA-Dirac equation. Adv. Appl. Clifford Algebras21(2), 429-440 (2011)
5. Wang, C: A remark on nonlinear Dirac equations. Proc. Am. Math. Soc.138(10), 3753-3758 (2010) 6. Wang, C, Xu, D: Remark on Dirac harmonic maps. Int. Math. Res. Not.20, 3759-3792 (2009) 7. Chen, Q, Jost, J, Li, J, Wang, G: Dirac-harmonic maps. Math. Z.254(2), 409-432 (2006)
8. Chen, Q, Jost, J, Li, J, Wang, G: Regularity theorems and energy identities for Dirac-harmonic maps. Math. Z.251(1), 61-84 (2005)
9. Chen, Q, Jost, J, Wang, G: Nonlinear Dirac equations on Riemann surfaces. Ann. Glob. Anal. Geom.33(3), 253-270 (2008)
10. Kaufman, R, Wu, JM: Removable singularities for analytic or subharmonic functions. Ark. Mat.18(1), 107-116 (1980) 11. Koskela, P, Martio, O: Removability theorems for solutions of degenerate elliptic partial differential equations. Ark.
Mat.31(2), 339-353 (1993)
12. Kilpelainen, T, Zhong, X: Removable sets for continuous solutions of quasilinear elliptic equations. Proc. Am. Math. Soc.130(6), 1681-1688 (2002)
13. Meyers, NG: Mean oscillation over cubes and Hölder continuity. Proc. Am. Math. Soc.15(5), 717-721 (1964) 14. Langmeyer, N: The quasihyperbolic metric, growth, and John domains. Ann. Acad. Sci. Fenn. Math.23(1), 205-224
(1998)
15. Stein, EM: Singular Integrals and Differentiablity Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)
10.1186/1029-242X-2013-463
Cite this article as:Sun and Chen:A-Harmonic operator in the Dirac system.Journal of Inequalities and Applications
