 Research
 Open access
 Published:
AHarmonic operator in the Dirac system
Journal of Inequalities and Applications volume 2013, Article number: 463 (2013)
Abstract
In this paper, we show how an Aharmonic operator arises from ADirac systems under natural growth condition. By the method of removable singularities for solutions to ADirac systems with natural growth conditions, we establish the fact that Aharmonic equations are the real part of the corresponding ADirac systems.
1 Introduction
In this paper, we derive the relation between an Aharmonic operator and ADirac systems under natural growth condition. The equation defined by an Aharmonic operator in the current paper is
where
x\to A(x,\xi ) is measurable for all ξ, and \xi \to A(x,\xi ) is continuous for a.e. x\in \mathrm{\Omega}. Further assume that A(x,\xi ) satisfies the following structure growth conditions for some constants a>0 and \lambda >0, with exponent p>1:
and f(x,\xi ) satisfies the following natural growth condition for certain constants {C}_{1} and {C}_{2}:
The exponent p represents an exponent throughout the paper.
Definition 1.1 We call the function u\in {W}_{\mathrm{loc}}^{1,p}(\mathrm{\Omega})\cap {L}^{\mathrm{\infty}}(\mathrm{\Omega}) a weak solution to (1.1) under structure growth conditions (1.3) and (1.4) if the equality
holds for all \varphi \in {W}^{1,p}(\mathrm{\Omega}) with compact support.
The ADirac systems in the paper are of the form
The main purpose of this paper is to show that under natural growth condition, an equation defined by an Aharmonic operator is the real part of the corresponding ADirac system. The properties of the Dirac operator are stated in Section 2. Further discussion on nonlinear Dirac equations can be found in [1–9] 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 an Aharmonic operator satisfying the Lipschitz condition or of bounded mean oscillation extends to Cliffordvalued solutions to corresponding Dirac equations.
The method of removability theorems was introduced in [1] for monogenic functions with modulus of continuity \omega (r), in which the sets of {r}^{n}\omega (r)Hausdorff measure are removable. The results were extended to Hölder continuous analytic functions by Kaufman and Wu in [10]. Then, the sets satisfying a certain geometric condition related to Minkowski dimension were shown to be removable for Aharmonic functions in Hölder and bounded mean oscillation classes in [11]. In the case of Hölder continuity, this was sharpened in [12] to a precise condition for removable sets for Aharmonic functions in terms of Hausdorff dimension. In [3], the sets satisfying a generalized Minkowskitype inequality were removable for solutions to the ADirac equation which satisfy a certain oscillation condition.
In this paper, we show that the above results even hold for inhomogeneity equations defined by an Aharmonic operator and ADirac systems, and we establish that under natural growth condition, the removable theorem holds for solutions to the corresponding ADirac equations and the main result can be stated as follows.
Theorem 1.2 Let E be a relatively closed subset of Ω. Suppose that u\in {L}_{\mathrm{loc}}^{p}(\mathrm{\Omega}) has distributional first derivatives in Ω, u is a solution to the scalar part of ADirac system (1.6) under natural growth condition in \mathrm{\Omega}\setminus E, and u is of p,koscillation in \mathrm{\Omega}\setminus E. If for each compact subset K of E,
then u extends to a solution of the ADirac equation in Ω.
2 ADirac operator and correspondence definition
In this section, we introduce an ADirac system. In order to definite the ADirac operator, we present the definitions and notations of Clifford algebra at first [3].
We write {\mathcal{U}}_{n} for the real universal Clifford algebra over {\mathbb{R}}^{n}. The Clifford algebra is generated over R by the basis of reduced products
where \{{e}_{1},{e}_{2},\dots ,{e}_{n}\} is an orthonormal basis of {\mathbb{R}}^{n} with the relation {e}_{i}{e}_{j}+{e}_{j}{e}_{i}=2{\delta}_{ij}. We write {e}_{0} for the identity. The dimension of {\mathcal{U}}_{n} is {\mathbb{R}}^{{2}^{n}}. We have an increasing tower \mathbb{R}\subset \mathbb{C}\subset \mathbb{H}\subset \mathcal{U}\subset \cdots . The Clifford algebra {\mathcal{U}}_{n} is graded algebra as {\mathcal{U}}_{n}={\u2a01}_{l}{\mathcal{U}}_{n}^{l}, where {\mathcal{U}}_{n}^{l} are those elements whose reduced Clifford products have length l.
For A\in {\mathcal{U}}_{n}, Sc(A) denotes the scalar part of A, that is, the coefficient of the element {e}_{0}.
Throughout, \mathrm{\Omega}\subset \mathbb{R} is a connected and open set with boundary ∂ Ω. A Cliffordvalued function u:\mathrm{\Omega}\to {\mathcal{U}}_{n} can be written as u={\sum}_{\alpha}{u}_{\alpha}{e}_{\alpha}, where each {u}_{\alpha} is realvalued and {e}_{\alpha} are reduced products. The norm used here is given by {\sum}_{\alpha}{u}_{\alpha}{e}_{\alpha}={({\sum}_{\alpha}{u}_{\alpha}^{2})}^{\frac{1}{2}}. This norm is submultiplicative, AB\le CAB.
The Dirac operator used here is
Also, {D}^{2}=\mathrm{\u25b3}. Here △ is the Laplace operator which operates only on coefficients. A function is monogenic when Du=0.
Throughout, Q is a cube in Ω with volume Q. We write σQ for the cube with the same center as Q and with side length σ times that of Q. For q>0, we write {L}^{q}(\mathrm{\Omega},{\mathcal{U}}_{n}) for the space of Cliffordvalued functions in Ω whose coefficients belong to the usual {L}^{q}(\mathrm{\Omega}) space. Also, {W}^{1,p}(\mathrm{\Omega},{\mathcal{U}}_{n}) is the space of Cliffordvalued functions in Ω whose coefficients as well as their first distributional derivatives are in {L}^{q}(\mathrm{\Omega}). We also write {L}_{\mathrm{loc}}^{q}(\mathrm{\Omega},{\mathcal{U}}_{n}) for \bigcap {L}^{q}({\mathrm{\Omega}}^{\prime},{\mathcal{U}}_{n}), where the intersection is over all {\mathrm{\Omega}}^{\prime} compactly contained in Ω. We similarly write {W}_{\mathrm{loc}}^{1,p}(\mathrm{\Omega},{\mathcal{U}}_{n}). Moreover, we write {\mathcal{M}}_{\mathrm{\Omega}}=\{u:\mathrm{\Omega}\to {\mathcal{U}}_{n}Du=0\} for the space of monogenic functions in Ω.
Furthermore, we define the DiracSobolev space
The local space {W}_{\mathrm{loc}}^{D,p} is similarly defined. Notice that if u is monogenic, then u\in {L}^{p}(\mathrm{\Omega}) if and only if u\in {W}^{D,p}(\mathrm{\Omega}). Also, it is immediate that {W}^{1,p}(\mathrm{\Omega})\subset {W}^{D,p}(\mathrm{\Omega}).
Under such definitions and notations, we can introduce the operator of ADirac. Define the linear isomorphism \theta :{\mathbb{R}}^{n}\to {\mathcal{U}}_{n}^{1} by
For x,y\in {\mathbb{R}}^{n}, with Du defined by \theta (\mathrm{\nabla}\varphi )=D\varphi for a realvalued function ϕ, we have
Here, \tilde{A}(x,\xi ):\mathrm{\Omega}\times {\mathcal{U}}_{1}\to {\mathcal{U}}_{1} is defined by
which means that (1.5) is equivalent to
For the Clifford conjugation \overline{({e}_{j1}\cdots {e}_{jl})}={(1)}^{l}{e}_{jl}\cdots {e}_{j1}, we define a Cliffordvalued inner product as \overline{\alpha}\beta. Moreover, the scalar part of this Clifford inner product Re(\overline{\alpha}\beta ) is the usual inner product in {\mathbb{R}}^{{2}^{n}}, \u3008\alpha ,\beta \u3009, when α and β are identified as vectors.
For convenience, we replace \tilde{A} with A and recast the structure equations above and define the operators:
where A preserves the grading of the Clifford algebra, x\to A(x,\xi ) is measurable for all ξ, and \xi \to A(x,\xi ) is continuous for a.e. x\in \mathrm{\Omega}. Furthermore, here A(x,\xi ) satisfies the structure conditions with p>1:
for some constant a>0. Hence we can definite the weak solution of equation (1.6) as follows.
Definition 2.1 A Cliffordvalued function u\in {W}_{\mathrm{loc}}^{D\cdot p}(\mathrm{\Omega},{\mathcal{U}}_{n}^{k}), for k=0,1,\dots ,n, is a weak solution to equation (1.6) under structure conditions (2.10) and (2.11), and further assume that the inhomogeneity term f(x,Du) satisfies the natural growth condition
for certain constants {a}_{1} and {a}_{2} and \lambda >0. Then for all \varphi \in {W}^{1,p}(\mathrm{\Omega},{\mathcal{U}}_{n}^{k}) with compact support, we have
Notice that when A is the identity, then the homogeneity part of (2.13),
is the Clifford Laplacian. Moreover, these equations generalize the important case of the PDirac equation
Here, A(x,\xi )={\xi }^{p2}\xi.
These equations were introduced and their conformal invariance was studied in [2].
Furthermore, when u is a realvalued function, (2.14) implies that A(x,\mathrm{\nabla}u) is a harmonic field and locally there exists a harmonic function H such that A(x,\mathrm{\nabla}u)=\mathrm{\nabla}H. If A(x,\xi ) is invertible, then \mathrm{\nabla}u={A}^{1}(x,\mathrm{\nabla}H). Hence regularity of A implies regularity of the solution u.
In general, Aharmonic functions do not have such regularity. This suggests the study of the scalar part of system equation (2.13) in general. Thus, a Caccioppoli estimate for solutions to the scalar part of (2.13) is necessary. This is the topic of the next section.
3 The proof of main results
In this section, we establish the main results. At first, a suitable Caccioppoli estimate for solutions to (2.13) is necessary. Just as which appears in [3, 4].
Theorem 3.1 Let u be a solution to the scalar part of (2.13) defined by Definition 2.1, and Q is a cube in Ω. Then, for \sigma >1, we have
Proof Let the cutoff function \eta \in {C}_{0}^{\mathrm{\infty}}(\mathrm{\Omega}), \eta >0, \eta =1 in Q and D\eta \le C{Q}^{1/n}. Choose \varphi =(u{u}_{\sigma Q}){\eta}^{p} as a test function in (2.13). Then
The solution definition (2.13) yields
Using the structure condition (2.10), we have
By natural growth condition (2.12) and then using Young’s inequality, we have
Notice that
Then, using (2.11), Hölder’s and then Young’s inequalities, we have
Combining (3.4), (3.5) with (3.6) in inequality (3.3), we obtain
Choosing {\epsilon}_{2} small enough such that \lambda {a}_{1}M{\epsilon}_{2}>0, we have
This completes the proof of Theorem 3.1. □
In order to remove singularity of solutions to the ADirac system, we also need the fact that realvalued functions satisfy various regularity properties. Thus we have Definition 5.1 [3].
Definition 3.2 Assume that u\in {L}_{\mathrm{loc}}^{1}(\mathrm{\Omega},{\mathcal{U}}_{n}), q>0 and that \mathrm{\infty}<k<1. We say that u is of q,koscillation in Ω when
The infimum over monogenic functions is natural since they are trivially solutions to an ADirac equation just as constants are solutions to an Aharmonic equation. If u is a function and q=1, then (3.7) is equivalent to the usual definition of the bounded mean oscillation when k=0 and (3.7) is equivalent to the usual local Lipschitz condition when 0<k\le 1 [13]. Moreover, at least when u is a solution to an Aharmonic equation, (3.7) is equivalent to a local order of growth condition when \mathrm{\infty}<k<1 [4, 14]. In these cases, the supremum is finite if we choose {u}_{Q} to be the average value of the function u over the cube Q. It is easy to see that in condition (3.7) the expansion factor ‘2’ can be replaced by any factor greater than ‘1’.
If the coefficients of an ADirac solution u are of bounded mean oscillation, local Hölder continuous, or of a certain local order of growth, then u is in an appropriate oscillation class [4].
Notice that monogenic functions satisfy (3.7) just as the space∖of constants is a subspace of the bounded mean oscillation and Lipschitz spaces of realvalued functions.
We remark that it follows from Hölder’s inequality that if s\le q and if u is of q,koscillation, then u is of s,koscillation.
The following lemma [3] shows that Definition 3.2 is independent of the expansion factor of the sphere.
Lemma 3.3 Suppose that F\in {L}_{\mathrm{loc}}^{1}(\mathrm{\Omega},\mathbb{R}), F>0 a.e., r\in \mathbb{R} and {\sigma}_{1},{\sigma}_{2}>1. If
then
Then we proceed to prove the main result, Theorem 1.2.
Proof of Theorem 1.2 Let Q be a cube in the Whitney decomposition of \mathrm{\Omega}\setminus E. The decomposition consists of closed dyadic cubes with disjoint interiors which satisfy

(a)
\mathrm{\Omega}\setminus E={\bigcup}_{Q\in \mathcal{W}}Q,

(b)
{Q}^{1/n}\le d(Q,\partial \mathrm{\Omega})\le 4{Q}^{1/n},

(c)
(1/4){{Q}_{1}}^{1/n}\le {{Q}_{2}}^{1/n}\le 4{{Q}_{1}}^{1/n} when {Q}_{1}\cap {Q}_{2} is not empty.
Here, d(Q,\partial \mathrm{\Omega}) is the Euclidean distance between Q and the boundary of Ω [15].
If A\subset {\mathbb{R}}^{n} and r>0, then we define the rinflation of A as
Using the Caccioppoli estimate and the p,koscillation condition, we have
Here, a=(n+pkp)/n and note that \mathrm{\infty}<k\le 1. Since the problem is local (use a partition of unity), we show that (2.13) holds whenever \varphi \in {W}_{0}^{1,p}(B({x}_{0},r)) with {x}_{0}\in E and r>0 sufficiently small. Choose r=(1/5\sqrt{n})min\{1,d({x}_{0},\partial \mathrm{\Omega})\}, and let K=E\cap \overline{B}({x}_{0},4r). Then K is a compact subset of E. Also, let {W}_{0} be those cubes in the Whitney decomposition of \mathrm{\Omega}\setminus E which meet B=B({x}_{0},r). Notice that each cube Q\in {W}_{0} lies in K(1)\setminus K. Let \gamma =p(k1)k. First, since \gamma \ge 1, from [2] we have m(K)=m(E)=0. Also, since an\ge \gamma, using (3.6) and (3.11), we obtain
Hence u\in {W}_{\mathrm{loc}}^{D,p}(\mathrm{\Omega}).
Next, let B=B({x}_{0},r) and assume that \psi \in {C}_{0}^{\mathrm{\infty}}(B). Also, let {W}_{j}, j=1,2,\dots , be those cubes Q\in {W}_{0} with l(Q)\le {2}^{j}.
Consider the scalar functions
Thus each {\varphi}_{j}, j=1,2,\dots , is Lipschitz, equal to 1 on K and as such \psi (1{\varphi}_{j})\in {W}^{1,p}(B\setminus E) with compact support. Hence
Let
Since u is a solution in B\setminus E, {J}_{1}=0.
Also, we have
Now there exists a constant C such that \psi \le C<\mathrm{\infty}. Hence, using Hölder’s inequality,
Next, using the estimate (3.11), the above becomes
Now, for x\in Q\in {W}_{j}, d(x,K) is bounded above and below by a multiple of {Q}^{1/n} and for Q\in {W}_{j}, {Q}^{1/n}\le {2}^{j}. Hence
Since \bigcup {W}_{j}\subset K(1)\setminus K and \bigcup {W}_{j}\to 0 as j\to \mathrm{\infty}, it follows that {J}_{2}^{\prime}\to 0 as j\to \mathrm{\infty}.
Next, again using Hölder’s inequality, we obtain
Since u\in {W}_{\mathrm{loc}}^{1,D}(\mathrm{\Omega}) and \bigcup {W}_{j}\to 0 as j\to \mathrm{\infty}, we have that {J}_{2}^{\u2033}\to 0 as j\to \mathrm{\infty}. Hence {J}_{2}\to 0.
In order to estimate {J}_{2}^{\u2034}, using (3.12), we have
Similar as the estimate of (3.18), using the Caccioppoli inequality and inequality (3.11), we get
and
Hence {J}_{3}\to 0.
Combining the estimates {J}_{1} and {J}_{2} in equation (3.14), we prove Theorem 1.2. □
References
AbreuBlaya R, BoryReyes J, PeñaPeña D: Jump problem and removable singularities for monogenic functions. J. Geom. Anal. 2007, 17(1):1–13. 10.1007/BF02922079
Nolder CA, Ryan J: p Dirac operators. Adv. Appl. Clifford Algebras 2009, 19(2):391–402. 10.1007/s0000600901627
Nolder CA: A Harmonic equations and the Dirac operator. J. Inequal. Appl. 2010., 2010: Article ID 124018
Nolder CA: Nonlinear A Dirac equation. Adv. Appl. Clifford Algebras 2011, 21(2):429–440. 10.1007/s0000601002535
Wang C: A remark on nonlinear Dirac equations. Proc. Am. Math. Soc. 2010, 138(10):3753–3758. 10.1090/S0002993910104389
Wang C, Xu D: Remark on Dirac harmonic maps. Int. Math. Res. Not. 2009, 20: 3759–3792.
Chen Q, Jost J, Li J, Wang G: Diracharmonic maps. Math. Z. 2006, 254(2):409–432. 10.1007/s0020900609617
Chen Q, Jost J, Li J, Wang G: Regularity theorems and energy identities for Diracharmonic maps. Math. Z. 2005, 251(1):61–84. 10.1007/s0020900507887
Chen Q, Jost J, Wang G: Nonlinear Dirac equations on Riemann surfaces. Ann. Glob. Anal. Geom. 2008, 33(3):253–270. 10.1007/s1045500790846
Kaufman R, Wu JM: Removable singularities for analytic or subharmonic functions. Ark. Mat. 1980, 18(1):107–116.
Koskela P, Martio O: Removability theorems for solutions of degenerate elliptic partial differential equations. Ark. Mat. 1993, 31(2):339–353. 10.1007/BF02559490
Kilpelainen T, Zhong X: Removable sets for continuous solutions of quasilinear elliptic equations. Proc. Am. Math. Soc. 2002, 130(6):1681–1688. 10.1090/S0002993901062372
Meyers NG: Mean oscillation over cubes and Hölder continuity. Proc. Am. Math. Soc. 1964, 15(5):717–721.
Langmeyer N: The quasihyperbolic metric, growth, and John domains. Ann. Acad. Sci. Fenn. Math. 1998, 23(1):205–224.
Stein EM Princeton Mathematical Series 30. In Singular Integrals and Differentiablity Properties of Functions. Princeton University Press, Princeton; 1970.
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Sun, F., Chen, S. AHarmonic operator in the Dirac system. J Inequal Appl 2013, 463 (2013). https://doi.org/10.1186/1029242X2013463
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X2013463