- Open Access
Regularity theory on A-harmonic system and A-Dirac system
© Sun and Chen; licensee Springer. 2014
- Received: 27 February 2014
- Accepted: 23 October 2014
- Published: 4 November 2014
In this paper, we show the regularity theory on an A-harmonic system and an A-Dirac system. By the method of the removability theorem, we explain how an A-harmonic system arises from an A-Dirac system and establish that an A-harmonic system is in fact the real part of the corresponding A-Dirac system.
- A-harmonic system
- A-Dirac system
- Caccioppoli estimate
- natural growth condition
- removable theorem
Here Ω is a bounded domain in (), and are measurable functions defined on , N is an integer with , is a vector valued function. Furthermore, and satisfy the following structural conditions with :
for all , , and . Without loss of generality, we take .
for all , and , where is a constant.
holds for all with compact support.
In this paper, we assume that the solutions of the A-harmonic system (1.1) and the A-Dirac system (1.2) exist  and establish the regularity result directly. In other words, the main purpose of this paper is to show the regularity theory on an A-harmonic system and the corresponding A-Dirac system. It means that we should know the properties of an A-harmonic operator and an A-Dirac operator. This main context will be stated in Section 2. Further discussion can be found in [2–10] and the references therein.
In order to prove the main result, we also need a suitable Caccioppoli estimation (see Theorem 3.1). Then by the technique of removable singularities, we can find that solutions to an A-harmonic system satisfying a Lipschitz condition or in the case of a bounded mean oscillation can be extended to Clifford valued solutions to the corresponding A-Dirac system.
The technique of removable singularities was used in  to remove singularities for monogenic functions with modulus of continuity , where the sets and Hausdorff measure are removable. Kaufman and Wu  used the method in the case of Hölder continuous analytic functions. In fact, under a certain geometric condition related to the Minkowski dimension, sets can be removable for A-harmonic functions in Hölder and bounded mean oscillation classes . Even in the case of Hölder continuity, a precise removable sets condition was stated . In , the author showed that under a certain oscillation condition, sets satisfying a generalized Minkowski-type inequality were removable for solutions to the A-Dirac system. The general result can be found in .
Motivated by these facts, one ask: Does a similar result hold for the more general case of the systems (1.1) and (1.2)? We will answer this question in this paper and obtain the following result.
then u extends to a solution of the A-Dirac system in Ω.
In this section, we would introduce the A-Dirac system. Thus the definition of the A-Dirac operator is necessary. We first present the definitions and notations as regards the Clifford algebra at first .
where is an orthonormal basis of with the relation . We write for the identity. The dimension of is . We have an increasing tower . The Clifford algebra is a graded algebra as , where are those elements whose reduced Clifford products have length l.
For , denotes the scalar part of A, that is, the coefficient of the element .
Throughout this paper, is a connected and open set with boundary ∂ Ω. A Clifford-valued function can be written as , where each is real-valued and are reduced products. The norm used here is given by , which is sub-multiplicative, .
Also . Here △ is Laplace operator.
Throughout, Q is a cube in Ω with volume . We write σQ for the cube with the same center as Q and with side length σ times that of Q. For , we write for the space of Clifford-valued functions in Ω whose coefficients belong to the usual space. Also, is the space of Clifford valued functions in Ω whose coefficients as well as their first distributional derivatives are in . We also write for , where the intersection is over all compactly contained in Ω. We similarly write . Moreover, we write for the space of monogenic functions in Ω.
The local space is similarly defined. Notice that if u is monogenic, then if and only if . Also it is immediate that .
For the Clifford conjugation , we define a Clifford-valued inner product as . Moreover, the scalar part of this Clifford inner product is the usual inner product in , when α and β are identified as vectors.
where A preserves the grading of the Clifford algebra, is measurable for all ξ, η, and , are continuous for a.e. .
after using (1.3), (H3), (H4).
Now let the domain of the left-hand side be Q, then we can get the right inequality immediately. □
In order to remove singularity of solutions to A-Dirac system, we also need the fact that real-valued functions satisfying various regularity properties. Thus we have the following.
Definition 3.2 
If and , then the inequality (3.4) is equivalent to the usual definition of the bounded mean oscillation; when and , then the inequality (3.4) is equivalent to the usual local Lipschitz condition . Further discussion of the inequality (3.4) can be found in [8, 17]. In these cases, the supremum is finite if we choose to be the average value of the function u over the cube Q.
We remark that it follows from Hölder’s inequality that if and if u is of the type of an -oscillation, then u is of the type of an -oscillation.
The following lemma shows that Definition 3.2 is independent of the expansion factor of the sphere.
Lemma 3.3 
Then we proceed to prove the main result, Theorem 1.2.
when is not empty.
Here is the Euclidean distance between Q and the boundary of Ω .
Next let and assume that . Also let , , be those cubes with .
Since u is a solution in , .
Since and as , it follows that as .
Combining estimates and in (3.11), we prove Theorem 1.2. □
Supported by National Natural Science Foundation of China (No: 11201415); Program for New Century Excellent Talents in Fujian Province University (No: JA14191).
- Giaquinta M: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton; 1983.MATHGoogle Scholar
- Abreu-Blaya R, Bory-Reyes J, Peña-Peña D: Jump problem and removable singularities for monogenic functions. J. Geom. Anal. 2007,17(1):1–13. 10.1007/BF02922079MathSciNetView ArticleMATHGoogle Scholar
- Chen Q, Jost J, Li J, Wang G: Regularity theorems and energy identities for Dirac-harmonic maps. Math. Z. 2005,251(1):61–84. 10.1007/s00209-005-0788-7MathSciNetView ArticleMATHGoogle Scholar
- Chen Q, Jost J, Li J, Wang G: Dirac-harmonic maps. Math. Z. 2006,254(2):409–432. 10.1007/s00209-006-0961-7MathSciNetView ArticleMATHGoogle Scholar
- Chen Q, Jost J, Wang G: Nonlinear Dirac equations on Riemann surfaces. Ann. Glob. Anal. Geom. 2008,33(3):253–270. 10.1007/s10455-007-9084-6MathSciNetView ArticleMATHGoogle Scholar
- Nolder CA, Ryan J: p -Dirac operators. Adv. Appl. Clifford Algebras 2009,19(2):391–402. 10.1007/s00006-009-0162-7MathSciNetView ArticleMATHGoogle Scholar
- Nolder CA: A -Harmonic equations and the Dirac operator. J. Inequal. Appl. 2010., 2010: Article ID 124018Google Scholar
- Nolder CA: Nonlinear A -Dirac equations. Adv. Appl. Clifford Algebras 2011,21(2):429–440. 10.1007/s00006-010-0253-5MathSciNetView ArticleMATHGoogle Scholar
- Wang C: A remark on nonlinear Dirac equations. Proc. Am. Math. Soc. 2010,138(10):3753–3758. 10.1090/S0002-9939-10-10438-9View ArticleMathSciNetMATHGoogle Scholar
- Wang C, Xu D: Regularity of Dirac-harmonic maps. Int. Math. Res. Not. 2009, 20: 3759–3792.MathSciNetMATHGoogle Scholar
- Kaufman R, Wu JM: Removable singularities for analytic or subharmonic functions. Ark. Mat. 1980,18(1):107–116.MathSciNetView ArticleMATHGoogle Scholar
- Koskela P, Martio O: Removability theorems for solutions of degenerate elliptic partial differential equations. Ark. Mat. 1993,31(2):339–353. 10.1007/BF02559490MathSciNetView ArticleMATHGoogle Scholar
- Kilpeläinen T, Zhong X: Removable sets for continuous solutions of quasilinear elliptic equations. Proc. Am. Math. Soc. 2002,130(6):1681–1688. 10.1090/S0002-9939-01-06237-2View ArticleMathSciNetMATHGoogle Scholar
- Sun F, Chen S: A -Harmonic operator in the Dirac system. J. Inequal. Appl. 2013., 2013: Article ID 463Google Scholar
- Chen S, Tan Z: Optimal interior partial regularity for nonlinear elliptic systems under the natural growth condition: the method of A -harmonic approximation. Acta Math. Sci. 2007,27(3):491–508. 10.1016/S0252-9602(07)60049-6MathSciNetView ArticleMATHGoogle Scholar
- Meyers NG: Mean oscillation over cubes and Hölder continuity. Proc. Am. Math. Soc. 1964,15(5):717–721.MathSciNetMATHGoogle Scholar
- Langmeyer N: The quasihyperbolic metric, growth, and John domains. Ann. Acad. Sci. Fenn., Math. 1998,23(1):205–224.MathSciNetMATHGoogle Scholar
- Stein EM Princeton Mathematical Series 30. In Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton; 1970.Google Scholar
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