- Open Access
The relation between A-harmonic operator and A-Dirac system
© Wang and Chen; licensee Springer 2013
- Received: 28 January 2013
- Accepted: 12 July 2013
- Published: 5 August 2013
In this paper, we show how an A-harmonic operator arises from Dirac systems under controllable growth condition. By the method of removable singularities for solutions to the A-Dirac system with controllable growth conditions, we establish the fact that an A-harmonic operator is a real part of the corresponding A-Dirac systems.
- A-harmonic operator
- A-Dirac system
- Caccioppoli estimate
- controllable growth condition
where for ; s is any exponent for .
holds for all with compact support.
The main purpose of this paper is to show that under controllable growth condition, an equation defined by the A-harmonic operator is a real part of the corresponding A-Dirac system. In order to obtain the desired result, we use the method of removability theorems, which proved that under suitable condition, a result concerning removable singularities for equations defined by the A-harmonic operator satisfying the Lipschitz condition or of bounded mean oscillation extends to Clifford-valued solutions to the corresponding Dirac equation. Further discussion on nonlinear Dirac equations can be found in [1–9] and their references.
The method of removability theorems was introduced by Abreu-Blaya et al. in , where they showed that -Hausdorff measure sets of monogenic functions with modulus of continuity can be removed. The results were extended to Hölder continuous analytic functions  by Kaufman and Wu immediately. And then, Koskela and Martio  established that in Hölder and bounded mean oscillation classes, the sets satisfying a certain geometric condition related to Minkowski dimension of an A-harmonic function can be removed. In terms of Hausdorff dimension, a precise condition for removable sets of A-harmonic functions in the case of Hölder continuity exists . The results were generalized  to the A-Dirac equation satisfying a certain oscillation condition. In the current paper, we further extend the results in  to discover the relation between the inhomogeneity A-harmonic equations and the inhomogeneity A-Dirac equations under controllable growth condition and obtain the main result as follows. It implies that under suitable condition, the solutions to the A-harmonic equation under controllable growth condition in fact is a real part of weak solutions to the corresponding A-Dirac systems.
then u extends to a solution of the A-Dirac equation in Ω.
In this section, we introduce an A-Dirac operator. In order to definite the A-Dirac operator, we should present the definition and notations about Clifford algebra at first .
where is an orthogonal basis of with the relation . We write for the identity. The dimension of is , which implies 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 , where 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 . This norm is sub-multiplicative, .
Also, . Here △ is the Laplace operator which operates only on coefficients. A function is monogenic when .
Q is a cube in Ω with volume throughout the paper. We write σQ for the cube with the same center as Q and with sidelength 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.
for some constant . We can definite the weak solution of equation (1.6) as follows.
where for ; s is any exponent for .
These equations were introduced and their conformal invariance was studied in .
Furthermore, when u is a real-valued function, (2.14) implies that is a harmonic field, and locally there exists a harmonic function H such that . If is invertible, then . Hence, the regularity of A implies the regularity of the solution u.
In general, A-harmonic functions do not have such regularity. This suggests the study of the scalar part of system (2.13) in general. Thus a Caccioppoli estimate for solutions to the scalar part of (2.13) is necessary.
In this section, we establish the main results. Thus, a suitable Caccioppoli estimate for solutions to (2.13) is necessary.
This completes the proof of Theorem 3.1. □
In order to discover the relation between A-harmonic equations and A-Dirac systems, we should remove singularity of solutions to A-Dirac systems at first. Thus, various regularity properties of real-valued functions, such as the following definition , are needed.
The infimum over monogenic functions is natural since they are trivial solutions to an A-Dirac equation just as constants are solutions to an A-harmonic equation. If u is a function and , then (3.7) is equivalent to the usual definition of the bounded mean oscillation when and (3.7) is equivalent to the usual local Lipschitz condition when . Moreover, at least when u is a solution to an A-harmonic equation, (3.7) is equivalent to a local order of growth condition when [5, 13]. In these cases, the supremum is finite if we choose 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 A-Dirac 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 .
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 real-valued functions. We remark that it follows from Hölder’s inequality that if and if u is of -oscillation, then u is of -oscillation. The following lemma shows that Definition 2.1 is independent of the expansion factor of the cube .
Then we can prove the main result.
Proof of Theorem 1.1 Let Q be a cube in the Whitney decomposition of .
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 .
where we have used the fact that for .
Since and as , it follows that as . □
Supported by the National Natural Science Foundation of China (No: 11201415); the Natural Science Foundation of Fujian Province (2012J01027) and Training Programme Foundation for Excellent Youth Researching Talents of Fujian’s Universities (JA12205).
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