A-Harmonic operator in the Dirac system
© Sun and Chen; licensee Springer. 2013
Received: 14 January 2013
Accepted: 12 September 2013
Published: 7 November 2013
In this paper, we show how an A-harmonic operator arises from A-Dirac systems under natural growth condition. By the method of removable singularities for solutions to A-Dirac systems with natural growth conditions, we establish the fact that A-harmonic equations are the real part of the corresponding A-Dirac systems.
KeywordsA-harmonic operator A-Dirac system Caccioppoli estimate natural growth condition
The exponent p represents an exponent throughout the paper.
holds for all with compact support.
The main purpose of this paper is to show that under natural growth condition, an equation defined by an A-harmonic operator is the real part of the corresponding A-Dirac 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 A-harmonic operator 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 continuity , in which the sets of -Hausdorff measure are removable. 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 for A-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 for A-harmonic functions in terms of Hausdorff dimension. In , the sets satisfying a generalized Minkowski-type inequality were removable for solutions to the A-Dirac equation which satisfy a certain oscillation condition.
In this paper, we show that the above results even hold for inhomogeneity equations defined by an A-harmonic operator and A-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.
then u extends to a solution of the A-Dirac equation in Ω.
2 A-Dirac operator and correspondence definition
In this section, we introduce an A-Dirac system. In order to definite the A-Dirac operator, we present the definitions and notations of 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 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, 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 .
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.
for some constant . Hence we can definite the weak solution of equation (1.6) as follows.
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 regularity of A implies 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 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
This completes the proof of Theorem 3.1. □
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 Definition 5.1 .
The infimum over monogenic functions is natural since they are trivially 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 [4, 14]. 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 3.2 is independent of the expansion factor of the sphere.
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 .
Since and as , we have that as . Hence .
Combining the estimates and in equation (3.14), we prove Theorem 1.2. □
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).
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