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# The relation between A-harmonic operator and A-Dirac system

- Zhanlei Wang
^{1}and - Shuhong Chen
^{1}Email author

**2013**:362

https://doi.org/10.1186/1029-242X-2013-362

© Wang and Chen; licensee Springer 2013

**Received:**28 January 2013**Accepted:**12 July 2013**Published:**5 August 2013

## Abstract

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.

## Keywords

- A-harmonic operator
- A-Dirac system
- Caccioppoli estimate
- controllable growth condition

## 1 Introduction

*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 conditions with $p>1$:

*p*represents an exponent throughout the paper. The inhomogeneity $f(x,\xi )$ satisfies the following controllable growth condition:

where $s=\frac{np}{n-p}$ for $n>p$; *s* is any exponent for $n=p$.

**Definition 1.1**We call a function $u\in {W}_{\mathrm{loc}}^{1,p}(\mathrm{\Omega})$ a weak solution to (1.1) under the structure conditions (1.3) and (1.4) if the equality

holds for all $\varphi \in {W}^{1,p}(\mathrm{\Omega})$ 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 [1], where they showed that ${r}^{n}\omega (r)$-Hausdorff measure sets of monogenic functions with modulus of continuity $\omega (r)$ can be removed. The results were extended to Hölder continuous analytic functions [10] by Kaufman and Wu immediately. And then, Koskela and Martio [11] 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 [12]. The results were generalized [9] to the A-Dirac equation satisfying a certain oscillation condition. In the current paper, we further extend the results in [9] 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.

**Theorem 1.1**

*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 A*-

*Dirac equation*(1.6)

*under controllable growth condition in*$\mathrm{\Omega}\setminus E$,

*and*

*u*

*is of*$p,k$-

*oscillation in*$\mathrm{\Omega}\setminus E$.

*If for each compact subset*

*K*

*of*

*E*

*then* *u* *extends to a solution of the A*-*Dirac equation in* Ω.

## 2 A-Dirac operator

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 [9].

*R*by the basis of reduced products

where $\{{e}_{1},{e}_{2},\dots ,{e}_{n}\}$ is an orthogonal basis of ${R}^{n}$ 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 ${R}^{{2}^{n}}$, which implies an increasing tower $R\subset C\subset H\subset {\mathcal{U}}_{n}\subset \cdots $ . The Clifford algebra ${\mathcal{U}}_{n}$ is a 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 $A\subset {\mathcal{U}}_{n}$, $\mathit{Sc}(A)$ denotes the scalar part of *A*, that is, the coefficient of the element ${e}_{0}$, where $\mathrm{\Omega}\subset {R}^{n}$ is a connected and open set with boundary *∂* Ω. A Clifford-valued function $u:\mathrm{\Omega}\to {\mathcal{U}}_{n}$ can be written as $u={\sum}_{\alpha}{u}_{\alpha}{e}_{\alpha}$, where each ${u}_{\alpha}$ is real-valued and ${e}_{\alpha}$ are reduced products. The norm used here is given by $|{\sum}_{\alpha}{u}_{\alpha}{e}_{\alpha}|={({\sum}_{\alpha}{u}_{\alpha}^{2})}^{1/2}$. This norm is sub-multiplicative, $|AB|\le C|AB|$.

Also, ${D}^{2}=-\mathrm{\u25b3}$. Here △ is the Laplace operator which operates only on coefficients. A function is monogenic when $Du=0$.

*Q* is a cube in Ω with volume $|Q|$ throughout the paper. We write *σQ* for the cube with the same center as *Q* and with sidelength *σ* times that of *Q*. For $q>0$, we write ${L}^{q}(\mathrm{\Omega},{\mathcal{U}}_{n})$ for the space of Clifford-valued functions in Ω whose coefficients belong to the usual ${L}^{q}(\mathrm{\Omega})$ space. Also, ${W}^{1,q}(\mathrm{\Omega},{\mathcal{U}}_{n})$ is the space of Clifford-valued 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 $\cap {L}^{q}({\mathrm{\Omega}}^{\prime},{\mathcal{U}}_{n})$, where the intersection is over all ${\mathrm{\Omega}}^{\prime}$ compactly contained in Ω. We similarly write ${W}_{\mathrm{loc}}^{1,q}(\mathrm{\Omega},{\mathcal{U}}_{n})$. Moreover, we write ${\mathcal{M}}_{\mathrm{\Omega}}=\{u:\mathrm{\Omega}\to {\mathcal{U}}_{n}|Du=0\}$ for the space of monogenic functions in Ω.

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 ${W}^{1,p}(\mathrm{\Omega})\subset {W}^{D,p}(\mathrm{\Omega})$.

For the Clifford conjugation $\overline{({e}_{j1}\cdots {e}_{jl})}={(-1)}^{l}{e}_{jl}\cdots {e}_{j1}$, we define a Clifford-valued 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.

*A*, recast the structure equations above and define the operator

*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 $a>0$. We can definite the weak solution of equation (1.6) as follows.

**Definition 2.1**A Clifford-valued function $u\in {W}_{\mathrm{loc}}^{D,p}(\mathrm{\Omega},{\mathcal{U}}_{n}^{k})$, for $k=0,1,2,\dots ,n$, is a weak solution to (1.6) under structure conditions (2.10) and (2.11), and further assume that the inhomogeneity term $f(x,Du)$ satisfies the following controllable growth condition:

where $s=\frac{np}{n-p}$ for $n>p$; *s* is any exponent for $n=p$.

*A*is identity, then the homogeneity part of (2.13)

*p*-Dirac equation

Here $A(x,\xi )={|\xi |}^{p-2}\xi $.

These equations were introduced and their conformal invariance was studied in [7].

Furthermore, when *u* is a real-valued 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, 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.

## 3 The proof of main results

In this section, we establish the main results. Thus, a suitable Caccioppoli estimate for solutions to (2.13) is necessary.

**Theorem 3.1**

*Let*

*u*

*be a solution to the scalar part of*(1.6)

*defined by*(2.13),

*and let Q be a cube with*$\sigma Q\subset \mathrm{\Omega}$,

*where*$\sigma >1$.

*Then*

*Proof*Let $\eta \in {C}_{\mathrm{\infty}}^{0}(\mathrm{\Omega})$ be a standard cut-off function, $\eta >0$, $\eta \equiv 1$ in

*Q*. Choose $\varphi =(u-{u}_{\sigma Q}){\eta}^{p}$ as a test function in (2.13). Then $D\varphi =p{\eta}^{p-1}(D\eta )(u-{u}_{\sigma Q})+{\eta}^{p}Du$. Using the structure conditions (2.10) and (2.11),

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 [9], are needed.

**Definition 3.1**Assume that $u\in {L}_{\mathrm{loc}}^{1}(\mathrm{\Omega},{\mathcal{U}}_{n})$, $q>0$ and that $-\mathrm{\infty}<k\le 1$. We say that

*u*is of $q,k$-oscillation in Ω when

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 $q=1$, 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 A-harmonic equation, (3.7) is equivalent to a local order of growth condition when $-\mathrm{\infty}<k<0$ [5, 13]. 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 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 [8].

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 $s\le q$ and if *u* is of $q,k$-oscillation, then *u* is of $s,k$-oscillation. The following lemma shows that Definition 2.1 is independent of the expansion factor of the cube [9].

**Lemma 3.1**

*Suppose that*$F\in {L}_{\mathrm{loc}}^{1}(\mathrm{\Omega},\mathbb{R})$, $F>0$

*a*.

*e*., $\gamma \in \mathbb{R}$,

*and*${\sigma}_{1},{\sigma}_{2}>1$.

*If*

*then*

Then we can prove the main result.

*Proof of Theorem 1.1* Let *Q* be a cube in the Whitney decomposition of $\mathrm{\Omega}\setminus E$.

- (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)
$(1/4){|Q|}^{1/n}\le {|{Q}_{2}|}^{1/n}\le 4{|{Q}_{1}|}^{1/n}$ 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 [14].

*r*-inflation of

*A*as

*K*is a compact subset of

*E*. Also, let ${W}_{0}$ be those cubes in the Whitney decomposition of $\mathrm{\Omega}\setminus E$. Notice that each cube $Q\in {W}_{0}$ lies in $K(1)\setminus K$. Let $\gamma =p(k-1)-k$. First, since $\gamma \ge 1$, it follows that $m(K)=m(E)=0$ [7]. Also, since $a-n\ge \gamma $ using (3.11) and (3.14), we obtain

Hence, $u\in {W}_{\mathrm{loc}}^{D,p}$.

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 $l(Q)\le {2}^{-j}$.

*K*and such that $\psi (1-{\varphi}_{j})\in {W}^{1,p}(B\setminus E)$ with compact support. Hence,

since *u* is a solution in $B\setminus E$, ${I}^{\prime}=0$.

*c*such that $|\psi |\le c<\mathrm{\infty}$. Hence, using Hölder’s inequality, we have

Since $\cup {W}_{j}\subset K(1)\setminus K$ and $|\cup {W}_{j}|\to 0$ as $j\to \mathrm{\infty}$, it follows that ${I}_{1}\to 0$ as $j\to \mathrm{\infty}$.

where we have used the fact that $a=(n+pk-p)/n$ for $-\mathrm{\infty}<k\le 1$.

Since $\cup {W}_{j}\subset K(1)\setminus K$ and $|{W}_{j}|\to 0$ as $j\to \mathrm{\infty}$, it follows that ${I}^{\u2034}\to 0$ as $j\to \mathrm{\infty}$. □

## Declarations

### Acknowledgements

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).

## Authors’ Affiliations

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