Regularity theory on A-harmonic system and A-Dirac system

Abstract

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.

1 Introduction

In this paper, we consider the regularity theory on an A-Dirac system,

$$-D\tilde{A}(x,u,Du)=f(x,u,Du),\text{in }\mathrm{\Omega },$$

(1.1)

and an A-harmonic system,

$$-divA(x,u,\mathrm{\nabla }u)=f(x,u,\mathrm{\nabla }u),\text{in }\mathrm{\Omega }.$$

(1.2)

Here Ω is a bounded domain in $R^n$ ($n\ge 2$), $A(x,u,\mathrm{\nabla }u)$ and $f(x,u,\mathrm{\nabla }u)$ are measurable functions defined on $\mathrm{\Omega }\times R^n\times R^{nN}$, N is an integer with $N>1$, $u:\mathrm{\Omega }\to R^n$ is a vector valued function. Furthermore, $A(x,u,\mathrm{\nabla }u)$ and $f(x,u,\mathrm{\nabla }u)$ satisfy the following structural conditions with $m>2$:

(H1) $A(x,u,p)$ are differentiable functions in p and there exists a constant $C>0$ such that

$$\left|\frac{\partial A(x,u,p)}{\partial p}\right|\le C{(1+{|p|}^2)}^{\frac{m-2}{2}}\text{for all }(x,u,p)\in \mathrm{\Omega }\times R^n\times R^{nN}.$$

(H2) $A(x,u,p)$ are uniformly strongly elliptic, that is, for some $\lambda >0$ we have

$$\left(\frac{\partial A(x,u,p)}{\partial p}{\nu }_i^{\alpha }\right){\nu }_j^{\beta }\ge \lambda {(1+{|p|}^2)}^{\frac{m-2}{2}}{|\nu |}^2.$$

(H3) There exist $\beta \in (0,1)$ and $K:[0,\mathrm{\infty })\mapsto [0,\mathrm{\infty })$ monotone nondecreasing such that

$$|A(x,u,p)-A(\tilde{x},\tilde{u},p)|\le K(|u|){({|x-\tilde{x}|}^m+{|u-\tilde{u}|}^m)}^{\frac{\beta }{m}}{(1+|p|)}^{\frac{m}{2}}$$

for all $x,\tilde{x}\in \mathrm{\Omega }$, $u,\tilde{u}\in R^n$, and $p\in R^{nN}$. Without loss of generality, we take $K\ge 1$.

(H4) There exist constants $C_1$ and $C_2$ such that

$$|f(x,u,p)|\le C_1{|p|}^m+C_2.$$

(H1) and (H2) imply

$$|A(x,u,p)-A(x,u,\xi )|\le C{(1+{|p|}^2+{|\xi |}^2)}^{\frac{m-2}{2}}|p-\xi |;$$

(1.3)

$$(A(x,u,p)-A(x,u,\xi ))(p-\xi )\ge \lambda {(1+{|p|}^2+{|\xi |}^2)}^{\frac{m-2}{2}}{|p-\xi |}^2$$

(1.4)

for all $x\in \mathrm{\Omega }$, $u\in R^n$ and $p,\xi \in R^{nN}$, where $\lambda >0$ is a constant.

Definition 1.1 We say that a function $u\in W_{\mathrm{loc}}^{1,m}(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega })$ is a weak solution to (1.2), if the equality

$${\int }_{\mathrm{\Omega }}A(x,u,\mathrm{\nabla }u)\mathrm{\nabla }\varphi dx={\int }_{\mathrm{\Omega }}f(x,u,\mathrm{\nabla }u)\varphi dx$$

(1.5)

holds for all $\varphi \in W_0^{1,m}(\mathrm{\Omega })$ 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 [1] 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 [2] to remove singularities for monogenic functions with modulus of continuity $\omega (\mathit{r})$, where the sets $r^n\omega (r)$ and Hausdorff measure are removable. Kaufman and Wu [11] 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 [12]. Even in the case of Hölder continuity, a precise removable sets condition was stated [13]. In [7], 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 [14].

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.

Theorem 1.2 Let E be a relatively closed subset of Ω. Suppose that $u\in L_{\mathrm{loc}}^m(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega })$ has distributional first derivatives in Ω, u is a solution to the scalar part of A-Dirac system (1.1) under the structure conditions (H1)-(H4) in $\mathrm{\Omega }\setminus E$, and u is of the type of an $m,k$-oscillation in $\mathrm{\Omega }\setminus E$. If for each compact subset K of E

$${\int }_{\mathrm{\Omega }\setminus K}d{(x,K)}^{m(k-1)-k}<\mathrm{\infty },$$

(1.6)

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

2 A-Dirac system

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 [7].

We write ${\mathcal{U}}_n$ for the real universal Clifford algebra over $R^n$. The Clifford algebra is generated over R by the basis of the reduced products

$$\{e_1,e_2,\dots ,e_1{e}_2,\dots ,e_1\cdots e_n\},$$

(2.1)

where $\{e_1,e_2,\dots ,e_n\}$ is an orthonormal 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 $2^n$. We have an increasing tower $R\subset C\subset H\subset {\mathcal{U}}_3\subset \cdots$ . The Clifford algebra ${\mathcal{U}}_n$ is a graded algebra as ${\mathcal{U}}_n={⨁}_l{\mathcal{U}}_n^l$, 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 this paper, $\mathrm{\Omega }\subset R^n$ 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 $|{\sum }_{\alpha }{u}_{\alpha }{e}_{\alpha }|={({\sum }_{\alpha }{u}_{\alpha }^2)}^{\frac{1}{2}}$, which is sub-multiplicative, $|AB|\le C|A||B|$.

The Dirac operator defined here is

$$D=\sum _{j=1}^n{e}_j\frac{\partial }{\partial x_j}.$$

(2.2)

Also $D^2=-\mathrm{△}$. Here △ is Laplace operator.

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 Clifford-valued functions in Ω whose coefficients belong to the usual $L^q(\mathrm{\Omega })$ space. Also, $W^{1,m}(\mathrm{\Omega },{\mathcal{U}}_n)$ 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 $\bigcap L^q({\mathrm{\Omega }}^{\prime },{\mathcal{U}}_n)$, where the intersection is over all ${\mathrm{\Omega }}^{\prime }$ compactly contained in Ω. We similarly write $W_{\mathrm{loc}}^{1,m}(\mathrm{\Omega },{\mathcal{U}}_n)$. 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 Dirac Sobolev space

$$W^{D,m}(\mathrm{\Omega })=\{u\in {\mathcal{U}}_n|{\int }_{\mathrm{\Omega }}{|u|}^m+{\int }_{\mathrm{\Omega }}{|Du|}^m<\mathrm{\infty }\}.$$

(2.3)

The local space $W_{\mathrm{loc}}^{D,m}$ is similarly defined. Notice that if u is monogenic, then $u\in L^m(\mathrm{\Omega })$ if and only if $u\in W^{D,m}(\mathrm{\Omega })$. Also it is immediate that $W^{1,m}(\mathrm{\Omega })\subset W^{D,m}(\mathrm{\Omega })$.

With those definitions and notations and also of the A-Dirac operator, we define the linear isomorphism $\theta :R^n\to {\mathcal{U}}_n^1$ by

$$\theta ({\omega }_1,\dots ,{\omega }_n)=\sum _{i=1}^n{\omega }_i{e}_i.$$

(2.4)

For $x,y\in R^n$, Du is defined by $\theta (\mathrm{\nabla }\varphi )=D\varphi$ for a real-valued function ϕ, and we have

$$-Sc(\theta (x)\theta (y))=〈x,y〉,$$

(2.5)

$$|\theta (x)|=|x|.$$

(2.6)

Here $\tilde{A}(x,\xi ,\eta ):\mathrm{\Omega }\times {\mathcal{U}}_1\times {\mathcal{U}}_n\to {\mathcal{U}}_n$ is defined by

$$\tilde{A}(x,u,\eta )=\theta A(x,u,{\theta }^{-1}\eta ),$$

(2.7)

which means that (1.5) is equivalent to

$$\begin{array}{rl}{\int }_{\mathrm{\Omega }}Sc(\theta A(x,u,\mathrm{\nabla }u)\theta (\mathrm{\nabla }\varphi ))dx& ={\int }_{\mathrm{\Omega }}Sc(\tilde{A}(x,u,Du)D\varphi )dx\\ ={\int }_{\mathrm{\Omega }}Sc(f(x,u,Du)\varphi )dx.\end{array}$$

(2.8)

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 $Sc(\overline{\alpha }\beta )$ is the usual inner product $〈\alpha ,\beta 〉$ in $R^{2^n}$, when α and β are identified as vectors.

For convenience, we replace $\tilde{A}$ with A and recast the structure systems above and define the operator:

$$A(x,\xi ,\eta ):\mathrm{\Omega }\times {\mathcal{U}}_1\times {\mathcal{U}}_n\to {\mathcal{U}}_n,$$

(2.9)

where A preserves the grading of the Clifford algebra, $x\to A(x,\xi ,\eta )$ is measurable for all ξ, η, and $\xi \to A(x,\xi ,\eta )$, $\eta \to A(x,\xi ,\eta )$ are continuous for a.e. $x\in \mathrm{\Omega }$.

Definition 2.1 A Clifford valued function $u\in W_{\mathrm{loc}}^{D,m}(\mathrm{\Omega },{\mathcal{U}}_n^k)\cap L^{\mathrm{\infty }}(\mathrm{\Omega },{\mathcal{U}}_n^k)$, for $k=0,1,\dots ,n$, is a weak solution to system (1.1) under conditions (H1)-(H4). If for all $\varphi \in W_0^{1,m}(\mathrm{\Omega },{\mathcal{U}}_n^k)$, then we have

$${\int }_{\mathrm{\Omega }}\overline{A(x,u,Du)}D\varphi dx={\int }_{\mathrm{\Omega }}\overline{f(x,u,Du)}\varphi dx.$$

(2.10)

3 Proof of the main results

In this section, we will establish the main results. At first, a suitable Caccioppoli estimate [7, 15] for solutions to (2.10) is necessary.

Theorem 3.1 Let u be weak solutions to the scalar part of system (1.1) with $\lambda >2C_{1}M$ and where (H1)-(H4) are satisfied. Then for every $x_0\in \mathrm{\Omega }$, $u_0\in {\mathcal{U}}_1^k$, $p_0\in {\mathcal{U}}_n^k$, and arbitrary $\sigma >1$ we have

$$\begin{array}{r}{\int }_Q[{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|Du-p_0|}^2+{|Du-p_0|}^m]dx\\ \le C\{\frac{1}{{(\sigma |Q|)}^{2/n}}{\int }_{\sigma Q}{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|u-P|}^{2}dx\\ +\frac{1}{{(\sigma |Q|)}^{m/n}}{\int }_{\sigma Q}{|u-P|}^{m}dx+{\int }_{\sigma Q}{G}^{2}dx\},\end{array}$$

(3.1)

where $P=u(x)-u_0+p_0(x-x_0)$ and

$$\begin{array}{rcl}{\int }_{\sigma Q}{G}^{2}dx& =& \sigma |Q|\{{\left[K(|u_0|+|p_0|){(1+|p_0|)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }}{(\sigma |Q|)}^{\frac{2\beta }{n}}\\ +(C_2^2+C_1^2{|p_0|}^{2m}){(\sigma |Q|)}^{\frac{2}{n}}\}.\end{array}$$

(3.2)

Proof Denote $u(x)-u_0-p_0(x-x_0)$ by $v(x)$ and $0<{|Q|}^{\frac{1}{n}}<\sigma {|Q|}^{\frac{1}{n}}<min\{1,dist(x_0,\partial \mathrm{\Omega })\}$ for $\sigma >1$, consider a standard cut-off function $\eta \in C_0^{\mathrm{\infty }}(\sigma Q(x_0))$ satisfying $0\le \eta \le 1$, $|\mathrm{\nabla }\eta |<\frac{1}{{|Q|}^{1/n}}$, $\eta \equiv 1$ on $Q(x_0)$. Then $\phi ={\eta }^{2}v$ is admissible as a test-function, and we obtain

$$\begin{array}{r}{\int }_{\sigma Q}A(x,u,Du)\cdot (Du-p_0){\eta }^{2}dx\\ =-2{\int }_{\sigma Q}A(x,u,Du)\eta v\cdot \mathrm{\nabla }\eta dx+{\int }_{\sigma Q}f(x,u,Du)\cdot \phi dx.\end{array}$$

We further have

$$\begin{array}{r}-{\int }_{\sigma Q}A(x,u,p_0)\cdot (Du-p_0){\eta }^{2}dx\\ =2{\int }_{\sigma Q}A(x,u,p_0)\eta v\cdot \mathrm{\nabla }\eta dx-{\int }_{\sigma Q}A(x,u,p_0)\cdot D\phi dx,\end{array}$$

and

$${\int }_{\sigma Q}A(x_0,u_0,p_0)\cdot D\phi dx=0.$$

Adding these equations yields

$$\begin{array}{r}{\int }_{\sigma Q}(A(x,u,Du)-A(x,u,p_0))(Du-p_0){\eta }^{2}dx\\ =-2{\int }_{\sigma Q}(A(x,u,Du)-A(x,u,p_0))(Du-p_0)\eta v\cdot \mathrm{\nabla }\eta dx\\ -{\int }_{\sigma Q}(A(x,u,p_0)-A(x,u_0+p_0(x-x_0),p_0))\cdot D\phi dx\\ -{\int }_{\sigma Q}(A(x,u_0+p_0(x-x_0),p_0)-A(x_0,u_0,p_0))\cdot D\phi dx+{\int }_{\sigma Q}f(x,u,Du)\cdot \phi dx\\ \le I+II+III+IV+V,\end{array}$$

(3.3)

where

$$\begin{array}{c}I=2C{\int }_{\sigma Q}{(1+{|Du|}^2+{|p_0|}^2)}^{\frac{m-2}{2}}|Du-p_0|\eta |v||\mathrm{\nabla }\eta |dx;\hfill \\ II=K(|u_0|+|p_0|){\int }_{\sigma Q}{|v|}^{\beta }|Du-p_0|{(1+|p_0|)}^{\frac{m}{2}}{\eta }^{2}dx;\hfill \\ III=2K(|u_0|+|p_0|){\int }_{\sigma Q}{|v|}^{\beta +1}|\mathrm{\nabla }\eta |{(1+|p_0|)}^{\frac{m}{2}}\eta dx;\hfill \\ \begin{array}{rl}IV=& K(|u_0|+|p_0|){\int }_{\sigma Q}{({|x-x_0|}^m+{|p_0(x-x_0)|}^m)}^{\frac{\beta }{m}}{(1+|p_0|)}^{\frac{m}{2}}\\ \cdot (2\eta |\mathrm{\nabla }\eta ||v|+{\eta }^2|Du-p_0|)dx;\end{array}\hfill \\ V={\int }_{\sigma Q}(C_1{|Du|}^m+C_2)|v|{\eta }^{2}dx,\hfill \end{array}$$

after using (1.3), (H3), (H4).

For positive ε, to be fixed later, using Young’s inequality, we have

$$\begin{array}{rl}I\le & 2C{\int }_{\sigma Q}{(1+2{|Du-p_0|}^2+3{|p_0|}^2)}^{\frac{m-2}{2}}|Du-p_0|\eta |v||\mathrm{\nabla }\eta |dx\\ \le & C[{\int }_{\sigma Q}{(1+{|p_0|}^2)}^{\frac{m-2}{2}}|Du-p_0|\eta |v||\mathrm{\nabla }\eta |dx+{\int }_{\sigma Q}{|Du-p_0|}^{m-1}\eta |v||\mathrm{\nabla }\eta |dx]\\ \le & C\epsilon {\int }_{\sigma Q}{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|Du-p_0|}^2{\eta }^{2}dx+C\frac{1}{\epsilon }{\int }_{\sigma Q}{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|v|}^2{|\mathrm{\nabla }\eta |}^{2}dx\\ +C\epsilon {\int }_{\sigma Q}{|Du-p_0|}^m{\eta }^{2}dx+C(\epsilon ){\int }_{\sigma Q}{|v|}^m{|\mathrm{\nabla }\eta |}^{m}dx.\end{array}$$

Using Young’s inequality twice in II, we have

$$\begin{array}{rl}II\le & \epsilon {\int }_{\sigma Q}{|Du-p_0|}^2{\eta }^{2}dx+\frac{1}{\epsilon }{K}^2(|u_0|+|p_0|){(1+|p_0|)}^m{\int }_{\sigma Q}{|v|}^{2\beta }dx\\ \le & \epsilon {\int }_{\sigma Q}{|Du-p_0|}^2{\eta }^{2}dx+\frac{1}{\epsilon }{\int }_{\sigma Q}{(\frac{1}{{(\sigma |Q|)}^{1/n}}|v|)}^{2}dx\\ +\frac{1}{\epsilon }{\left[K(|u_0|+|p_0|){(1+|p_0|)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }}{(\sigma |Q|)}^{(\frac{2\beta }{1-\beta }+n)/n}\\ \le & \epsilon {\int }_{\sigma Q}{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|Du-p_0|}^2{\eta }^{2}dx+\frac{1}{\epsilon }{\int }_{\sigma Q}{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{\left(\frac{1}{{(\sigma |Q|)}^{1/n}}\right)}^2{|v|}^{2}dx\\ +\frac{1}{\epsilon }{\left[K(|u_0|+|p_0|){(1+|p_0|)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }}{(\sigma |Q|)}^{(\frac{2\beta }{1-\beta }+n)/n},\end{array}$$

and similarly we see

$$\begin{array}{rcl}III& \le & \frac{1}{2}{\int }_{\sigma Q}{|v|}^2{|\mathrm{\nabla }\eta |}^{2}dx\\ +4K^2(|u_0|+|p_0|){(1+|p_0|)}^m{\int }_{\sigma Q}{(\sigma |Q|)}^{\frac{2\beta }{n}}{\left(\frac{|v|}{{(\sigma |Q|)}^{1/n}}\right)}^{2\beta }{\eta }^{2}dx\\ \le & {\int }_{\sigma Q}{\left(\frac{1}{{(\sigma |Q|)}^{1/n}}\right)}^2{|v|}^{2}dx+{\left[4K(|u_0|+|p_0|){(1+|p_0|)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }}{(\sigma |Q|)}^{(\frac{2\beta }{1-\beta }+n)/n}\\ \le & {(1+{|p_0|}^2)}^{\frac{m-2}{2}}{\int }_{\sigma Q}{\left(\frac{1}{{(\sigma |Q|)}^{1/n}}\right)}^2{|v|}^{2}dx\\ +{\left[4K(|u_0|+|p_0|){(1+|p_0|)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }}{(\sigma |Q|)}^{(\frac{2\beta }{1-\beta }+n)/n}\end{array}$$

and

$$\begin{array}{rcl}IV& \le & {\int }_{\sigma Q}K(|u_0|+|p_0|){(1+|p_0|)}^{\frac{m}{2}}{(\sigma |Q|)}^{\frac{\beta }{n}}{(1+{|p_0|}^m)}^{\frac{\beta }{m}}(\eta |Du-p_0|+2\eta |\mathrm{\nabla }\eta ||v|)dx\\ \le & {\int }_{\sigma Q}K(|u_0|+|p_0|){(1+|p_0|)}^{\frac{m}{2}}{(\sigma |Q|)}^{\frac{\beta }{n}}{(1+|p_0|)}^{\beta }(\eta |Du-p_0|+2\eta |\mathrm{\nabla }\eta ||v|)dx\\ \le & \epsilon {\int }_{\sigma Q}{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|Du-p_0|}^2{\eta }^{2}dx+{\int }_{\sigma Q}{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|\mathrm{\nabla }\eta |}^2{|v|}^{2}dx\\ +(4+\frac{1}{\epsilon })K^2(|u_0|+|p_0|){(1+|p_0|)}^{2(\frac{m}{2}+\beta )}{(\sigma |Q|)}^{\frac{n+2\beta }{n}},\end{array}$$

and for positive μ, to be fixed later, this yields

$$\begin{array}{rcl}V& =& {\int }_{\sigma Q}{C}_1{|Du|}^m|u-u_0-p_0(x-x_0)|{\eta }^{2}dx+{\int }_{\sigma Q}(\frac{1}{{(\sigma |Q|)}^{1/n}}|v|\eta )\left(C_2{(\sigma |Q|)}^{\frac{1}{n}}\eta \right)dx\\ \le & {\int }_{\sigma Q}{C}_1[(1+\mu ){|Du-p_0|}^m+(1+\frac{1}{\mu }){|p_0|}^m]|u-u_0-p_0(x-x_0)|{\eta }^{2}dx\\ +\frac{1}{2}\epsilon C_2^2{(\sigma |Q|)}^{\frac{n+2}{n}}+\frac{1}{2\epsilon {(\sigma |Q|)}^{2/n}}{\int }_{\sigma Q}{|v|}^{2}dx\\ \le & C_1(1+\mu )(2M+p_0{(\sigma |Q|)}^{\frac{1}{n}}){\int }_{\sigma Q}{|Du-p_0|}^m{\eta }^{2}dx+C_1(1+\frac{1}{\mu }){|p_0|}^m{\int }_{\sigma Q}|v|{\eta }^{2}dx\\ +\frac{1}{2}\epsilon C_2^2{(\sigma |Q|)}^{\frac{n+2}{n}}+\frac{1}{2\epsilon }{\int }_{\sigma Q}\frac{1}{{(\sigma |Q|)}^{2/n}}{|v|}^{2}dx\\ \le & C_1(1+\mu )(2M+p_0{(\sigma |Q|)}^{\frac{1}{n}}){\int }_{\sigma Q}{|Du-p_0|}^m{\eta }^{2}dx+\frac{1}{\epsilon }{\int }_{\sigma Q}\frac{1}{{(\sigma |Q|)}^{2/n}}{|v|}^{2}dx\\ +\frac{\epsilon }{2}[C_2^2+C_1^2{(1+\frac{1}{\mu })}^2{|p_0|}^{2m}]{(\sigma |Q|)}^{\frac{n+2}{n}}.\end{array}$$

By (1.4), we obtain

$$\begin{array}{r}{\int }_{\sigma Q}(A(x,u,Du)-A(x,u,p_0))(Du-p_0){\eta }^{2}dx\\ \ge \lambda {\int }_{\sigma Q}{(1+{|Du|}^2+{|p_0|}^2)}^{\frac{m-2}{2}}{|Du-p_0|}^2{\eta }^{2}dx\\ \ge \lambda {\int }_{\sigma Q}{(1+{|Du-p_0|}^2+{|p_0|}^2)}^{\frac{m-2}{2}}{|Du-p_0|}^2{\eta }^{2}dx\\ \ge \lambda \{{\int }_{\sigma Q}{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|Du-p_0|}^2{\eta }^{2}dx+{\int }_{\sigma Q}{|Du-p_0|}^m{\eta }^{2}dx\}.\end{array}$$

Combining these estimates in (3.3) and noting that $K^2\le K^{\frac{2}{1-\beta }}$ (as $K\ge 1$), ${(\sigma |Q|)}^{\frac{2\beta }{(1-\beta )n}}\le {(\sigma |Q|)}^{\frac{2\beta }{n}}$ for $\sigma >1$, ${[{(1+|p_0|)}^{\frac{m}{2}}]}^{\frac{2}{1-\beta }}\ge {(1+|p_0|)}^{2(\frac{m}{2}+\beta )}$, and $4\le 4^{\frac{2}{1-\beta }}$, we can estimate

$$\begin{array}{r}[\lambda -2C\epsilon -2\epsilon -C_1(1+\mu )(2M+p_0{(\sigma |Q|)}^{\frac{1}{n}})]\\ \cdot \{{\int }_{\sigma Q}{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|Du-p_0|}^2{\eta }^{2}dx+{\int }_{\sigma Q}{|Du-p_0|}^m{\eta }^{2}dx\}\\ \le (\frac{C}{\epsilon }+\frac{2}{\epsilon }+C(\epsilon )+2)\{\frac{1}{{(\sigma |Q|)}^{2/n}}{\int }_{\sigma Q}{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|u-P|}^{2}dx\\ +\frac{1}{{(\sigma |Q|)}^{m/n}}{\int }_{\sigma Q}{|u-P|}^{m}dx\}+2(\frac{1}{\epsilon }+4^{\frac{2}{1-\beta }})\\ \cdot {\left[K(|u_0|+|p_0|){(1+|p_0|)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }}{(\sigma |Q|)}^{\frac{n+2\beta }{n}}\\ +\frac{\epsilon }{2}[C_2^2+C_1^2{(1+\frac{1}{\mu })}^2{|p_0|}^{2m}]{(\sigma |Q|)}^{\frac{n+2}{n}}.\end{array}$$

Define $\epsilon =\epsilon (\lambda ,m)$, $\mu =\mu (C_1,M,m,\lambda )$ small enough, we obtain

$$\begin{array}{r}{\int }_{\sigma Q}{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|Du-p_0|}^2{\eta }^{2}dx+{\int }_{\sigma Q}{|Du-p_0|}^m{\eta }^{2}dx\\ \le C\{\frac{1}{{(\sigma |Q|)}^{2/n}}{\int }_{\sigma Q}{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|u-P|}^{2}dx\\ +\frac{1}{{(\sigma |Q|)}^{m/n}}{\int }_{\sigma Q}{|u-P|}^{m}dx+{\int }_{\sigma Q}{G}^{2}dx\},\end{array}$$

where $C=C(m,\lambda ,\beta ,M)$ and

$${\int }_{\sigma Q}{G}^{2}dx=\sigma |Q|\{{\left[K(|u_0|+|p_0|){(1+|p_0|)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }}{(\sigma |Q|)}^{\frac{2\beta }{n}}+(C_2^2+C_1^2{|p_0|}^{2m}){(\sigma |Q|)}^{\frac{2}{n}}\}.$$

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 [7]

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 the type of a $q,k$-oscillation in Ω when

$$\underset{2Q\subset \mathrm{\Omega }}{sup}{|Q|}^{-(qk+n)/qn}\underset{u_Q\in {\mathcal{M}}_Q}{inf}{\left({\int }_Q{|u-u_Q|}^q\right)}^{1/q}<\mathrm{\infty }.$$

(3.4)

If $q=1$ and $k=0$, then the inequality (3.4) is equivalent to the usual definition of the bounded mean oscillation; when $q=1$ and $0<k\le 1$, then the inequality (3.4) is equivalent to the usual local Lipschitz condition [16]. Further discussion of the inequality (3.4) can be found in [8, 17]. 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.

We remark that it follows from Hölder’s inequality that if $s\le q$ and if u is of the type of an $q,k$-oscillation, then u is of the type of an $s,k$-oscillation.

The following lemma shows that Definition 3.2 is independent of the expansion factor of the sphere.

Lemma 3.3 [7]

Suppose that $F\in L_{\mathrm{loc}}^1(\mathrm{\Omega },R)$, $F>0$ a.e., $r\in R$ and ${\sigma }_1,{\sigma }_2>1$. If

$$\underset{{\sigma }_{1}Q\subset Q}{sup}{|Q|}^r{\int }_{Q}F<\mathrm{\infty },$$

then

$$\underset{{\sigma }_{2}Q\subset Q}{sup}{|Q|}^r{\int }_{Q}F<\mathrm{\infty }.$$

(3.5)

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 [18].

If $A\subset R^n$ and $r>0$, then we define the r-inflation of A as

$$A(r)=\bigcup B(x,r).$$

(3.6)

Let Q be a cube in the Whitney decomposition of $\mathrm{\Omega }\setminus E$. Using the Caccioppoli estimate (3.1), we have

$$\begin{array}{r}{\int }_Q[{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|Du-p_0|}^2+{|Du-p_0|}^m]dx\\ \le C\{\frac{1}{{(\sigma Q)}^{2/n}}{\int }_{\sigma Q}{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|u-P|}^{2}dx\\ +\frac{1}{{(\sigma Q)}^{m/n}}{\int }_{\sigma Q}{|u-P|}^{m}dx+{\int }_{\sigma Q}{G}^{2}dx\},\end{array}$$

with (3.2)

$${\int }_{\sigma Q}{G}^{2}dx\le C{|\sigma Q|}^{\frac{n+2\beta }{n}}{H}^2(1+|u_Q|+|p_0|),$$

(3.7)

where

$$H(t)={[\tilde{K}(t){(1+t)}^{\frac{m}{2}}]}^{\frac{2}{1-\beta }},\tilde{K}(t)=max\{K(t),C_1,C_2\},$$

and choose $|Q|$ small enough such that

$${|Q|}^{\frac{\beta }{n}}H(1+|u_Q|+|p_0|)\le 1.$$

By the definition of the $q,k$-oscillation condition, we have

$$\begin{array}{r}{\int }_Q[{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|Du-p_0|}^2+{|Du-p_0|}^m]dx\\ \le C_1{|Q|}^{-\frac{2}{n}}{|Q|}^{\frac{2k+n}{n}}+C_2{|Q|}^{-\frac{m}{n}}{|Q|}^{(mk+n)/n}+C_3|Q|\\ \le C{|Q|}^a.\end{array}$$

(3.8)

Here $a=(n+mk-m)/n$. Since the problem is local (use a partition of unity), we show that (2.10) holds whenever $\varphi \in W_0^{1,m}(B(x_0,r))$ 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 $\mathrm{\Omega }\setminus K$. Let $\gamma =m(k-1)-k$. First, since $\gamma \ge -1$, from [12] we have $m(K)=m(E)=0$. Also since $na-n\ge \gamma$, using (1.6) and (3.8), we obtain

$$\begin{array}{r}{\int }_{B(x_0,r)}[{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|Du-p_0|}^2+{|Du-p_0|}^m]dx\\ \le C\sum _{Q\in W_0}{|Q|}^a\le C\sum _{Q\in W_0}d{(Q,K)}^{na}\\ \le C\sum _{Q\in W_0}{\int }_{Q}d{(x,K)}^{na-n}dx\le C{\int }_{K(1)\setminus K}d{(x,K)}^{na-n}dx\\ \le C{\int }_{K(1)\setminus K}d{(x,K)}^{\gamma }dx<\mathrm{\infty }.\end{array}$$

(3.9)

Hence $u\in W_{\mathrm{loc}}^{D,m}(\mathrm{\Omega })$.

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

$${\varphi }_j=max\{(2^{-j}-d(x,K))2^j,0\}.$$

(3.10)

Thus each ${\varphi }_j$, $j=1,2,\dots$ , is Lipschitz, equal to 1 on K and as such $\psi (1-{\varphi }_j)\in W^{1,m}(B\setminus E)$ with compact support. Hence

$$\begin{array}{r}{\int }_B[\overline{A(x,u,Du)}D\psi -\overline{f(x,u,Du)}\psi ]dx\\ ={\int }_{B\setminus E}[\overline{A(x,u,Du)}D(\psi (1-{\varphi }_j))-\overline{f(x,u,Du)}\psi (1-{\varphi }_j)]dx\\ +{\int }_B[\overline{A(x,u,Du)}D(\psi {\varphi }_j)-\overline{f(x,u,Du)}\psi {\varphi }_j]dx.\end{array}$$

(3.11)

Let

$$\begin{array}{r}{J}_1={\int }_{B\setminus E}[\overline{A(x,u,Du)}D(\psi (1-{\varphi }_j))-\overline{f(x,u,Du)}\psi (1-{\varphi }_j)]dx,\\ J_2={\int }_B[\overline{A(x,u,Du)}D(\psi {\varphi }_j)-\overline{f(x,u,Du)}\psi {\varphi }_j]dx.\end{array}$$

Since u is a solution in $B\setminus E$, $J_1=0$.

Next we estimate $J_2$ as

$$\begin{array}{rcl}{J}_2& =& {\int }_{B}A(x,u,Du)\psi D{\varphi }_{j}dx+{\int }_B{\varphi }_{j}A(x,u,Du)D\psi dx-{\int }_B\overline{f(x,u,Du)}\psi {\varphi }_{j}dx\\ =& J_2^{\prime }+J_2^{″}+J_2^{‴}.\end{array}$$

(3.12)

Noting that there exists a constant C such that $|\psi |\le C<\mathrm{\infty }$,

$$\left|J_2^{\prime }\right|\le C\sum _{Q\in W_j}{\int }_B|A(x,u,Du)||D{\varphi }_j|dx.$$

Recalling that ${|Q|}^{\frac{\beta }{n}}K(t)\le 1$, we have

$$\begin{array}{r}{\int }_B|A(x,u,Du)||D{\varphi }_j|dx\\ \le {\int }_B|A(x,u,Du)-A(x,u,p_0)||D{\varphi }_j|dx\\ +{\int }_B|A(x,u,p_0)-A(x_0,u_0,p_0)||D{\varphi }_j|dx\\ \le C{\int }_B{(1+{|Du|}^2+{|p_0|}^2)}^{\frac{m-2}{2}}|Du-p_0||D{\varphi }_j|dx\\ +C{\int }_{B}K(|u|){({|x-x_0|}^m+{|u-u_0|}^m)}^{\frac{\beta }{m}}{(1+|p_0|)}^{\frac{m}{2}}|D{\varphi }_j|dx\\ \le C{\int }_B({(1+{|p_0|}^2)}^{\frac{m-2}{2}}+{|Du-p_0|}^{m-2})|Du-p_0||D{\varphi }_j|dx\\ +C{\int }_{B}K(|u|)({|x-x_0|}^{\beta }+{|u-u_0|}^{\beta }){(1+|p_0|)}^{\frac{m}{2}}|D{\varphi }_j|dx\\ \le C{\int }_B{(1+{|p_0|}^2)}^{\frac{m-2}{2}}|Du-p_0||D{\varphi }_j|dx+C{\int }_B{|Du-p_0|}^{m-1}|D{\varphi }_j|dx\\ +C{\int }_{B}K(|u|){|x-x_0|}^{\beta }{(1+|p_0|)}^{\frac{m}{2}}|D{\varphi }_j|dx\\ +C{\int }_{B}K(|u|){|u-u_0|}^{\beta }{(1+|p_0|)}^{\frac{m}{2}}|D{\varphi }_j|dx\\ \le C{\int }_B[{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|Du-p_0|}^2+{|Du-p_0|}^m]dx\\ +C{\int }_B[{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|D{\varphi }_j|}^2+{|D{\varphi }_j|}^m]dx\\ +C{\int }_{B}K(|u|){|Q|}^{\frac{\beta }{n}}{(1+|p_0|)}^{\frac{m}{2}}|D{\varphi }_j|dx\\ +C{\int }_{B}K(|u|){|Q|}^{\frac{\beta }{n}}{(1+|p_0|)}^{\frac{m}{2}}|D{\varphi }_j|dx\\ \le C{|Q|}^a+C{\int }_B[{|D{\varphi }_j|}^2+{|D{\varphi }_j|}^m]dx+C{\int }_B|D{\varphi }_j|dx+C{\int }_B|D{\varphi }_j|dx\\ \le C{|Q|}^a+C{\int }_B(2^{2j}+2^{mj})dx+C{\int }_B{2}^{j}dx.\end{array}$$

(3.13)

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

$$\begin{array}{rl}\left|J_2^{\prime }\right|& \le C\sum _{Q\in W_j}({|Q|}^a+{|Q|}^{-\frac{m}{n}}|Q|+C{|Q|}^{-\frac{2}{n}}|Q|+{|Q|}^{-\frac{1}{n}}{|Q|}^n)\\ \le C\sum _{Q\in W_j}{|Q|}^a\le C{\int }_{\bigcup W_j}d{(x,K)}^{m(k-1)-k}.\end{array}$$

(3.14)

Since $\bigcup W_j\subset \mathrm{\Omega }\setminus K$ and $|\bigcup W_j|\to 0$ as $j\to \mathrm{\infty }$, it follows that $J_2^{\prime }\to 0$ as $j\to \mathrm{\infty }$.

For

$$\left|J_2^{″}\right|\le C\sum _{Q\in W_j}{\int }_B{\varphi }_{j}A(x,u,Du)D\psi dx.$$

Similarly, we get

$$\begin{array}{r}{\int }_B{\varphi }_{j}A(x,u,Du)D\psi dx\\ \le {\int }_B(A(x,u,Du)-A(x,u,p_0))D\psi dx+{\int }_B(A(x,u,p_0)-A(x_0,u_0,p_0))D\psi dx\\ \le C{\int }_B{(1+{|Du|}^2+{|p_0|}^2)}^{\frac{m-2}{2}}|Du-p_0|dx\\ +C{\int }_{B}K(|u|){({|x-x_0|}^m+{|u-u_0|}^m)}^{\frac{\beta }{m}}{(1+|p_0|)}^{\frac{m}{2}}|D\psi |dx\\ \le C{\int }_B({(1+{|p_0|}^2)}^{\frac{m-2}{2}}+{|Du-p_0|}^{m-2})|Du-p_0||D\psi |dx\\ +C{\int }_{B}K(|u|)({|x-x_0|}^{\beta }+{|u-u_0|}^{\beta }){(1+|p_0|)}^{\frac{m}{2}}|D\psi |dx\\ \le C{\int }_B{(1+{|p_0|}^2)}^{\frac{m-2}{2}}|Du-p_0||D\psi |dx+C{\int }_B{|Du-p_0|}^{m-1}|D\psi |dx\\ +C{\int }_{B}K(|u|){|x-x_0|}^{\beta }{(1+|p_0|)}^{\frac{m}{2}}|D\psi |dx\\ +C{\int }_{B}K(|u|){|u-u_0|}^{\beta }{(1+|p_0|)}^{\frac{m}{2}}|D\psi |dx\\ \le C{\int }_B[{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|Du-p_0|}^2+{|Du-p_0|}^m]dx\\ +C{\int }_B[{(1+{|p_0|}^2)}^{\frac{m-2}{2}}{|D\psi |}^2+{|D\psi |}^m]dx\\ +C{\int }_B|D\psi |dx+C{\int }_{B}K(|u|){|Q|}^{\frac{\beta }{n}}|D\psi |dx\\ \le C{|Q|}^a+C{\int }_B({|D\psi |}^2+{|D\psi |}^m)dx+C{\int }_B|D\psi |dx\\ \le C{|Q|}^a+C{\int }_{B}dx.\end{array}$$

Thus,

$$\left|J_2^{″}\right|\le C\sum _{Q\in W_j}({|Q|}^a+|Q|)\le C\sum _{Q\in W_j}{|Q|}^a\le C{\int }_{\bigcup W_j}d{(x,K)}^{m(k-1)-k}.$$

(3.15)

Since $u\in W_{\mathrm{loc}}^{1,D}(\mathrm{\Omega })$ and $|\bigcup W_j|\to 0$ as $j\to \mathrm{\infty }$, we have $J_2^{″}\to 0$ as $j\to \mathrm{\infty }$. In order to estimate $J_2^{‴}$, we should use (H4):

$$J_2^{‴}={\int }_B\overline{f(x,u,Du)}\psi {\varphi }_{j}dx\le C{\int }_B{|Du-p_0|}^{m}dx+C{\int }_B{|p_0|}^{m}dx=J_3^{\prime }+J_3^{″}.$$

(3.16)

Similar to the estimate of (3.14), using the Caccioppoli inequality (3.1) and the inequality (3.8), we get

$$\begin{array}{rl}{J}_3^{\prime }& \le C\sum _{Q\in W_j}{\int }_Q{|Du-p_0|}^{m}dx\le C\sum _{Q\in W_j}{|Q|}^{\frac{(n+mk-m)}{n}}\\ \le C{\int }_{\bigcup W_j}d{(x,K)}^{n+mk-m}dx\le C{\int }_{\bigcup W_j}d{(x,K)}^{m(k-1)-k}dx.\\ \to 0(j\to \mathrm{\infty }),\end{array}$$

and

$$\begin{array}{rl}{J}_3^{″}& \le C\sum _{Q\in W_j}{\int }_{Q}dx=C\sum _{Q\in W_j}|Q|\\ \le C{\int }_{\bigcup W_j}d{(x,K)}^{n}dx\le C{\int }_{\bigcup W_j}d{(x,K)}^{n+mk-m}dx\\ \le C{\int }_{\bigcup W_j}d{(x,K)}^{m(k-1)-k}dx\\ \to 0(j\to \mathrm{\infty }).\end{array}$$

Hence $J_2\to 0$.

Combining estimates $J_1$ and $J_2$ in (3.11), we prove Theorem 1.2. □