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Research | Open | Published:

A-Harmonic operator in the Dirac system

Abstract

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.

1 Introduction

In this paper, we derive the relation between an A-harmonic operator and A-Dirac systems under natural growth condition. The equation defined by an A-harmonic operator in the current paper is

divA(x,u)=f(x,u),
(1.1)

where

A(x,ξ):Ω× R n R n ,
(1.2)

xA(x,ξ) is measurable for all ξ, and ξA(x,ξ) is continuous for a.e. xΩ. Further assume that A(x,ξ) satisfies the following structure growth conditions for some constants a>0 and λ>0, with exponent p>1:

A ( x , ξ ) , ξ λ | ξ | p , | A ( x , ξ ) | a | ξ | p 1 ,
(1.3)

and f(x,ξ) satisfies the following natural growth condition for certain constants C 1 and C 2 :

f(x,u) C 1 | u | p + C 2 ,|u|<M<+ and λ> C 1 M.
(1.4)

The exponent p represents an exponent throughout the paper.

Definition 1.1 We call the function u W loc 1 , p (Ω) L (Ω) a weak solution to (1.1) under structure growth conditions (1.3) and (1.4) if the equality

Ω A ( x , u ) , ϕ = Ω f ( x , u ) , ϕ
(1.5)

holds for all ϕ W 1 , p (Ω) with compact support.

The A-Dirac systems in the paper are of the form

D A ˜ (x,Du)=f(x,Du).
(1.6)

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 [19] 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 [1] for monogenic functions with modulus of continuity ω(r), in which the sets of r n ω(r)-Hausdorff measure are removable. The results were extended to Hölder continuous analytic functions by Kaufman and Wu in [10]. 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 [11]. In the case of Hölder continuity, this was sharpened in [12] to a precise condition for removable sets for A-harmonic functions in terms of Hausdorff dimension. In [3], 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.

Theorem 1.2 Let E be a relatively closed subset of Ω. Suppose that u L loc p (Ω) has distributional first derivatives in Ω, u is a solution to the scalar part of A-Dirac system (1.6) under natural growth condition in ΩE, and u is of p,k-oscillation in ΩE. If for each compact subset K of E,

K ( 1 ) K d ( x , K ) p ( k 1 ) k <,
(1.7)

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 [3].

We write U n for the real universal Clifford algebra over R n . The Clifford algebra is generated over R by the basis of reduced products

{ e 1 , e 2 ,, e 1 e 2 ,, e 1 e n },
(2.1)

where { e 1 , e 2 ,, e n } is an orthonormal basis of R n with the relation e i e j + e j e i =2 δ i j . We write e 0 for the identity. The dimension of U n is R 2 n . We have an increasing tower RCHU . The Clifford algebra U n is graded algebra as U n = l U n l , where U n l are those elements whose reduced Clifford products have length l.

For A U n , Sc(A) denotes the scalar part of A, that is, the coefficient of the element e 0 .

Throughout, ΩR is a connected and open set with boundary Ω. A Clifford-valued function u:Ω U n can be written as u= α u α e α , where each u α is real-valued and e α are reduced products. The norm used here is given by | α u α e α |= ( α u α 2 ) 1 2 . This norm is sub-multiplicative, |AB|C|A||B|.

The Dirac operator used here is

D= j = 1 n e j x j .
(2.2)

Also, D 2 =. Here is the Laplace operator which operates only on coefficients. A function is monogenic when Du=0.

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 (Ω, U n ) for the space of Clifford-valued functions in Ω whose coefficients belong to the usual L q (Ω) space. Also, W 1 , p (Ω, U n ) is the space of Clifford-valued functions in Ω whose coefficients as well as their first distributional derivatives are in L q (Ω). We also write L loc q (Ω, U n ) for L q ( Ω , U n ), where the intersection is over all Ω compactly contained in Ω. We similarly write W loc 1 , p (Ω, U n ). Moreover, we write M Ω ={u:Ω U n |Du=0} for the space of monogenic functions in Ω.

Furthermore, we define the Dirac-Sobolev space

W D , p (Ω)= { u U n | Ω | u | p + Ω | D u | p < } .
(2.3)

The local space W loc D , p is similarly defined. Notice that if u is monogenic, then u L p (Ω) if and only if u W D , p (Ω). Also, it is immediate that W 1 , p (Ω) W D , p (Ω).

Under such definitions and notations, we can introduce the operator of A-Dirac. Define the linear isomorphism θ: R n U n 1 by

θ( ω 1 ,, ω n )= i = 1 n ω i e i .
(2.4)

For x,y R n , with Du defined by θ(ϕ)=Dϕ for a real-valued function ϕ, we have

Re ( θ ( x ) θ ( y ) ) =x,y,
(2.5)
| θ ( x ) | =|x|.
(2.6)

Here, A ˜ (x,ξ):Ω× U 1 U 1 is defined by

A ˜ (x,ξ)=θA ( x , θ 1 ξ ) ,
(2.7)

which means that (1.5) is equivalent to

Ω Re ( θ A ( x , u ) θ ( ϕ ) ) dx= Ω Re ( A ˜ ( x , D u ) D ϕ ) dx= Ω Re ( f ( x , D u ) θ ϕ ) dx.
(2.8)

For the Clifford conjugation ( e j 1 e j l ) ¯ = ( 1 ) l e j l e j 1 , we define a Clifford-valued inner product as α ¯ β. Moreover, the scalar part of this Clifford inner product Re( α ¯ β) is the usual inner product in R 2 n , α,β, when α and β are identified as vectors.

For convenience, we replace A ˜ with A and recast the structure equations above and define the operators:

A(x,ξ):Ω× U n U n ,
(2.9)

where A preserves the grading of the Clifford algebra, xA(x,ξ) is measurable for all ξ, and ξA(x,ξ) is continuous for a.e. xΩ. Furthermore, here A(x,ξ) satisfies the structure conditions with p>1:

Re ( A ( x , ξ ) ¯ ξ ) λ | ξ | p ,
(2.10)
| A ( x , ξ ) | a | ξ | p 1
(2.11)

for some constant a>0. Hence we can definite the weak solution of equation (1.6) as follows.

Definition 2.1 A Clifford-valued function u W loc D p (Ω, U n k ), for k=0,1,,n, is a weak solution to equation (1.6) under structure conditions (2.10) and (2.11), and further assume that the inhomogeneity term f(x,Du) satisfies the natural growth condition

f(x,Du) a 1 | D u | p + a 2 ,|u|<M<+ and λ> a 1 M
(2.12)

for certain constants a 1 and a 2 and λ>0. Then for all ϕ W 1 , p (Ω, U n k ) with compact support, we have

Ω A ( x , D u ) ¯ Dϕdx= Ω f ( x , D u ) ¯ ϕdx.
(2.13)

Notice that when A is the identity, then the homogeneity part of (2.13),

Ω A ( x , D u ) ¯ Dϕdx=0,
(2.14)

is the Clifford Laplacian. Moreover, these equations generalize the important case of the P-Dirac equation

D ( | D u | n 2 D u ) =0.
(2.15)

Here, A(x,ξ)= | ξ | p 2 ξ.

These equations were introduced and their conformal invariance was studied in [2].

Furthermore, when u is a real-valued function, (2.14) implies that A(x,u) is a harmonic field and locally there exists a harmonic function H such that A(x,u)=H. If A(x,ξ) is invertible, then u= A 1 (x,H). 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

In this section, we establish the main results. At first, a suitable Caccioppoli estimate for solutions to (2.13) is necessary. Just as which appears in [3, 4].

Theorem 3.1 Let u be a solution to the scalar part of (2.13) defined by Definition  2.1, and Q is a cube in Ω. Then, for σ>1, we have

Q | D u | p dxC | Q | p n σ Q | u u σ Q | p dx+C | Q | n ( p 1 ) + p n ( p 1 ) .
(3.1)

Proof Let the cut-off function η C 0 (Ω), η>0, η=1 in Q and |Dη|C | Q | 1 / n . Choose ϕ=(u u σ Q ) η p as a test function in (2.13). Then

Dϕ=p η p 1 (Dη)(u u σ Q )+ η p Du.

The solution definition (2.13) yields

σ Q A ( x , D u ) ¯ p η p 1 ( D η ) ( u u σ Q ) d x + σ Q A ( x , D u ) ¯ η p D u d x = σ Q f ( x , D u ) ¯ ( u u σ Q ) η p d x .
(3.2)

Using the structure condition (2.10), we have

λ σ Q | D u | p η p d x σ Q f ( x , D u ) ¯ ( u u σ Q ) η p d x + [ σ Q A ( x , D u ) ¯ p η p 1 ( D η ) ( u u σ Q ) d x ] = I 1 + I 2 .
(3.3)

By natural growth condition (2.12) and then using Young’s inequality, we have

I 1 σ Q a 1 | D u | p ( u u σ Q ) η p d x + σ Q a 2 η p ( u u σ Q ) d x a 1 M σ Q | D u | p η p d x + σ Q u u σ Q | Q | 1 n ( a 2 | Q | 1 n η p ) d x a 1 M σ Q | D u | p η p d x + ε 1 σ Q ( u u σ Q ) p | Q | p n d x + C ( ε 1 ) σ Q ( a 2 | Q | 1 n η p ) p p 1 d x .
(3.4)

Notice that

σ Q ( a 2 | Q | 1 n η p ) p p 1 dx C 1 | Q | n ( p 1 ) + p n ( p 1 ) .
(3.5)

Then, using (2.11), Hölder’s and then Young’s inequalities, we have

| I 2 | p a σ Q | u u σ Q | | η | | D u | p 1 | η | p 1 d x C ( σ Q | u u σ Q | p | η | p d x ) 1 p ( σ Q | D u | p η p d x ) p 1 p C ( ε 2 ) ( σ Q | u u σ Q | p | η | p d x ) 1 p p + ε 2 σ Q | D u | p η p d x C 2 σ Q | u u σ Q | p | Q | p n d x + ε 2 σ Q | D u | p η p d x .
(3.6)

Combining (3.4), (3.5) with (3.6) in inequality (3.3), we obtain

( λ a 1 M ε 2 ) Ω | D u | p | η | p d x ( C 2 + C ( ε 1 ) ) | Q | p n σ Q | u u σ Q | p d x + ε 1 C 1 | Q | n ( p 1 ) + p n ( p 1 ) .

Choosing ε 2 small enough such that λ a 1 M ε 2 >0, we have

Q | D u | p dxC | Q | p n σ Q | u u σ Q | p dx+C | Q | n ( p 1 ) + p n ( p 1 ) .

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 [3].

Definition 3.2 Assume that u L loc 1 (Ω, U n ), q>0 and that <k<1. We say that u is of q,k-oscillation in Ω when

sup 2 Q Ω | Q | ( q k + n ) / q n inf u Q M Q ( Q | u u Q | q ) 1 / q <.
(3.7)

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 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<k1 [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 <k<1 [4, 14]. 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 [4].

Notice that monogenic functions satisfy (3.7) just as the spaceof 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 sq and if u is of q,k-oscillation, then u is of s,k-oscillation.

The following lemma [3] shows that Definition 3.2 is independent of the expansion factor of the sphere.

Lemma 3.3 Suppose that F L loc 1 (Ω,R), F>0 a.e., rR and σ 1 , σ 2 >1. If

sup σ 1 Q Q | Q | r Q Fdx<,
(3.8)

then

sup σ 2 Q Q | Q | r Q Fdx<.
(3.9)

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 ΩE. The decomposition consists of closed dyadic cubes with disjoint interiors which satisfy

  1. (a)

    ΩE= Q W Q,

  2. (b)

    | Q | 1 / n d(Q,Ω)4 | Q | 1 / n ,

  3. (c)

    (1/4) | Q 1 | 1 / n | Q 2 | 1 / n 4 | Q 1 | 1 / n when Q 1 Q 2 is not empty.

Here, d(Q,Ω) is the Euclidean distance between Q and the boundary of Ω [15].

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

A(r)=B(x,r).
(3.10)

Using the Caccioppoli estimate and the p,k-oscillation condition, we have

Q | D u | p d x C | Q | p n σ Q | u u σ Q | p d x + C | Q | n ( p 1 ) + p n ( p 1 ) C inf u Q M σ Q | Q | p n σ Q | u u σ Q | p d x + C | Q | n ( p 1 ) + p n ( p 1 ) C | Q | p n | Q | p k + n n + C | Q | n ( p 1 ) + p n ( p 1 ) C | Q | a .
(3.11)

Here, a=(n+pkp)/n and note that <k1. Since the problem is local (use a partition of unity), we show that (2.13) holds whenever ϕ W 0 1 , p (B( x 0 ,r)) with x 0 E and r>0 sufficiently small. Choose r=(1/5 n )min{1,d( x 0 ,Ω)}, and let K=E B ¯ ( x 0 ,4r). Then K is a compact subset of E. Also, let W 0 be those cubes in the Whitney decomposition of ΩE which meet B=B( x 0 ,r). Notice that each cube Q W 0 lies in K(1)K. Let γ=p(k1)k. First, since γ1, from [2] we have m(K)=m(E)=0. Also, since anγ, using (3.6) and (3.11), we obtain

B ( x 0 , r ) | D u | p d x C Q W 0 | Q | a / n C Q W 0 d ( Q , K ) a C Q W 0 Q d ( x , K ) a n d x C K ( 1 ) K d ( x , K ) a n d x C K ( 1 ) K d ( x , K ) γ d x < .
(3.12)

Hence u W loc D , p (Ω).

Next, let B=B( x 0 ,r) and assume that ψ C 0 (B). Also, let W j , j=1,2, , be those cubes Q W 0 with l(Q) 2 j .

Consider the scalar functions

ϕ j =max { ( 2 j d ( x , K ) ) 2 j , 0 } .
(3.13)

Thus each ϕ j , j=1,2, , is Lipschitz, equal to 1 on K and as such ψ(1 ϕ j ) W 1 , p (BE) with compact support. Hence

B [ A ( x , D u ) ¯ D ψ f ( x , D u ) ¯ ψ ] d x = B E [ A ( x , D u ) ¯ D ( ψ ( 1 ϕ j ) ) f ( x , D u ) ¯ ψ ( 1 ϕ j ) ] d x + B [ A ( x , D u ) ¯ D ( ψ ϕ j ) f ( x , D u ) ¯ ψ ϕ j ] d x .
(3.14)

Let

J 1 = B E [ A ( x , D u ) ¯ D ( ψ ( 1 ϕ j ) ) f ( x , D u ) ¯ ψ ( 1 ϕ j ) ] d x , J 2 = B [ A ( x , D u ) ¯ D ( ψ ϕ j ) f ( x , D u ) ¯ ψ ϕ j ] d x .

Since u is a solution in BE, J 1 =0.

Also, we have

J 2 = B A ( x , D u ) ψ D ϕ j d x + B ϕ j A ( x , D u ) D ψ d x B f ( x , D u ) ¯ ψ ϕ j d x = J 2 + J 2 + J 2 .
(3.15)

Now there exists a constant C such that |ψ|C<. Hence, using Hölder’s inequality,

| J 2 | C Q W j Q | A ( X , D u ) | | D ϕ j | d x C Q W j Q | D u | p 1 | D ϕ j | d x C Q W j ( Q | D u | p d x ) ( p 1 ) / p ( Q | D ϕ j | p d x ) 1 / p .
(3.16)

Next, using the estimate (3.11), the above becomes

| J 2 | C Q W j | Q | ( p ( k 1 ) + n ) ( p 1 ) / n p 2 j | Q | 1 / p .
(3.17)

Now, for xQ W j , d(x,K) is bounded above and below by a multiple of | Q | 1 / n and for Q W j , | Q | 1 / n 2 j . Hence

| J 2 | C Q W j | Q | 1 / n + 1 / p + ( p ( k 1 ) + n ) ( p 1 ) / n p C W j d ( x , K ) p ( k 1 ) k dx.
(3.18)

Since W j K(1)K and | W j |0 as j, it follows that J 2 0 as j.

Next, again using Hölder’s inequality, we obtain

| J 2 | sup B | D ψ | ( W j | D u | p d x ) ( p 1 ) / p | W j | 1 / p C ( K K ( 1 ) | D u | p d x ) ( p 1 ) / p | W j | 1 / p .
(3.19)

Since u W loc 1 , D (Ω) and | W j |0 as j, we have that J 2 0 as j. Hence J 2 0.

In order to estimate J 2 , using (3.12), we have

J 2 = B f ( x , D u ) ¯ ψ ϕ j dxC B | D u | p dx+C B dx= J 3 + J 3 .
(3.20)

Similar as the estimate of (3.18), using the Caccioppoli inequality and inequality (3.11), we get

J 3 C Q W j Q | D u | p d x C Q W j | Q | ( n + p k p ) / n C W j d ( x , K ) n + p k p d x C W j d ( x , K ) p ( k 1 ) k d x . 0 ( j )

and

J 3 C Q W j Q d x = C Q W j | Q | C W j d ( x , K ) n d x C W j d ( x , K ) n + p k p d x C W j d ( x , K ) p ( k 1 ) k d x 0 ( j ) .

Hence J 3 0.

Combining the estimates J 1 and J 2 in equation (3.14), we prove Theorem 1.2. □

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Acknowledgements

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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Correspondence to Shuhong Chen.

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Authors’ contributions

FS participated in design of the study and drafted the manuscript. SC participated in conceived of the study and the amendment of the paper. All authors read and approved the final manuscript.

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Keywords

  • A-harmonic operator
  • A-Dirac system
  • Caccioppoli estimate
  • natural growth condition