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

Journal of Inequalities and Applications20142014:443

https://doi.org/10.1186/1029-242X-2014-443

Received: 27 February 2014

Accepted: 23 October 2014

Published: 4 November 2014

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.

Keywords

A-harmonic systemA-Dirac systemCaccioppoli estimatenatural growth conditionremovable theorem

1 Introduction

In this paper, we consider the regularity theory on an A-Dirac system,
D A ˜ ( x , u , D u ) = f ( x , u , D u ) , in  Ω ,
(1.1)
and an A-harmonic system,
div A ( x , u , u ) = f ( x , u , u ) , in  Ω .
(1.2)

Here Ω is a bounded domain in R n ( n 2 ), A ( x , u , u ) and f ( x , u , u ) are measurable functions defined on Ω × R n × R n N , N is an integer with N > 1 , u : Ω R n is a vector valued function. Furthermore, A ( x , u , u ) and f ( x , u , 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
| A ( x , u , p ) p | C ( 1 + | p | 2 ) m 2 2 for all  ( x , u , p ) Ω × R n × R n N .
(H2) A ( x , u , p ) are uniformly strongly elliptic, that is, for some λ > 0 we have
( A ( x , u , p ) p ν i α ) ν j β λ ( 1 + | p | 2 ) m 2 2 | ν | 2 .
(H3) There exist β ( 0 , 1 ) and K : [ 0 , ) [ 0 , ) monotone nondecreasing such that
| A ( x , u , p ) A ( x ˜ , u ˜ , p ) | K ( | u | ) ( | x x ˜ | m + | u u ˜ | m ) β m ( 1 + | p | ) m 2

for all x , x ˜ Ω , u , u ˜ R n , and p R n N . Without loss of generality, we take K 1 .

(H4) There exist constants C 1 and C 2 such that
| f ( x , u , p ) | C 1 | p | m + C 2 .
(H1) and (H2) imply
| A ( x , u , p ) A ( x , u , ξ ) | C ( 1 + | p | 2 + | ξ | 2 ) m 2 2 | p ξ | ;
(1.3)
( A ( x , u , p ) A ( x , u , ξ ) ) ( p ξ ) λ ( 1 + | p | 2 + | ξ | 2 ) m 2 2 | p ξ | 2
(1.4)

for all x Ω , u R n and p , ξ R n N , where λ > 0 is a constant.

Definition 1.1 We say that a function u W loc 1 , m ( Ω ) L ( Ω ) is a weak solution to (1.2), if the equality
Ω A ( x , u , u ) ϕ d x = Ω f ( x , u , u ) ϕ d x
(1.5)

holds for all ϕ W 0 1 , m ( Ω ) 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 [210] 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 ω ( r ) , where the sets r n ω ( 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 L loc m ( Ω ) L ( Ω ) 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 Ω E , and u is of the type of an m , k -oscillation in Ω E . If for each compact subset K of E
Ω K d ( x , K ) m ( k 1 ) k < ,
(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 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 , , 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 2 n . We have an increasing tower R C H U 3  . The Clifford algebra U n is a 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 , S c ( A ) denotes the scalar part of A, that is, the coefficient of the element e 0 .

Throughout this paper, Ω R n 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 , which is sub-multiplicative, | A B | C | A | | B | .

The Dirac operator defined here is
D = j = 1 n e j x j .
(2.2)

Also D 2 = . 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 ( 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 , m ( 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 , m ( Ω , U n ) . Moreover, we write M Ω = { u : Ω U n | D u = 0 } for the space of monogenic functions in Ω.

Furthermore, we define the Dirac Sobolev space
W D , m ( Ω ) = { u U n | Ω | u | m + Ω | D u | m < } .
(2.3)

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

With those definitions and notations and also of the A-Dirac operator, we 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 , Du is defined by θ ( ϕ ) = D ϕ for a real-valued function ϕ, and we have
S c ( θ ( x ) θ ( y ) ) = x , y ,
(2.5)
| θ ( x ) | = | x | .
(2.6)
Here A ˜ ( x , ξ , η ) : Ω × U 1 × U n U n is defined by
A ˜ ( x , u , η ) = θ A ( x , u , θ 1 η ) ,
(2.7)
which means that (1.5) is equivalent to
Ω S c ( θ A ( x , u , u ) θ ( ϕ ) ) d x = Ω S c ( A ˜ ( x , u , D u ) D ϕ ) d x = Ω S c ( f ( x , u , D u ) ϕ ) d x .
(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 S c ( α ¯ β ) is the usual inner product α , β in R 2 n , when α and β are identified as vectors.

For convenience, we replace A ˜ with A and recast the structure systems above and define the operator:
A ( x , ξ , η ) : Ω × U 1 × U n U n ,
(2.9)

where A preserves the grading of the Clifford algebra, x A ( x , ξ , η ) is measurable for all ξ, η, and ξ A ( x , ξ , η ) , η A ( x , ξ , η ) are continuous for a.e. x Ω .

Definition 2.1 A Clifford valued function u W loc D , m ( Ω , U n k ) L ( Ω , U n k ) , for k = 0 , 1 , , n , is a weak solution to system (1.1) under conditions (H1)-(H4). If for all ϕ W 0 1 , m ( Ω , U n k ) , then we have
Ω A ( x , u , D u ) ¯ D ϕ d x = Ω f ( x , u , D u ) ¯ ϕ d x .
(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 λ > 2 C 1 M and where (H1)-(H4) are satisfied. Then for every x 0 Ω , u 0 U 1 k , p 0 U n k , and arbitrary σ > 1 we have
Q [ ( 1 + | p 0 | 2 ) m 2 2 | D u p 0 | 2 + | D u p 0 | m ] d x C { 1 ( σ | Q | ) 2 / n σ Q ( 1 + | p 0 | 2 ) m 2 2 | u P | 2 d x + 1 ( σ | Q | ) m / n σ Q | u P | m d x + σ Q G 2 d x } ,
(3.1)
where P = u ( x ) u 0 + p 0 ( x x 0 ) and
σ Q G 2 d x = σ | Q | { [ K ( | u 0 | + | p 0 | ) ( 1 + | p 0 | ) m 2 ] 2 1 β ( σ | Q | ) 2 β n + ( C 2 2 + C 1 2 | p 0 | 2 m ) ( σ | Q | ) 2 n } .
(3.2)
Proof Denote u ( x ) u 0 p 0 ( x x 0 ) by v ( x ) and 0 < | Q | 1 n < σ | Q | 1 n < min { 1 , dist ( x 0 , Ω ) } for σ > 1 , consider a standard cut-off function η C 0 ( σ Q ( x 0 ) ) satisfying 0 η 1 , | η | < 1 | Q | 1 / n , η 1 on Q ( x 0 ) . Then φ = η 2 v is admissible as a test-function, and we obtain
σ Q A ( x , u , D u ) ( D u p 0 ) η 2 d x = 2 σ Q A ( x , u , D u ) η v η d x + σ Q f ( x , u , D u ) φ d x .
We further have
σ Q A ( x , u , p 0 ) ( D u p 0 ) η 2 d x = 2 σ Q A ( x , u , p 0 ) η v η d x σ Q A ( x , u , p 0 ) D φ d x ,
and
σ Q A ( x 0 , u 0 , p 0 ) D φ d x = 0 .
Adding these equations yields
σ Q ( A ( x , u , D u ) A ( x , u , p 0 ) ) ( D u p 0 ) η 2 d x = 2 σ Q ( A ( x , u , D u ) A ( x , u , p 0 ) ) ( D u p 0 ) η v η d x σ Q ( A ( x , u , p 0 ) A ( x , u 0 + p 0 ( x x 0 ) , p 0 ) ) D φ d x σ Q ( A ( x , u 0 + p 0 ( x x 0 ) , p 0 ) A ( x 0 , u 0 , p 0 ) ) D φ d x + σ Q f ( x , u , D u ) φ d x I + I I + I I I + I V + V ,
(3.3)
where
I = 2 C σ Q ( 1 + | D u | 2 + | p 0 | 2 ) m 2 2 | D u p 0 | η | v | | η | d x ; I I = K ( | u 0 | + | p 0 | ) σ Q | v | β | D u p 0 | ( 1 + | p 0 | ) m 2 η 2 d x ; I I I = 2 K ( | u 0 | + | p 0 | ) σ Q | v | β + 1 | η | ( 1 + | p 0 | ) m 2 η d x ; I V = K ( | u 0 | + | p 0 | ) σ Q ( | x x 0 | m + | p 0 ( x x 0 ) | m ) β m ( 1 + | p 0 | ) m 2 ( 2 η | η | | v | + η 2 | D u p 0 | ) d x ; V = σ Q ( C 1 | D u | m + C 2 ) | v | η 2 d x ,

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

For positive ε, to be fixed later, using Young’s inequality, we have
I 2 C σ Q ( 1 + 2 | D u p 0 | 2 + 3 | p 0 | 2 ) m 2 2 | D u p 0 | η | v | | η | d x C [ σ Q ( 1 + | p 0 | 2 ) m 2 2 | D u p 0 | η | v | | η | d x + σ Q | D u p 0 | m 1 η | v | | η | d x ] C ε σ Q ( 1 + | p 0 | 2 ) m 2 2 | D u p 0 | 2 η 2 d x + C 1 ε σ Q ( 1 + | p 0 | 2 ) m 2 2 | v | 2 | η | 2 d x + C ε σ Q | D u p 0 | m η 2 d x + C ( ε ) σ Q | v | m | η | m d x .
Using Young’s inequality twice in II, we have
I I ε σ Q | D u p 0 | 2 η 2 d x + 1 ε K 2 ( | u 0 | + | p 0 | ) ( 1 + | p 0 | ) m σ Q | v | 2 β d x ε σ Q | D u p 0 | 2 η 2 d x + 1 ε σ Q ( 1 ( σ | Q | ) 1 / n | v | ) 2 d x + 1 ε [ K ( | u 0 | + | p 0 | ) ( 1 + | p 0 | ) m 2 ] 2 1 β ( σ | Q | ) ( 2 β 1 β + n ) / n ε σ Q ( 1 + | p 0 | 2 ) m 2 2 | D u p 0 | 2 η 2 d x + 1 ε σ Q ( 1 + | p 0 | 2 ) m 2 2 ( 1 ( σ | Q | ) 1 / n ) 2 | v | 2 d x + 1 ε [ K ( | u 0 | + | p 0 | ) ( 1 + | p 0 | ) m 2 ] 2 1 β ( σ | Q | ) ( 2 β 1 β + n ) / n ,
and similarly we see
I I I 1 2 σ Q | v | 2 | η | 2 d x + 4 K 2 ( | u 0 | + | p 0 | ) ( 1 + | p 0 | ) m σ Q ( σ | Q | ) 2 β n ( | v | ( σ | Q | ) 1 / n ) 2 β η 2 d x σ Q ( 1 ( σ | Q | ) 1 / n ) 2 | v | 2 d x + [ 4 K ( | u 0 | + | p 0 | ) ( 1 + | p 0 | ) m 2 ] 2 1 β ( σ | Q | ) ( 2 β 1 β + n ) / n ( 1 + | p 0 | 2 ) m 2 2 σ Q ( 1 ( σ | Q | ) 1 / n ) 2 | v | 2 d x + [ 4 K ( | u 0 | + | p 0 | ) ( 1 + | p 0 | ) m 2 ] 2 1 β ( σ | Q | ) ( 2 β 1 β + n ) / n
and
I V σ Q K ( | u 0 | + | p 0 | ) ( 1 + | p 0 | ) m 2 ( σ | Q | ) β n ( 1 + | p 0 | m ) β m ( η | D u p 0 | + 2 η | η | | v | ) d x σ Q K ( | u 0 | + | p 0 | ) ( 1 + | p 0 | ) m 2 ( σ | Q | ) β n ( 1 + | p 0 | ) β ( η | D u p 0 | + 2 η | η | | v | ) d x ε σ Q ( 1 + | p 0 | 2 ) m 2 2 | D u p 0 | 2 η 2 d x + σ Q ( 1 + | p 0 | 2 ) m 2 2 | η | 2 | v | 2 d x + ( 4 + 1 ε ) K 2 ( | u 0 | + | p 0 | ) ( 1 + | p 0 | ) 2 ( m 2 + β ) ( σ | Q | ) n + 2 β n ,
and for positive μ, to be fixed later, this yields
V = σ Q C 1 | D u | m | u u 0 p 0 ( x x 0 ) | η 2 d x + σ Q ( 1 ( σ | Q | ) 1 / n | v | η ) ( C 2 ( σ | Q | ) 1 n η ) d x σ Q C 1 [ ( 1 + μ ) | D u p 0 | m + ( 1 + 1 μ ) | p 0 | m ] | u u 0 p 0 ( x x 0 ) | η 2 d x + 1 2 ε C 2 2 ( σ | Q | ) n + 2 n + 1 2 ε ( σ | Q | ) 2 / n σ Q | v | 2 d x C 1 ( 1 + μ ) ( 2 M + p 0 ( σ | Q | ) 1 n ) σ Q | D u p 0 | m η 2 d x + C 1 ( 1 + 1 μ ) | p 0 | m σ Q | v | η 2 d x + 1 2 ε C 2 2 ( σ | Q | ) n + 2 n + 1 2 ε σ Q 1 ( σ | Q | ) 2 / n | v | 2 d x C 1 ( 1 + μ ) ( 2 M + p 0 ( σ | Q | ) 1 n ) σ Q | D u p 0 | m η 2 d x + 1 ε σ Q 1 ( σ | Q | ) 2 / n | v | 2 d x + ε 2 [ C 2 2 + C 1 2 ( 1 + 1 μ ) 2 | p 0 | 2 m ] ( σ | Q | ) n + 2 n .
By (1.4), we obtain
σ Q ( A ( x , u , D u ) A ( x , u , p 0 ) ) ( D u p 0 ) η 2 d x λ σ Q ( 1 + | D u | 2 + | p 0 | 2 ) m 2 2 | D u p 0 | 2 η 2 d x λ σ Q ( 1 + | D u p 0 | 2 + | p 0 | 2 ) m 2 2 | D u p 0 | 2 η 2 d x λ { σ Q ( 1 + | p 0 | 2 ) m 2 2 | D u p 0 | 2 η 2 d x + σ Q | D u p 0 | m η 2 d x } .
Combining these estimates in (3.3) and noting that K 2 K 2 1 β (as K 1 ), ( σ | Q | ) 2 β ( 1 β ) n ( σ | Q | ) 2 β n for σ > 1 , [ ( 1 + | p 0 | ) m 2 ] 2 1 β ( 1 + | p 0 | ) 2 ( m 2 + β ) , and 4 4 2 1 β , we can estimate
[ λ 2 C ε 2 ε C 1 ( 1 + μ ) ( 2 M + p 0 ( σ | Q | ) 1 n ) ] { σ Q ( 1 + | p 0 | 2 ) m 2 2 | D u p 0 | 2 η 2 d x + σ Q | D u p 0 | m η 2 d x } ( C ε + 2 ε + C ( ε ) + 2 ) { 1 ( σ | Q | ) 2 / n σ Q ( 1 + | p 0 | 2 ) m 2 2 | u P | 2 d x + 1 ( σ | Q | ) m / n σ Q | u P | m d x } + 2 ( 1 ε + 4 2 1 β ) [ K ( | u 0 | + | p 0 | ) ( 1 + | p 0 | ) m 2 ] 2 1 β ( σ | Q | ) n + 2 β n + ε 2 [ C 2 2 + C 1 2 ( 1 + 1 μ ) 2 | p 0 | 2 m ] ( σ | Q | ) n + 2 n .
Define ε = ε ( λ , m ) , μ = μ ( C 1 , M , m , λ ) small enough, we obtain
σ Q ( 1 + | p 0 | 2 ) m 2 2 | D u p 0 | 2 η 2 d x + σ Q | D u p 0 | m η 2 d x C { 1 ( σ | Q | ) 2 / n σ Q ( 1 + | p 0 | 2 ) m 2 2 | u P | 2 d x + 1 ( σ | Q | ) m / n σ Q | u P | m d x + σ Q G 2 d x } ,
where C = C ( m , λ , β , M ) and
σ Q G 2 d x = σ | Q | { [ K ( | u 0 | + | p 0 | ) ( 1 + | p 0 | ) m 2 ] 2 1 β ( σ | Q | ) 2 β n + ( C 2 2 + C 1 2 | p 0 | 2 m ) ( σ | Q | ) 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 L loc 1 ( Ω , U n ) , q > 0 , and that < k < 1 . We say that u is of the type of a 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 < .
(3.4)

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 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 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 L loc 1 ( Ω , R ) , F > 0 a.e., r R and σ 1 , σ 2 > 1 . If
sup σ 1 Q Q | Q | r Q F < ,
then
sup σ 2 Q Q | Q | r Q F < .
(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 Ω E . The decomposition consists of closed dyadic cubes with disjoint interiors which satisfy
  1. (a)

    Ω E = 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 [18].

If A R n and r > 0 , then we define the r-inflation of A as
A ( r ) = B ( x , r ) .
(3.6)
Let Q be a cube in the Whitney decomposition of Ω E . Using the Caccioppoli estimate (3.1), we have
Q [ ( 1 + | p 0 | 2 ) m 2 2 | D u p 0 | 2 + | D u p 0 | m ] d x C { 1 ( σ Q ) 2 / n σ Q ( 1 + | p 0 | 2 ) m 2 2 | u P | 2 d x + 1 ( σ Q ) m / n σ Q | u P | m d x + σ Q G 2 d x } ,
with (3.2)
σ Q G 2 d x C | σ Q | n + 2 β n H 2 ( 1 + | u Q | + | p 0 | ) ,
(3.7)
where
H ( t ) = [ K ˜ ( t ) ( 1 + t ) m 2 ] 2 1 β , K ˜ ( t ) = max { K ( t ) , C 1 , C 2 } ,
and choose | Q | small enough such that
| Q | β n H ( 1 + | u Q | + | p 0 | ) 1 .
By the definition of the q , k -oscillation condition, we have
Q [ ( 1 + | p 0 | 2 ) m 2 2 | D u p 0 | 2 + | D u p 0 | m ] d x C 1 | Q | 2 n | Q | 2 k + n n + C 2 | Q | m n | Q | ( m k + n ) / n + C 3 | Q | C | Q | a .
(3.8)
Here a = ( n + m k m ) / n . Since the problem is local (use a partition of unity), we show that (2.10) holds whenever ϕ W 0 1 , m ( 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 , 4 r ) . Then K is a compact subset of E. Also let W 0 be those cubes in the Whitney decomposition of Ω E which meet B = B ( x 0 , r ) . Notice that each cube Q W 0 lies in Ω K . Let γ = m ( k 1 ) k . First, since γ 1 , from [12] we have m ( K ) = m ( E ) = 0 . Also since n a n γ , using (1.6) and (3.8), we obtain
B ( x 0 , r ) [ ( 1 + | p 0 | 2 ) m 2 2 | D u p 0 | 2 + | D u p 0 | m ] d x C Q W 0 | Q | a C Q W 0 d ( Q , K ) n a C Q W 0 Q d ( x , K ) n a n d x C K ( 1 ) K d ( x , K ) n a n d x C K ( 1 ) K d ( x , K ) γ d x < .
(3.9)

Hence u W loc D , m ( 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 } .
(3.10)
Thus each ϕ j , j = 1 , 2 ,  , is Lipschitz, equal to 1 on K and as such ψ ( 1 ϕ j ) W 1 , m ( B E ) with compact support. Hence
B [ A ( x , u , D u ) ¯ D ψ f ( x , u , D u ) ¯ ψ ] d x = B E [ A ( x , u , D u ) ¯ D ( ψ ( 1 ϕ j ) ) f ( x , u , D u ) ¯ ψ ( 1 ϕ j ) ] d x + B [ A ( x , u , D u ) ¯ D ( ψ ϕ j ) f ( x , u , D u ) ¯ ψ ϕ j ] d x .
(3.11)
Let
J 1 = B E [ A ( x , u , D u ) ¯ D ( ψ ( 1 ϕ j ) ) f ( x , u , D u ) ¯ ψ ( 1 ϕ j ) ] d x , J 2 = B [ A ( x , u , D u ) ¯ D ( ψ ϕ j ) f ( x , u , D u ) ¯ ψ ϕ j ] d x .

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

Next we estimate J 2 as
J 2 = B A ( x , u , D u ) ψ D ϕ j d x + B ϕ j A ( x , u , D u ) D ψ d x B f ( x , u , D u ) ¯ ψ ϕ j d x = J 2 + J 2 + J 2 .
(3.12)
Noting that there exists a constant C such that | ψ | C < ,
| J 2 | C Q W j B | A ( x , u , D u ) | | D ϕ j | d x .
Recalling that | Q | β n K ( t ) 1 , we have
B | A ( x , u , D u ) | | D ϕ j | d x B | A ( x , u , D u ) A ( x , u , p 0 ) | | D ϕ j | d x + B | A ( x , u , p 0 ) A ( x 0 , u 0 , p 0 ) | | D ϕ j | d x C B ( 1 + | D u | 2 + | p 0 | 2 ) m 2 2 | D u p 0 | | D ϕ j | d x + C B K ( | u | ) ( | x x 0 | m + | u u 0 | m ) β m ( 1 + | p 0 | ) m 2 | D ϕ j | d x C B ( ( 1 + | p 0 | 2 ) m 2 2 + | D u p 0 | m 2 ) | D u p 0 | | D ϕ j | d x + C B K ( | u | ) ( | x x 0 | β + | u u 0 | β ) ( 1 + | p 0 | ) m 2 | D ϕ j | d x C B ( 1 + | p 0 | 2 ) m 2 2 | D u p 0 | | D ϕ j | d x + C B | D u p 0 | m 1 | D ϕ j | d x + C B K ( | u | ) | x x 0 | β ( 1 + | p 0 | ) m 2 | D ϕ j | d x + C B K ( | u | ) | u u 0 | β ( 1 + | p 0 | ) m 2 | D ϕ j | d x C B [ ( 1 + | p 0 | 2 ) m 2 2 | D u p 0 | 2 + | D u p 0 | m ] d x + C B [ ( 1 + | p 0 | 2 ) m 2 2 | D ϕ j | 2 + | D ϕ j | m ] d x + C B K ( | u | ) | Q | β n ( 1 + | p 0 | ) m 2 | D ϕ j | d x + C B K ( | u | ) | Q | β n ( 1 + | p 0 | ) m 2 | D ϕ j | d x C | Q | a + C B [ | D ϕ j | 2 + | D ϕ j | m ] d x + C B | D ϕ j | d x + C B | D ϕ j | d x C | Q | a + C B ( 2 2 j + 2 m j ) d x + C B 2 j d x .
(3.13)
Now for x Q 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 | a + | Q | m n | Q | + C | Q | 2 n | Q | + | Q | 1 n | Q | n ) C Q W j | Q | a C W j d ( x , K ) m ( k 1 ) k .
(3.14)

Since W j Ω K and | W j | 0 as j , it follows that J 2 0 as j .

For
| J 2 | C Q W j B ϕ j A ( x , u , D u ) D ψ d x .
Similarly, we get
B ϕ j A ( x , u , D u ) D ψ d x B ( A ( x , u , D u ) A ( x , u , p 0 ) ) D ψ d x + B ( A ( x , u , p 0 ) A ( x 0 , u 0 , p 0 ) ) D ψ d x C B ( 1 + | D u | 2 + | p 0 | 2 ) m 2 2 | D u p 0 | d x + C B K ( | u | ) ( | x x 0 | m + | u u 0 | m ) β m ( 1 + | p 0 | ) m 2 | D ψ | d x C B ( ( 1 + | p 0 | 2 ) m 2 2 + | D u p 0 | m 2 ) | D u p 0 | | D ψ | d x + C B K ( | u | ) ( | x x 0 | β + | u u 0 | β ) ( 1 + | p 0 | ) m 2 | D ψ | d x C B ( 1 + | p 0 | 2 ) m 2 2 | D u p 0 | | D ψ | d x + C B | D u p 0 | m 1 | D ψ | d x + C B K ( | u | ) | x x 0 | β ( 1 + | p 0 | ) m 2 | D ψ | d x + C B K ( | u | ) | u u 0 | β ( 1 + | p 0 | ) m 2 | D ψ | d x C B [ ( 1 + | p 0 | 2 ) m 2 2 | D u p 0 | 2 + | D u p 0 | m ] d x + C B [ ( 1 + | p 0 | 2 ) m 2 2 | D ψ | 2 + | D ψ | m ] d x + C B | D ψ | d x + C B K ( | u | ) | Q | β n | D ψ | d x C | Q | a + C B ( | D ψ | 2 + | D ψ | m ) d x + C B | D ψ | d x C | Q | a + C B d x .
Thus,
| J 2 | C Q W j ( | Q | a + | Q | ) C Q W j | Q | a C W j d ( x , K ) m ( k 1 ) k .
(3.15)
Since u W loc 1 , D ( Ω ) and | W j | 0 as j , we have J 2 0 as j . In order to estimate J 2 , we should use (H4):
J 2 = B f ( x , u , D u ) ¯ ψ ϕ j d x C B | D u p 0 | m d x + C B | p 0 | m d x = J 3 + J 3 .
(3.16)
Similar to the estimate of (3.14), using the Caccioppoli inequality (3.1) and the inequality (3.8), we get
J 3 C Q W j Q | D u p 0 | m d x C Q W j | Q | ( n + m k m ) n C W j d ( x , K ) n + m k m d x C W j d ( x , K ) m ( 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 + m k m d x C W j d ( x , K ) m ( k 1 ) k d x 0 ( j ) .

Hence J 2 0 .

Combining estimates J 1 and J 2 in (3.11), we prove Theorem 1.2. □

Declarations

Acknowledgements

Supported by National Natural Science Foundation of China (No: 11201415); Program for New Century Excellent Talents in Fujian Province University (No: JA14191).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Minnan Normal University, Fujian, China

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© Sun and Chen; licensee Springer. 2014

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