SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Boundary regularity for quasilinear elliptic systems with super quadratic controllable growth condition

Shuhong Chen1* and Zhong Tan2

Author Affiliations

1 School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, Fujian, 363000, China

2 School of Mathematical Science, Xiamen University, Xiamen, Fujian, 361005, China

For all author emails, please log on.

Boundary Value Problems 2014, 2014:88  doi:10.1186/1687-2770-2014-88


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/88


Received:3 November 2013
Accepted:10 April 2014
Published:6 May 2014

© 2014 Chen and Tan; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

We consider the boundary regularity for weak solutions to quasilinear elliptic systems under a super quadratic controllable growth condition, and we obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. Combined with existing results on the interior partial regularity, this result yields an upper bound on the Hausdorff dimension of a singular set at the boundary.

Keywords:
quasilinear elliptic systems; controllable growth condition; A-harmonic approximation technique; boundary partial regularity

1 Introduction

In this paper we are concerned with partial regularity for weak solutions of quasilinear elliptic systems:

D α ( A i j α β ( x , u ) D β u j ) = B i ( x , u , D u ) , (1.1)

where Ω is a bounded domain in R n , n 2 , N > 1 , and u and B i take values in R N . Here each A i j α β maps Ω × R N into R, and each B i maps Ω × R N × R n N into R. For m > 2 , we have the following.

(H1) There exists L > 0 such that

A i j α β ( x , ξ ) ( ν , ν ˜ ) L ( 1 + | ξ | 2 ) m 2 2 | ν | | ν ˜ | for all  ( x , ξ ) Ω ¯ × R N , ν , ν ˜ R n N .

(H2) A i j α β ( x , ξ ) is uniformly strongly elliptic, that is, for some λ > 0 we have

A i j α β ( x , ξ ) ( ν , ν ) λ ( 1 + | ξ | 2 ) m 2 2 | ν | 2 for all  ( x , ξ ) Ω ¯ × R N , ν R n N .

(H3) There exists a monotone nondecreasing concave function ω ( t , s ) : [ 0 , ) [ 0 , ) with ω ( t , 0 ) = 0 , continuous at 0, such that

| A i j α β ( x , u ) A i j α β ( y , v ) | ω ( | x y | m + | u v | m )

for all x , y Ω ¯ , u , v R N .

(H4) The B i fulfill the following controllable growth condition:

| B i ( x , ξ , ν ) | C ( | ν | m ( 1 1 r ) + | ξ | r 1 + 1 ) ,

where r = n m n m if n > m , or any exponent if n = m ; for all x Ω ¯ , ξ R N and ν R n N .

(H5) There exist s with s > n and a function g H 1 , s ( Ω , R N ) , such that we have

u | Ω = g | Ω .

Note that we trivially have g H 1 , m ( Ω , R N ) . Further, by Sobolev’s embedding theorem we have g C 0 , κ ( Ω , R N ) for any κ [ 0 , 1 n s ] . If g | Ω 0 , we will take g 0 on Ω.

If the domain we consider is an upper half unit ball B + , the boundary condition is the following.

(H5)′ There exist s with s > n and a function g H 1 , s ( B + , R N ) , such that we have

u | D = g | D .

Here we write B ρ ( x 0 ) = { x R n : | x x 0 | < ρ } , and further B ρ = B ρ ( 0 ) , B = B 1 . For x 0 R n 1 × { 0 } we write B ρ + ( x 0 ) for { x R n : x n > 0 , | x x 0 | < ρ } , and we set B ρ + = B ρ + ( 0 ) , B + = B 1 + . We further write D ρ ( x 0 ) = { x R n : x n = 0 , | x x 0 | < ρ } , and we set D ρ = D ρ ( 0 ) , D = D 1 . For bounded X R n with L n ( X ) > 0 we denote the average of a given function g L 1 ( X ) by X g d x , i.e. X g d x = 1 L n ( X ) X g d x . For ν L 1 ( Ω ) , x 0 Ω we set ν x 0 , R = Ω B ¯ R ( x 0 ) ν d H n 1 . In particular, for ν L 1 ( D ρ ( x 0 ) ) , x 0 D , we write ν x 0 , ρ = D ρ ( x 0 ) ν d H n 1 .

Now we can definite weak solutions to systems (1.1). Because there is a very large literature on the existence of weak solutions [1,2], we assume that a weak solution exists [3] and deal with the problem of regularity directly.

Definition 1.1 By a weak solution of (1.1) we mean a vector-valued function u W 1 , m ( Ω ¯ , R N ) such that

Ω A i j α β ( x , u ) ( D β u j , D α φ i ) d x = Ω B i ( x , u , D u ) φ i d x , (1.2)

holds for all test-functions φ C 0 ( Ω ¯ , R N ) and, by approximation, for all φ W 0 1 , m ( Ω ¯ , R N ) , where we have introduced the notation

A i j α β ( x , ξ ) ( ν , ν ˜ ) = ( A i j α β ( x , ξ ) ν ) ν ˜ . (1.3)

In the current situation, Sobolev’s embedding theorem yields the existence of a constant C s depending only on s, n, and N such that we have

sup B ρ + ( x 0 ) | g g x 0 , ρ | C s ρ 1 n s g H 1 , s ( B ρ + ( x 0 ) , R N ) (1.4)

for x 0 D , ρ 1 | x 0 | . Obviously, the inequality remains true if we replace g H 1 , s ( B ρ + ( x 0 ) , R N ) by g H 1 , s ( B + , R N ) , which we will henceforth abbreviate simply as g H 1 , s .

We also note here that Poincaré’s inequality in this setting yields

B ρ + ( x 0 ) | g g x 0 , ρ | m d x C p ρ m B ρ + ( x 0 ) | D g | m d x (1.5)

for a constant C p depending only on n.

Finally, we fix an exponent σ ( 0 , 1 ) as follows: if g 0 , σ can be chosen arbitrary (but henceforth fixed); otherwise we take σ fixed in ( 0 , 1 n s ] .

Under such assumptions, one cannot expect that weak solutions to (1.1) will be classical [4]. This was first shown by De Giorgi [5]. Thus, our goal is to establish a partial regularity for weak solutions of systems (1.1).

There are some previous partial regularity results at boundary for inhomogeneous quasilinear systems. Arkhipova has studied regularity up to the boundary for nonlinear and quasilinear systems [6-8]. For systems in diagonal form, boundary regularity was first established by Wiegner [9], and the proof was generalized and extended by Hildebrandt-Widman [10]. Jost-Meier [11] established full regularity in a neighborhood of boundary for minima of functionals with the form Ω A ( x , u ) | D u | 2 d x .

The results which are most closely related to that given here were shown in [12] and [13]. In this paper, we would get the desired conclusions by the method of A-harmonic approximation. The A-harmonic approximation technique is a natural extension of harmonic approximation technique. In [14] Simon used harmonic approximation method to simplify Allard’s [15] regularity theorem and later on Schoen and Uhlenbeck’s [16] regularity result for harmonic maps. The idea was generalized to more general linear operators by Duzaar and Steffen [17], in order to deal with the regularity of almost minimizers to elliptic variational integrals in the setting of geometric measure theory. As a by-product Duzaar and Grotowski [18] were able to use the idea of A-harmonic approximation to deal with elliptic systems under quadratic growth, even to the boundary points for nonlinear elliptic systems [19] and variational problems [20].

In this context, we use an A-harmonic approximation method to establish boundary regularity results.

Theorem 1.1Let Ω be a bounded domain in R N , with boundary of class C 1 . Letube a weak solution of (1.1) satisfying the structural conditions (H1)-(H5). Consider a fixed γ ( 0 , σ ] . Then there exist positive R 1 and ε 0 (depending only onn, N, λ, L, ω ( ) andγ) with the property that

B R ( x 0 ) Ω [ | u u x 0 , R | 2 + | u u x 0 , R | m R m 2 ] d x + [ g H 1 , s 2 + g H 1 , s m ] R 2 ( 1 n s ) + R 2 ε 0 2

for some R ( 0 , R 1 ] , x 0 Ω , which implies u C 0 , γ ( B ¯ R 2 ( x 0 ) Ω ¯ , R N ) .

Note in particular that the boundary condition (H5) means that u x 0 , R makes sense: in fact, we have u x 0 , R = g x 0 , R .

A standard covering argument [3] allows us to obtain the following.

Corollary 1.1Under the assumptions of Theorem 1.1 the singular set of the weak solutionuhas ( n m ) -dimensional Hausdorff measure zero in Ω ¯ .

If the domain of the main step in proving Theorem 1.1 is a half ball, the result then is the following.

Theorem 1.2Consider a weak solution of (1.1) on the upper half unit ball B + which satisfies the structural conditions (H1)-(H4) and (H5)′. Then there exist positive R 0 and ε 0 (depending only onn, N, λ, L, ω ( ) andγ) with the property that

B R ( x 0 ) Ω [ | u u x 0 , R | 2 + | u u x 0 , R | m R m 2 ] d x + [ g H 1 , s 2 + g H 1 , s m ] R 2 ( 1 n s ) + R 2 ε 0 2

for some R ( 0 , R 0 ] , x 0 Ω , which implies u C 0 , γ ( B ¯ R 2 ( x 0 ) , R N ) .

Analogously to above, the boundary condition (H5)′ ensures that u x 0 , R exists, and u x 0 , R = g x 0 , R .

2 The A-harmonic approximation technique

In this section we present an A-harmonic approximation lemma [12], and some standard results due to Campanato [21,22].

Lemma 2.1 (A-harmonic approximation lemma)

Consider fixed positiveλandL, and n , N N with n 2 . Then for any given ε > 0 there exists δ = δ ( n , N , λ , L , ε ) ( 0 , 1 ] with the following property: for any A Bil ( R n N ) satisfying

A ( ν , ν ) λ | ν | 2 for all  ν R n N (2.1)

and

| A ( ν , ν ¯ ) | L | ν | | ν ¯ | for all  ν , ν ¯ R n N (2.2)

for any w H 1 , 2 ( B ρ + ( x 0 ) , R N ) (for some ρ > 0 , x 0 R n ) satisfying

ρ 2 n B ρ + ( x 0 ) | D w | 2 d x 1 (2.3)

and

| ρ 2 n B ρ + ( x 0 ) A ( D g , D φ ) d x | δ ρ sup B ρ + ( x 0 ) | D φ | , (2.4)

and

w | D ρ ( x 0 ) = 0 (2.5)

for all φ C 0 1 ( B ρ + ( x 0 ) , R N ) , there exists anA-harmonic function

v H ˜ = { w ˜ H 1 , 2 ( B ρ + ( x 0 ) , R N ) | ρ 2 n B ρ + ( x 0 ) | D w ˜ | 2 d x 1 , w ˜ | D ρ ( x 0 ) 0 } ,

with

ρ n B ρ + ( x 0 ) | v w | 2 d x ε . (2.6)

Next we recall a characterization of Hölder continuous functions with a slight modification [21].

Lemma 2.2Consider n N , n 2 , and x 0 R n 1 × { 0 } . Suppose that there are positive constantsκandα, with α ( 0 , 1 ] such that, for some ν L 2 ( B 6 R + ( x 0 ) ) , we have the following:

inf μ R { ρ n B ρ + ( y ) | ν μ | 2 d x } κ 2 ( ρ R ) 2 α (2.7)

for all y D 2 R ( x 0 ) and ρ 4 R ; and

inf μ R ρ n { B ρ ( y ) | ν μ | 2 d x } κ 2 ( ρ R ) 2 α (2.8)

for all y B 2 R + ( x 0 ) and B ρ ( y ) B 2 R + ( x 0 ) .

Then there exists a Hölder continuous representative of the L 2 -class ofνon B ¯ R + ( x 0 ) , and for this representative ν ¯ we have

| ν ¯ ( x ) ν ¯ ( z ) | C κ ( | x z | R ) α (2.9)

for all x , z B ¯ R + ( x 0 ) , with the constant C κ depending only onnandα.

We close this section by a standard estimate for the solutions to homogeneous second order elliptic systems with constant coefficients, due originally to Campanato [22].

Lemma 2.3Consider fixed positiveλandL, and n , N N with n 2 . Then there exists C 0 depending only onn, N, λ, andL (without loss of generality we take C 0 1 ) such that for A Bil ( R n N ) satisfying (2.1) and (2.2), anyA-harmonic functionhon B ρ + ( x 0 ) with h | D ρ ( x 0 ) 0 satisfies

ρ 2 sup B ρ 2 + ( x 0 ) | D h | 2 C 0 ρ 2 n B ρ + ( x 0 ) | D h | 2 d x .

3 Caccioppoli inequality

In this section we prove Caccioppoli’s inequality.

Theorem 3.1 (Caccioppoli inequality)

Let u W 1 , m ( Ω ¯ , R N ) be a weak solution of systems (1.1) under the conditions (H1)-(H5). Then there exists ρ 0 > 0 depending only onL, s, and g H 1 , s , such that for all B ρ + ( x 0 ) B R + ( x 0 ) , with x 0 D + , ρ < R < ρ 0 , we have

B ρ + ( x 0 ) ( | D u | 2 + | D | u | m 2 | 2 ) d x C 1 B R + ( x 0 ) [ 1 ( R ρ ) 2 | u ( x ) u x 0 , R | 2 + 1 ( R ρ ) m | u ( x ) u x 0 , R | m ] d x + C 2 [ g H 1 , s 2 R n ( 1 2 s ) + g H 1 , s m R n ( 1 m s ) ] + C 3 α n R n , (3.1)

where C 1 , C 2 , C 3 depend only onλ, L, m, and g L ( B , R N ) , and C 2 additionally on C p , C s , and also ons.

Proof Now we consider a cut off function η C 0 ( B ρ + ( x 0 ) ) , satisfying 0 η 1 , η 0 on B ρ + ( x 0 ) , and | η | < 1 R ρ . Then the function φ = ( u g ) η 2 is in W 0 1 , m ( B ρ + ( x 0 ) , R N ) , and thus it can be taken as a test-function.

Using conditions (H1), (H4), (H5), and the definition of weak solutions (1.2), we have

B R + ( x 0 ) A i j α β ( , u ) ( D u , D u ) η 2 d x 2 L B R + ( x 0 ) ( 1 + | u | 2 ) m 2 2 | D u | | D η | | u g | η d x + L B R + ( x 0 ) ( 1 + | u | 2 ) m 2 2 | D u | | D g | η 2 d x + C B R + ( x 0 ) ( | D u | m ( 1 1 r ) + | u | r 1 + 1 ) | φ | d x . (3.2)

By Young’s, Hölder’s and then Sobolev’s inequalities, together with the estimate inequalities (1.4), (1.5), and the fact that u x 0 , R = g x 0 , R , yields

B R + ( x 0 ) A i j α β ( x , u ) ( D u , D u ) η 2 d x 2 ε B R + ( x 0 ) ( 1 + | u | 2 ) m 2 2 | D u | 2 η 2 d x + 2 L ε B R + ( x 0 ) ( 1 + | u | 2 ) m 2 2 | D η | 2 | u u x 0 , R | 2 d x + 2 L 2 ε B R + ( x 0 ) ( 1 + | u | 2 ) m 2 2 | D η | 2 | g g x 0 , R | 2 d x + L 2 ε B R + ( x 0 ) ( 1 + | u | 2 ) m 2 2 | D g | 2 η 2 d x + C ( B R + ( x 0 ) ( | D u | m + | u | r + 1 ) d x ) 1 1 r ( B R + ( x 0 ) | D φ | m d x ) 1 m . (3.3)

Using Young’s inequality again,

B R + ( x 0 ) ( 1 + | u | 2 ) m 2 2 | D η | 2 | u u x 0 , R | 2 d x = B R + ( x 0 ) ( 1 + | u u x 0 , R + u x 0 , R | 2 ) m 2 2 | D η | 2 | u u x 0 , R | 2 d x 2 m 2 2 B R + ( x 0 ) ( 1 + | u x 0 , R | 2 ) m 2 2 | D η | 2 | u u x 0 , R | 2 d x + 2 m 2 2 B R + ( x 0 ) | u u x 0 , R | m ( R ρ ) m d x ,

here we have used the fact that | D η | < 1 R ρ and 0 < ρ < R < 1 .

Similarly, we have

B R + ( x 0 ) ( 1 + | u | 2 ) m 2 2 | D η | 2 | g g x 0 , R | 2 d x = B R + ( x 0 ) ( 1 + | u u x 0 , R + u x 0 , R | 2 ) m 2 2 | D η | 2 | g g x 0 , R | 2 d x 2 m 2 2 B R + ( x 0 ) ( 1 + | u x 0 , R | 2 ) m 2 2 | D η | 2 | g g x 0 , R | 2 d x + 2 m 2 2 B R + ( x 0 ) | u u x 0 , R | m 2 | D η | 2 | g g x 0 , R | 2 d x 2 m 2 2 B R + ( x 0 ) ( 1 + | u x 0 , R | 2 ) m 2 2 | D η | 2 | g g x 0 , R | 2 d x + 2 m 2 2 B R + ( x 0 ) | u u x 0 , R | m d x + 2 m 2 2 B R + ( x 0 ) | D η | m | g g x 0 , R | m d x 2 m 2 2 ( 1 + | u x 0 , R | 2 ) m 2 2 C p B R + ( x 0 ) | D g | 2 d x + 2 m 2 2 C p B R + ( x 0 ) | D g | m d x + 2 m 2 2 B R + ( x 0 ) | u u x 0 , R | m d x , B R + ( x 0 ) ( 1 + | u | 2 ) m 2 2 η 2 | D g | 2 d x 2 m 2 2 B R + ( x 0 ) ( 1 + | u x 0 , R | 2 ) m 2 2 η 2 | D g | 2 d x + 2 m 2 2 B R + ( x 0 ) | u u x 0 , R | m 2 η 2 | D g | 2 d x 2 m 2 2 B R + ( x 0 ) ( 1 + | u x 0 , R | 2 ) m 2 2 η 2 | D g | 2 d x + 2 m 2 2 B R + ( x 0 ) | D g | m η m d x + 2 m 2 2 B R + ( x 0 ) | u u x 0 , R | m d x

and

( B R + ( x 0 ) ( | D u | m + | u | r + 1 ) d x ) 1 1 r ( B R + ( x 0 ) | D φ | m d x ) 1 m ε B R + ( x 0 ) | D φ | m d x + C ( ε ) ( B R + ( x 0 ) ( | D u | m + | u | r + 1 ) d x ) ( 1 1 r ) ( m m 1 ) .

Combining these estimates in (3.3), we have

B R + ( x 0 ) A i j α β ( x , u ) ( D u , D u ) η 2 d x 2 ε B R + ( x 0 ) ( 1 + | u | 2 ) m 2 2 | D u | 2 η 2 d x + 2 m 2 2 B R + ( x 0 ) ( 1 + | u x 0 , R | 2 ) m 2 2 | D η | 2 | u u x 0 , R | 2 d x + 2 m + 2 2 B R + ( x 0 ) | u u x 0 , R | m ( R ρ ) m d x + 2 m 2 2 B R + ( x 0 ) ( 1 + | u x 0 , R | 2 ) m 2 2 ( 1 + C p ) | D g | 2 d x + 2 m 2 2 ( 1 + C p ) B R + ( x 0 ) | D g | m d x + ε B R + ( x 0 ) | D φ | m d x + C ( ε ) ( B R + ( x 0 ) ( | D u | m + | u | r + 1 ) d x ) ( 1 1 r ) ( m m 1 ) .

Noting that u W 1 , m ( B R + ( x 0 ) ¯ , R N ) , we have

( B R + ( x 0 ) ( | D u | m + | u | r + 1 ) d x ) m m 1 ( 1 1 r ) ( B R + ( x 0 ) | D u | m d x ) m m 1 ( 1 1 r ) + ( B R + ( x 0 ) | u u x 0 , R | r d x ) m m 1 ( 1 1 r ) + ( | u x 0 , R | r 1 + 1 ) ( α n R n / 2 ) m m 1 ( 1 1 r ) ( B R + ( x 0 ) | D u | m d x ) m m 1 ( 1 1 r ) + ( B R + ( x 0 ) | D u | m d x ) r 1 m 1 + ( | u x 0 , R | r 1 + 1 ) ( α n R n / 2 ) m m 1 ( 1 1 r ) C ( ε , u W 1 , m ) ( | u x 0 , R | r 1 + 1 ) ( α n R n / 2 ) . (3.4)

Using (H2), we thus have

( λ 2 ε ) B R + ( x 0 ) [ | D u | 2 + | D | u | m 2 | 2 ] d x ( λ 2 ε ) B R + ( x 0 ) ( 1 + | u | 2 ) m 2 2 | D u | 2 d x C ( ε , m , L ) B R + ( x 0 ) ( | D η | 2 | u u x 0 , R | 2 + | D η | m | u u x 0 , R | m ) d x + C ( m , L , ε ) B R + ( x 0 ) ( | D g | 2 + | D g | m ) d x + C ( ε , u W 1 , m ) ( α n R n / 2 ) .

Recalling that u x 0 , R = g x 0 , R and φ = ( u g ) η 2 , we get

( λ 2 ε ) B R + ( x 0 ) [ | D u | 2 + | D | u | m 2 | 2 ] d x C ( ε , L , m ) B R + ( x 0 ) [ 1 ( R ρ ) 2 | u u x 0 , R | 2 + 1 ( R ρ ) m | u u x 0 , R | m ] d x + C ( ε , L , m , C s , C p ) [ g H 1 , s 2 ( α n R n / 2 ) R ( 1 2 s ) + ( α n 2 R n ) 1 m s g H 1 , s m ] + C ( ε , u W 1 , m ) ( α n R n / 2 ) .

Fixing ε small enough yields the desired inequality immediately. □

4 Proof of the main theorem

In this section we proceed to the proof of partial regularity result.

Lemma 4.1Consider u W 1 , m ( Ω ¯ , R N ) to be a weak solution of (1.1), x 0 D , y D R ( x 0 ) , D ρ ( y ) D R ( x 0 ) , R < 1 | x 0 | , and φ C 0 ( B ρ 2 + ( y ) , R N ) with sup B ρ + ( y ) | D φ | 1 . Then

| ( ρ 2 ) 2 n B ρ 2 + ( y ) A ( y , u y , ρ + ) ( D u , D φ ) d x | C 5 I [ I + ω ( I ) ] ρ sup B ρ 2 + ( x 0 ) | D φ | , (4.1)

where

I ( z , r 0 ) = B r 0 + ( z ) [ | u u z , r 0 | 2 + 1 r 0 m 2 | u u z , r 0 | m ] d x + [ g H 1 , s 2 + g H 1 , s m ] r 0 2 ( 1 n s ) + r 0 2

for z D , r ( 0 , 1 | z | ) .

Proof From the definition of weak solution we have

B ρ 2 + ( y ) A i j α β ( y , u y , ρ ) ( D u , D φ ) d x C [ B ρ 2 + ( y ) ( | D u | m ( 1 1 r ) + | u | r 1 + 1 ) d x ] ρ sup B ρ 2 + ( y ) | D φ | + B ρ 2 + ( y ) | A i j α β ( y , u y , ρ ) A i j α β ( x , u ) | | D u | d x sup B ρ 2 + ( y ) | D φ | . (4.2)

Similarly as (3.4), by Hölder’s inequality and Sobolev’s embedding theorem, we have

C B ρ 2 + ( y ) ( | D u | m ( 1 1 r ) + | u | r 1 + 1 ) d x C ( | u y , ρ / 2 | r 1 + 1 ) α n ρ n / 2 .

Henceforth we restrict ρ to be sufficiently small. Applying in turn Young’s inequality, (H3), Caccioppoli’s inequality (Theorem 3.1) and then Jensen’s inequality we calculate that

B ρ 2 + ( y ) A i j α β ( y , u y , ρ ) ( D u , D φ ) d x C ( | u y , ρ / 2 | r 1 + 1 ) α n ρ n / 2 ρ sup B ρ 2 + ( x 0 ) | D φ | + ω ( ρ m + B ρ + ( y ) | u u y , ρ | m d x ) ( α n ρ n / 2 ) ( B ρ + ( y ) | D u | 2 d x ) 1 2 ρ sup B ρ 2 + ( x 0 ) | D φ | . (4.3)

For z D , r 0 ( 0 , 1 | z | ) , we introduce the notation

I ( z , r 0 ) = B r 0 + ( y ) [ | u u y , r 0 | 2 + 1 r 0 m 2 | u u y , r 0 | m ] d x + [ g H 1 , s 2 + g H 1 , s m ] r 0 2 ( 1 n s ) + r 0 2 ,

and further write I for I ( y , ρ ) . Defining the constant C 4 by C 4 = m a x { C 1 , C 2 , C 3 } , from (4.3) and Theorem 3.1, we have

B ρ 2 + ( y ) A i j α β ( y , u y , ρ ) ( D u , D φ ) d x 1 2 α n ρ n 1 C 4 I + C 4 α n ρ n 1 / 2 ω ( I ) I .

For arbitrary φ C 0 ( Ω , R N ) we thus have

B ρ 2 + ( y ) A i j α β ( y , u y , ρ ) ( D u , D φ ) d x 1 2 α n ρ n 1 max { C 4 , C 4 } I ( I + ω ( I ) ) sup B ρ 2 + ( x 0 ) | D φ | .

Multiplying through by ( ρ 2 ) 2 n , this yields

| ( ρ 2 ) 2 n B ρ 2 + ( y ) A i j α β ( y , u y , ρ ) ( D u , D φ ) d x | C 5 I ( I + ω ( I ) ) ρ sup B ρ 2 + ( x 0 ) | D φ | (4.4)

for C 5 defined by C 5 = 2 n 2 max { C 4 , C 4 } . □

Lemma 4.2Considerusatisfying the conditions of Theorem 1.1 andσfixed, then we can findδand s 0 together with positive constant C 11 such that the smallness conditions 0 < ω ( | u | , s 0 ) δ 2 and I ( x 0 , R ) C 9 1 min { δ 2 4 , s 0 } together imply the growth condition

I ( y , θ ρ ) θ 2 σ I ( y , ρ ) .

Proof Set w = u g , using in turn (H1), Young’s inequality and Hölder’s inequality, from (4.4) we can see that for C 6 = C 5 + L ( α n 2 ) 1 1 s ( 1 + | u y , ρ | 2 ) m 2 2 :

| ( ρ 2 ) 2 n B ρ 2 + ( y ) A i j α β ( y , u y , ρ ) ( D w , D φ ) d x | C 6 I ( I + ω ( I ) ) ρ sup B ρ 2 + ( x 0 ) | D φ | . (4.5)

We now set v = w γ , for γ = C 6 I . From (4.5) yields

| ( ρ 2 ) 2 n B ρ 2 + ( y ) A i j α β ( y , u y , ρ ) ( D v , D φ ) d x | ( I + ω ( I ) ) ρ sup B ρ 2 + ( x 0 ) | D φ | , (4.6)

and we observe from the definition of γ, (4.6) means that

( ρ 2 ) 2 n B ρ 2 + ( y ) | D v | 2 d x < 1 . (4.7)

Further we note that

v | D ρ ( y ) = 1 γ w | D ρ ( y ) = 1 γ ( u g ) | D ρ ( y ) 0 . (4.8)

For ε > 0 we take δ = δ ( n , N , λ , L , ε ) to be the corresponding δ from the A-harmonic approximation lemma. Suppose that we could ensure that the smallness condition

I + ω ( I ) δ (4.9)

holds. Then in view of (4.6), (4.7), (4.8) we would be able to apply A-harmonic approximation lemma to conclude that the existence of a function h H 1 , 2 ( B ρ 2 + ( y ) , R N ) , which is A i j α β ( y , u y , ρ ) -harmonic, with h | D ρ 2 ( y ) 0 such that

( ρ 2 ) 2 n B ρ 2 + ( y ) | D h | 2 d x 1 , (4.10)

( ρ 2 ) n B ρ 2 + ( y ) | v h | 2 d x ε . (4.11)

For θ ( 0 , 1 4 ] arbitrary (to be fixed later), from Lemma 2.3 and (4.10), recalling also that h ( y ) = 0 , we have

sup B θ ρ + ( y ) | h | 2 θ 2 ρ 2 sup B ρ 4 + ( y ) | D h | 2 4 C 0 θ 2 . (4.12)

Using (4.11) and (4.12) we observe

( θ ρ ) n B θ ρ + ( y ) | v | 2 d x 2 ( θ ρ ) n [ ( ρ 2 ) n ε + 1 2 α n ( θ ρ ) n sup B θ ρ + ( y ) | h | 2 ] 2 1 n θ n ε + 4 α n C 0 θ 2

and hence, on multiplying this through by γ 2 , we obtain the estimate

( θ ρ ) n B θ ρ + ( y ) | w | 2 d x ( 2 1 n θ n ε + 4 α n C 0 θ 2 ) γ 2 . (4.13)

For the time being, we restrict ourselves to the case that g does not vanish identically. Recalling that w = u g , (4.13) yields, using in turn Poincaré’s, Sobolev’s and then Hölder’s inequalities, noting also that u y , θ ρ = g y , θ ρ :

( θ ρ ) n B θ ρ + ( y ) | u u y , θ ρ | 2 d x 2 ( 2 1 n θ n ε + 4 α n C 0 θ 2 ) γ 2 + 2 C p ( θ ρ ) 2 n [ 1 2 α n ( θ ρ ) n ] 1 2 s g H 1 , s 2 C 7 ( θ n ε + θ 2 ) I + C 7 θ 2 ( 1 n s ) I (4.14)

for C 7 = max { 8 α n C 0 C 6 2 , 2 2 s C p α n 1 2 s } and provided ε = θ n + 2 together with θ 2 θ 2 ( 1 n s ) , we have

( θ ρ ) n B θ ρ + ( y ) | u u y , θ ρ | 2 d x 3 C 7 θ 2 ( 1 n s ) I . (4.15)

For 2 < m < n ( n 3 ) we have 2 < m < r , where r = { m n n m , if  n > m , a n y n u m b e r > m , if  m n , with 1 r < 1 m < 1 2 . Therefore we can find t [ 0 , 1 ] such that 1 m = 1 t 2 + t r .

Using Sobolev’s, Caccioppoli’s, and Young’s inequalities together with (4.14), we have

( θ ρ ) n ( θ ρ ) m + 2 B θ ρ + ( y ) | u u y , θ ρ | m d x ( θ ρ ) n ( θ ρ ) m 2 ( B θ ρ + ( y ) | u u y , θ ρ | 2 d x ) ( 1 t ) m 2 ( B θ ρ + ( y ) | u u y , θ ρ | r d x ) m t r ( θ ρ ) n ( θ ρ ) m 2 ( C 7 ( θ ρ ) n ( θ n ε + θ 2 + θ 2 ( 1 n s ) ) I ) ( 1 t ) m 2 ( ( θ ρ ) m B θ ρ + ( y ) | D u | m d x ) t ( θ ρ ) n ( θ ρ ) m 2 [ C 7 ( θ ρ ) n ( θ n ε + θ 2 + θ 2 ( 1 n s ) ) I ] ( 1 t ) m 2 [ α n 2 ( θ ρ ) ( m + n ) C ] t ( θ ρ ) n ( ( 1 t ) m 2 ( θ ρ ) n C 7 ( θ n ε + θ 2 + θ 2 ( 1 n s ) ) I + m t r C r m t ( α n 2 ) r m ( θ ρ ) ( m t + n t ) r m t ) C 8 ( θ n ε + θ 2 + θ 2 ( 1 n s ) ) I (4.16)

for C 8 = ( 1 t ) m 2 C 7 + m t r C r m t ( α n 2 ) r m .

We then fix ε = θ n + 2 , note that this also fixes δ. Since ρ 1 , we see from the definition of γ: [ g H 1 , s 2 + g H 1 , s m ] ( θ ρ ) 2 ( 1 n s ) θ 2 ( 1 n s ) I , and further ( θ ρ ) 2 θ 2 I .

Combining these estimates with (4.15) and (4.16), we have

I ( y , θ ρ ) 3 ( C 7 + C 8 + 1 ) θ 2 ( 1 n s ) I . (4.17)

We choose θ ( 0 , 1 4 ] small enough, such that we have 3 ( C 7 + C 8 + 1 ) θ 2 ( 1 n s ) θ 2 σ .

Thus from (4.17) we can see

I ( y , θ ρ ) θ 2 σ I ( y , ρ ) . (4.18)

We now choose s 0 > 0 such that 0 < ω ( s 0 ) δ 2 , and we define C 9 by C 9 = 2 n + 2 m + 1 ( 2 C s 2 + 2 m C s m ) . Suppose that

I ( x 0 , R ) C 9 1 min { δ 2 4 , s 0 } (4.19)

for some R ( 0 , R 0 ] , where R 0 = min { 2 s 0 , 1 | x 0 | } .

For any y D R 2 ( x 0 ) we use Sobolev’s inequality to calculate

α n R n 2 n + 1 | u x 0 , R u y , R 2 | 2 = B R 2 + ( y ) | u x 0 , R u y , R 2 | 2 d x = B R 2 + ( y ) | g x 0 , R g y , R 2 | 2 d x 2 B R 2 + ( y ) | g g x 0 , R | 2 d x + 2 B R 2 + ( y ) | g g y , R 2 | 2 d x 2 α n C s 2 g H 1 , s 2 R n + 2 ( 1 n s ) (4.20)

and

α n R n 2 n + 1 | u x 0 , R u y , R 2 | m = B R 2 + ( y ) | u x 0 , R u y , R 2 | m d x = B R 2 + ( y ) | g x 0 , R g y , R 2 | m d x 2 m 1 B R 2 + ( y ) | g g x 0 , R | m d x + 2 m 1 B R 2 + ( y ) | g g y , R 2 | m d x 2 m α n C s m g H 1 , s m R n + m ( 1 n s ) 2 m α n C s m g H 1 , s m R n + 2 ( 1 n s ) . (4.21)

Using these estimations, we can obtain

I ( y , ( 1 / 2 ) R ) 2 n + 2 m + 1 [ B R + ( y ) ( | u u x 0 , R | 2 + 1 R m 2 | u u x 0 , R | m ) d x ] + 1 4 R 2 + ( 2 C s 2 + 2 m C s m ) [ g H 1 , s 2 + g H 1 , s m ] R 2 ( 1 n s ) C 10 I ( x 0 , R ) . (4.22)

Thus we have

I ( y , ( 1 / 2 ) R ) + ω ( I ( y , ( 1 / 2 ) R ) ) δ 2 + ω ( s 0 ) δ 2 + δ 2 δ ,

which means that the condition (4.18) is sufficient to guarantee the smallness condition (4.9) for ρ = R / 2 , for all y D R / 2 ( x 0 ) . Thus, we can conclude that (4.17) holds in this situation. From (4.18) we thus have

I ( y , ( 1 / 2 ) θ R ) + ω ( I ( y , ( 1 / 2 ) θ R ) ) δ 2 + ω ( s 0 ) δ 2 + δ 2 δ ,

which means that we can apply (4.18) on B θ R / 2 + ( y ) as well, and this yields

I ( y , θ 2 R / 2 ) θ 4 σ I ( y , ( 1 / 2 ) R ) ,

and inductively we find

I ( y , θ k R / 2 ) θ 2 σ k I ( y , ( 1 / 2 ) R ) . (4.23)

The next step is to go from a discrete to a continuous version of the decay estimate. Given ρ ( 0 , R / 2 ] , we can find k N 0 such that θ k + 1 R / 2 < ρ < θ k R / 2 . Then we calculate in a similar manner to above. Firstly, we use Sobolev’s inequality (1.4) to see that

B ρ + ( y ) | u y , ρ u y , θ k R / 2 | 2 d x 2 α n C s 2 g H 1 , s 2 ( 1 2 θ k R ) n + 2 ( 1 n s )

and

B ρ + ( y ) | u y , ρ u y , θ k R / 2 | m d x 2 m α n C s m g H 1 , s m ( 1 2 θ k R ) n + m ( 1 n s ) ,

which allows us to deduce that

B ρ + ( y ) ( | u u y , ρ | 2 + 1 ρ m 2 | u u y , ρ | m ) d x 2 2 m 3 [ B ρ + ( y ) ( | u u y , θ k R / 2 | 2 + 1 ( θ k R / 2 ) m 2 | u u y , θ k R / 2 | m ) d x ] + 4 α n C s 2 g H 1 , s 2 ( 1 2 θ k R ) n + 2 ( 1 n s ) + 2 m + 1 α n C s m g H 1 , s m ( 1 2 θ k R ) n + m ( 1 n s ) ,

and hence, finally

I ( y , ρ ) C 11 I ( y , θ k R / 2 )

for C 11 = max { 2 2 m 3 , 8 C s 2 θ n , 2 m + 1 C s m θ n } . We combine these estimates with (4.22) and (4.23):

I ( y , ρ ) C 11 θ 2 σ k I ( y , R / 2 ) C 10 C 11 θ 2 σ ( 2 ρ R ) 2 σ I ( x 0 , R ) C 10 C 11 ( 2 θ ) 2 σ ( ρ R ) 2 σ I ( x 0 , R ) ,

and more particularly

inf μ R N B ρ + ( y ) | u μ | 2 d x C 12 I ( x 0 , R ) ( ρ R ) 2 σ (4.24)

for C 12 given by C 12 = C 10 C 11 ( 2 θ ) 2 σ . We recall that this estimate is valid for all y D and ρ with D ρ ( y ) D R / 2 ( x 0 ) , and we assume only the smallness condition (4.19) on I ( x 0 , R ) . This yields after replacing R by 6R the boundary estimate required to apply Lemma 2.2.

Similarly, one can get the analogous interior estimate as (4.24). Applying Lemma 2.2, we can conclude the desired Hölder continuity. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

This article was supported by National Natural Science Foundation of China (Nos: 11201415, 11271305); Natural Science Foundation of Fujian Province (2012J01027).

References

  1. Landes, R: Quasilinear elliptic operators and weak solutions of the Euler equations. Manuscr. Math.. 27, 47–72 (1979)

  2. Visik, IM: Quasilinear strongly elliptic systems of differential equations in divergence form. Tr. Mosk. Mat. Obŝ.. 12, 140–208 (1963)

  3. Giaquinta, M: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton (1983)

  4. Giaquinta, M: A counter-example to the boundary regularity of solutions to elliptic quasilinear systems. Manuscr. Math.. 24, 217–220 (1978)

  5. De Giorgi, E: Frontiere orientate di misura minima. Semin. Math. Sc. Norm. Super. Pisa. 57, 1–56 (1961)

  6. Arkhipova, AA: Regularity results for quasilinear elliptic systems with nonlinear boundary conditions. J. Math. Sci.. 77(4), 3277–3294 (1995)

  7. Arkhipova, AA: On the regularity of solutions of boundary-value problem for quasilinear elliptic systems with quadratic nonlinearity. J. Math. Sci.. 80(6), 2208–2225 (1995)

  8. Arkhipova, AA: On the regularity of the oblique derivative problem for quasilinear elliptic systems. J. Math. Sci.. 84(1), 817–822 (1997)

  9. Wiegner, M: A priori schranken für lösungen gewisser elliptischer systeme. Manuscr. Math.. 18, 279–297 (1976)

  10. Hildebrandt, S, Widman, KO: On the Hölder continuity of weak solutions of quasilinear elliptic systems of second order. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4). 4, 144–178 (1977)

  11. Jost, J, Meier, M: Boundary regularity for minima of certain quadratic functionals. Math. Ann.. 262, 549–561 (1983)

  12. Grotowski, JF: Boundary regularity for quasilinear elliptic systems. Commun. Partial Differ. Equ.. 27(11-12), 2491–2512 (2002)

  13. Grotowski, JF: Boundary regularity for nonlinear elliptic systems. Calc. Var. Partial Differ. Equ.. 15(3), 353–388 (2002)

  14. Simon, L: Lectures on Geometric Measure Theory, Australian National University Press, Canberra (1983)

  15. Allard, WK: On the first variation of a varifold. Ann. Math.. 95, 225–254 (1972)

  16. Schoen, R, Uhlenbeck, K: A regularity theorem for harmonic maps. J. Differ. Geom.. 17, 307–335 (1982)

  17. Duzaar, F, Steffen, K: Optimal interior and boundary regularity for almost minimal currents to elliptic integrands. J. Reine Angew. Math.. 546, 73–138 (2002)

  18. Duzaar, F, Grotowski, JF: Partial regularity for nonlinear elliptic systems: the method of A-harmonic approximation. Manuscr. Math.. 103, 267–298 (2000)

  19. Duzaar, F, Kristensen, J, Mingione, G: The existence of regular boundary points for non-linear elliptic systems. J. Reine Angew. Math.. 602, 17–58 (2007)

  20. Kristensen, J, Mingione, G: Boundary regularity in variational problems. Arch. Ration. Mech. Anal.. 198, 369–455 (2010)

  21. Campanato, S: Proprietà di Hölderianità di alcune classi di funzioni. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (3). 17, 175–188 (1963)

  22. Campanato, S: Equazioni ellittiche del I I e ordine e spazi L 2 , λ . Ann. Mat. Pura Appl.. 69, 321–381 (1965)