Research Article Optimal Interior Partial Regularity for Nonlinear Elliptic Systems for the Case 1 <m<2 under Natural Growth Condition

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1 Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 010, Article ID , 3 pages doi: /010/ Research Article Optimal Interior Partial Regularity for Nonlinear Elliptic Systems for the Case 1 <m< under Natural Growth Condition Shuhong Chen 1, and Zhong Tan 1 Department of Information and Mathematics Sciences, China Jiliang University, Hangzhou, Zhejiang , China School of Mathematical Science, Xiamen University, Xiamen, Fujian , China Correspondence should be addressed to Shuhong Chen, shiny030@163.com Received 16 November 009; Accepted 18 March 010 Academic Editor: Shusen Ding Copyright q 010 S. Chen and Z. Tan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the interior regularity for weak solutions of second-order nonlinear elliptic systems with subquadratic growth under natural growth condition. We obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particularly the regularity we obtained is optimal. 1. Introduction In this paper we consider optimal interior partial regularity for the weak solutions of nonlinear elliptic systems with subquadratic growth under natural growth condition of the following type: n D α A α i x, u, Du B i x, u, Du, i 1,...,N in Ω, 1.1 α 1 where Ω is a bounded domain in R n, u and B i taking values in R N,andA α i,, has value in R nn. N > 1, u : Ω R N Du {D α u i }, 1 α n, 1 i N stand for the div of u and 1 <m<. To define weak solution to 1.1, one needs to impose certain structural and regularity conditions on A α i and the inhomogeneity B i, as well as to restrict u to a particular class of functions as follows, for 1 <m<, E1 A α i x, u, p are differentiable functions in p and there exists L>0such that A α ) i x, u, p L 1 p ) m / ), x, u, p Ω R N R nn, 1. p j β

2 Journal of Inequalities and Applications E A α i is uniformly strongly elliptic, that is, for some λ>0, we have A α i x, u, p ) p j β ξ i αξ j β λ 1 p ) m / ξ, x Ω, u R N,p,ξ R nn, 1.3 E3 There exists β 0, 1 and K : 0, 0, monotone nondecreasing such that A α i ) x, ξ, p A α i x, ξ, ) p K ξ x x m ξ ξ m ) β/m 1 p ) m/ 1.4 for all x, x Ω, ξ, ξ R N,andp R nn ; without loss of generality, we take K 1. Furthermore E1 allows us to deduce the existence of a function ω t, s : 0, 0, 0, with ω t, 0 0 for all t such that t ω t, s is monotone nondecreasing for fixed s, s ω t, s is concave and monotone nondecreasing for fixed t, and such that for all x, u Ω R N and p, q R nn, we have Aα ip j β ) ) x, u, p A α x, u, q C 1 p ) q m / ) ω p, p q, 1.5 ip j β E4 there exist constants a and b, such that Bi x, u, p ) a p m b, 1.6 or E4 ) ) Bi x, u, p C p m ε b, ε > Definition 1.1. By a weak solution of 1.1 with structure assumptions E1 E4 or E4, we mean a vector valued function u W 1,m Ω,R N L Ω,R N such that A α i x, u, Du D αϕ i dx B i x, u, Du ϕ i dx, 1.8 for all ϕ C 0 Ω,RN. Ω Even under reasonable assumptions on A α i and B i, in the case of systems i.e., N > 1 one cannot, in general, expect that weak solutions of 1.1 will be classical, that is, C - solutions. This was first shown by De Giorgi 1,. The goal, then, is to establish partial regularity theory. We refer the reader to monographs of Giaquinta 3, 4 for an extensive treatment of partial regularity theory for systems of the form 1.1, as well as more general elliptic systems. Ω

3 Journal of Inequalities and Applications 3 In the class direct proofs, one freezes the coefficients with constant coefficients. The solution of the Dirichlet problem associated to these coefficients with boundary data u and the solution itself can then be compared. This procedure was first carried out by Giaquinta and Modica 5. But the technique of harmonic approximation is to show that a function which is approximately-harmonic lies L close to some harmonic function. This technique has its origins in Simon s proof 6 of the regularity theorem of Allard 7. Which also be used in 8 to find a so-called ε-regularity theorem for energy minimizing harmonic maps. The technique of harmonic approximation allows the author to simplify the original ε-regularity theorem due to Schoen and Uhlenbeck 9. In the remarkable proof when m given by Duzaar and Grotowski in 10, the key difference is that the solution is compared not to the solution of the Dirichlet problem for the system with frozen coefficients, but rather to an A-harmonic function which is close to w in L, where w is a function corresponding from weak solutions. In particular, the optimal regularity result can be obtained. In 11, 1, we deal with the optimal partial regularity of the weak solution to 1.1 for the case m > by the method of A-harmonic approximation technique, which is advantage to the result of 13. The extension of A- harmonic approximation technique also can be found in 14, 15. The purpose of this paper is to establish the optimal partial regularity of weak solution to 1.1 under natural growth condition with subquadratic growth, that is, the case of 1 < m <, directly. Indeed the main difficulty in our setting is that the exponent of the integral function is negative 1/ < m / < 0, which means we cannot use the amplify technique as usually. Motivated by the technique used in 16, where the authors considered the minimizers of nonquadratic functional, we removed the hinder at last. And then with the help of A-harmonic approximation technique, one can find a A α i / pj β x 0,u x0,ρ, Du x0,ρ harmonic function, which is close to a function w in sense of L, the function w is which we defined in Lemma 4. and which is a corresponding function from the weak solution u. Thanks to the standard results of linear theory presented in Section and the elementary inequalities, we obtain the decay estimate of Φ x 0,ρ,p 0 ) V Du V p0 dx and the optimal regularity. Now we may state the main result. Theorem 1.. Let u W 1,m Ω,R N L Ω,R N m 1, be a weak solution of 1.1 with sup Ω u M. Suppose that the natural growth conditions E1) E4) or E4 )) and am < λ hold. Then there exists Ω 0 that is open in Ω and u C 1,β Ω 0,R N for β is defined in E3). Furthermore, 1.9 where Ω \ Ω 0 Σ 1 Σ, { m Σ 1 x 0 Ω : lim inf Du Du ρ 0 x0,ρ dx > 0 }, { } ) Du x0,ρ Σ x 0 Ω : lim sup. ρ In particular, meas Ω \ Ω 0 0.

4 4 Journal of Inequalities and Applications. The A-Harmonic Approximation Technique and Preliminary Lemmas In this section, we present the A-harmonic approximation lemma, the key ingredient in proving our regularity result, and some useful preliminaries will be need in later. At first, we introduce two new functions. Throughout the paper we will use the functions V V p : R n R n and W W p : R n R n defined by ξ V ξ 1 ξ ), W ξ ξ, m /4 1 ξ m.1 for each ξ R n and for any m>1. From the elementary inequality x / m x 1 1 m / x / m,. applied to the vector x 1, ξ m R we deduce that 1 ξ ) m / 1 ξ m m/ 1 ξ ) m /,.3 which immediately yields W ξ V ξ C m W ξ..4 The purpose of introducing W is the fact that in contrast to V /m, the function W /m is a convex function on R k. This can easily be shown as follows. Firstly a direct computation yields that t W /m t t /m 1 t m 1/m is convex and monotone increasing on 0, with W /m 0 0. Secondly we have ) ξ η /m W W ξ η ) /m ) ξ η /m W ξ /m W ) η /m W W ξ /m ) W η /m,.5 for any ξ, η R n. We use a number of properties of V V p which can be found in 17, Lemma.1. Lemma.1. Let m 1, and V, W : R n R n be the functions defined in.1. Then for any ξ, η R n and t>0there holds: i 1/ min ξ, ξ m/ V ξ min ξ, ξ m/ ; ii V tξ max t, t m/ V ξ ;

5 Journal of Inequalities and Applications 5 iii V ξ η c m V ξ V η ; iv m/ ξ η V ξ V η / 1 ξ η m /4 C k, m ξ η ; v V ξ V η c k, m V ξ η ; vi V ξ η C m, M V ξ V η,for all η with η M. The inequalities i iii also hold if we replace V by W. For later purposes we state the following two simple estimates which can easily be deduced from Lemma.1 i and vi. For ξ, η R n with η M we have for ξ η 1the estimate ξ η C m, M V ξ V η ) ;.6 as for ξ η > 1 we have ξ η m C m, M V ξ V η..7 The next result we would state is the A-harmonic approximation lemma, which is prove in 18. Lemma. A-harmonic approximation lemma. Let κ, K be positive constants. Then for any ε>0 there exist δ δ n, N, κ, K, ε 0, 1 with the following property. For any bilinear form A on R nn which is elliptic in the sense of Legendre-Hadamard with ellipticity constant κ and upper bound K, for any v W 1,m,R N satisfying W Dv dx γ 1, A Dv, Dϕ ) dx γδ sup B ρ x 0 Dϕ,.8 for all ϕ C 1 0 B ρ x 0,R N, there exists an A-harmonic function h satisfying W Dh dx 1, ) v γh W dx γ ε. ρ.9 Definition.3. Here a function h is called A-harmonic if it satisfies A Dh, Dϕ ) dx 0,.10 for all ϕ C 1 0 B ρ,r N. Then we would recall a simple consequence of the a prior estimates for solutions of linear elliptic systems of second order with constant coefficients; see 17, Proposition.10 for a similar result.

6 6 Journal of Inequalities and Applications Lemma.4. Let h W 1,1,R N be such that A Dh, Dϕ ) dx 0,.11 for any ϕ C 1 0 B ρ x 0,R N,whereA R nn is elliptic in the sense of Legendre-Hadamard with ellipticity constant κ and upper bound K. Thenh C,R N and ρ sup B ρ/ x 0 D h sup B ρ/ x 0 Dh C a Dh dx, B ρ.1 where the constant C a depends only on n, N, κ, and K. The next lemma is a more general version of 17, Lemma.7, which itself is an extension of 3, Lemma 3.1, Chapter V. The proof in which can easily be adapted to the present situation by replacing the condition of homogeneity by Lemma.1 ii. Lemma.5. Let 0 ν<1, a,b 0, v L p, and g be a nonnegative bounded function satisfying g t νg s a ) v V dx b, s t.13 for all ρ/ t<s ρ. Then there exists a constant C C ν such that ρ ) g C ν a ) ) v V dx b..14 ρ And then we state a Poincare type inequality involving the function V, which have been found in 17 and, in a sharp way, in 18. Lemma.6 Poincare-type inequality. Let m 1, and u W 1,m B ρ,r N,B ρ Ω,then B ρ ) u V ux0,ρ m 1/m ) 1/ dx) C P n, N, m V Du dx,.15 ρ B ρ where m n/ n m. In particular, the previous inequality is valid with m replaced by. We conclude the section with an algebraic fact can be retrieved again from 16, Lemma.1.

7 Journal of Inequalities and Applications 7 Lemma.7. For every t 1/, 0 and μ 0, one has ) μ A s à ) t A ds 8 μ A à ) t t 1,.16 for any A, à R nn, not both zero if μ A Caccioppoli Second Inequality For x 0 Ω, u 0 R N, p 0 R nn, we define P {p i x }, i 1,,N, p i u i 0 pi 0α x α x 0α and we simply write P u 0 p 0 x x 0. In order to prove the main result, our first aim is to establish a suitable Caccioppoli inequality. Lemma 3.1 Caccioppoli second inequality. Let u W 1,m Ω,R N L Ω,R N 1 <m< be a weak solution of 1.1 with sup Ω u M and am < λ hold under natural growth conditions E1) E4) or E4 )). Then for every x 0 Ω, u 0 R N,p 0 R nn, and arbitrary ρ with 0 <ρ< min{1, dist x 0, Ω }, one has ) V Du p0 dx Cc B ρ/ x0 ) ] u V u0 p 0 x x 0 dx G, 3.1 ρ for G K u 0 p 0 ) 1 p 0 ) m/ ] σρ β max{ a p 0 m b ], a p 0 m b ] m/ m 1 } ρ. 3. where σ max{m/ m β, m β / m 1 } > and the constant C c C c n, N, m, L, λ, M. Proof. Let Ω. Choose ρ/ t < s ρ and a standard cut off function η C 1 0 B ρ x 0, 0, 1 with η 1onB t x 0, which satisfies η 1/ s t. For u 0 R N and p 0 R nn,let v u u 0 p 0 x x and define ϕ ηv, ψ 1 η ) v. 3.4 Then Dϕ Dψ Dv Du p 0, 3.5

8 8 Journal of Inequalities and Applications and further there holds Dϕ m C m Dv m v m) s t ; Dψ m C m Dv m v m) s t. 3.6 Using hypothesis E, fromlemma.7, and as the elementary inequality 1 1 b a ) 1 a b a 3 1 a b ), we can get A α i x, u, p0 Dϕ ) A α )] i x, u, p0 Dα ϕ i dx 1 A α i 0 λ 1 0 x, u, p0 θdϕ ) dθd β ϕ j D α ϕ i dx P j β 1 p0 θdϕ ) m / dθ Dϕ dx 3 m / λ 1 p0 ) Dϕ m / Dϕ dx, 3.8 A simple calculation yields 3 m / λ 1 p 0 Dϕ ) m / Dϕ dx 1 A α ) i x, u, Du θdψ dθd 0 P j β ψ j D α ϕ i dx β A α i A α i x, u, p0 ) A α i x, P, p0 )) Dα ϕ i dx x, P, p0 ) A α i x0,u 0,p 0 )) Dα ϕ i dx B i x, u, Du ϕ i dx 3.9 I II III IV.

9 Journal of Inequalities and Applications 9 By E1, Lemma.7 and 3.7, there holds I C 1 Du ) Du Dψ m / Dψ Dϕdx Noting that suppdψ B t \ B s and 1/ < m / < 0, one can take the domain into { Dψ > 1} { Dϕ > 1}, { Dψ > 1} { Dϕ 1}, { Dψ 1} { Dϕ > 1},and { Dψ 1} { Dϕ 1}, four parts, and then by Young inequality and the estimations.6 and.7, thus there is I C 1 \B t x 0 V Du p0 dx ) ) v V dx ρ From the structure condition E3 yields II K u 0 ) ) p0 1 p0 m/ v β Dϕdx. B s x Similar to I, we split the domain of integration into four parts as follows. And on the part { v/s > 1} { Dϕ 1}, wesee K u 0 p0 ) 1 p0 ) m/ v β Dϕ ε Dϕ C ε K u 0 ) ) p ] 0 1 p 0 m/ v β ε Dϕ C ε v m s ) εc V Dv C ε v V s C ε K u 0 ) ) p ] 0 1 p 0 m/s m/ m β β C ε K u 0 ) ) ] p0 1 p0 m/ m/ m β s β, 3.13 as on the set { v/s 1} { Dϕ > 1}, there are K u 0 p0 ) 1 p0 ) m/ v β Dϕ K u 0 p0 ) 1 p0 ) m/s β Dϕ ε Dϕ m C ε K u 0 p0 ) 1 p0 ) m/s β ] m/ m 1 ) εc V Dv C ε v V s C ε K u 0 ) ) ] p0 1 p0 m/ m/ m 1 s β, 3.14

10 10 Journal of Inequalities and Applications and on the case { v/s 1} { Dϕ 1}, one can get K u 0 ) ) p0 1 p0 m/ v β Dϕ K u 0 ) ) p0 1 p0 m/s β Dϕ ε Dϕ C ε K u 0 ) ) ] p0 1 p0 m/s β ) εc V Dv C ε v V s C ε K u 0 ) ) ] p0 1 p0 m/ s β, 3.15 Finally, noting that sup Ω u M, then for the case { v/s > 1} { Dϕ > 1}, there exists a constant 0 < m 1 /m β <1 such that K u 0 p0 ) 1 p0 ) m/ v β Dϕ K u 0 p0 ) 1 p0 ) m/ v β) m 1 / m β M p0 s ) m β / m β Dϕ ε Dϕ m C ε K u 0 p0 ) 1 p0 ) m/ ] m/ m 1 v mβ/ m β M p 0 s ) m β / m β m/ m 1 εdϕ v m C ε s C ε K u 0 ) ) ] p0 1 p0 m/ m β / m 1 s β M p 0 s ) m β / m 1 ) ε V Dv C ε v V s C ε K u 0 ) ) ] p0 1 p0 m/ m β / m 1 M p0 s ) m β / m 1 s β Combining these estimations on II, we have II Cε V Dv dx C ε C ε ) v V dx s 1 M p 0 s ) m β / m 1 ] K u 0 p 0 ) 1 p 0 ) m/ ] σαn s n β, 3.17 for σ max{,m/ m 1, m β / m 1, m/ m β } max{ m β / m 1, m/ m β }. And noting that K 1, and that m/ m 1, and similarly as II,wesee III Cε V Dv dx C ε ) v V dx s C ε K u 0 ) ) ] p0 1 p0 m/ β m/ m 1 αn s n β, 3.18

11 Journal of Inequalities and Applications 11 and for μ positive to be fixed later, we have IV a Du m u u0 p 0 x x 0 ηdx v ) s bsη dx On the part {} { Du p 0 1} { v/s 1}, argue anginous as II and III, by Young s inequality and.6 and.7, we have a Du m u u0 p 0 x x 0 v η ) s bsη 1 ) a μ Du p0 m 1 1 ) ] p0 m u u0 p 0 x x 0 η μ εb s η C ε v s a 1 μ ) M p 0 s ) V Dv a 1 μ) 1 ) p0 m v η εb s η C ε v V s a 1 μ ) M p 0 s ) V Dv εa 1 μ) 1 ) p0 m s η εb s η C ε v V. s 3.0 Similarly, on the part {} { Du p 0 1} { v/s 1}, wesee a Du mu u 0 p 0 x x 0 η v ) s bsη a 1 μ ) M p 0 s ) V Dv εs m/ m 1 η m/ m 1 b m/ m 1 a m/ m ) m/ m 1 p0 m / m 1 ] μ 3.1 ) C ε v V, s and on the part {} { Du p 0 1} { v/s 1}, a Du m u u0 p 0 x x 0 v η ) s bsη a 1 μ ) Du p0 m v η a 1 1 ) ] p0 m b v η μ

12 1 Journal of Inequalities and Applications εs Du p0 C ε M p0 s ) m 1 / m ) / m v a 1 μ s ε a 1 μ) 1 ] p0 m b s η C ε v s ε V Dv C ε M p 0 s ) m 1 / m a 1 μ ) / m V ε a 1 μ) 1 ] p0 ) m b s η C ε v V, s and on the part {} { Du p 0 1} { v/s 1}, a Du mu u 0 p 0 x x 0 η v ) s bsη εs Du p0 C ε M p 0 s ) m m / m ) / m v a 1 μ s ε a 1 μ) 1 m/ m 1 p 0 m b] s m/ m 1 η m/ m 1 C ε v m s ) v s m ε V Dv C ε M p 0 s ) ) m m / m ) / m v a 1 μ V s ε a 1 μ) 1 ] p0 m/ m 1 ) m b s m/ m 1 η m/ m 1 C ε v V. s have Combining these estimates in IV, and noting that m/ m 1 > ands 1, η 1, we ) IV C ε, M, a V Dv dx C ε, M, m, a v V dx s max{ a 1 μ) 1 ] p0 m b, a 1 1 ) ] p0 m/ m 1 } m b α n s n. μ 3.4 Finally, on B t x 0 we use Lemma.1 iv and vi to bound the integrand of the lefthand side of 3.9 from below: λ 1 p0 ) Dϕ m / Dϕ λ 1 ) p0 m / Dv Dv C m λ 1 p 0 Du ) m / Du p 0 C n, N, m, λ V Du V p0 3.5 C n, N, m, λ, M V Dv.

13 Journal of Inequalities and Applications 13 C 1 Using this in 3.9 together with the estimates I, II, III,andIV we finally arrive at B t x 0 V Dv dx C C 3 \B t x 0 ) V Dv ) v V dx s t ) V Dv ) v V dx C 4 K u 0 ) ) ] p0 1 p0 m/ σαn s n β s t C 5 max{ a 1 μ) 1 ] p0 m b, a 1 1 ) ] p0 m/ m 1 } m b α n s n. μ 3.6 The proof is now completed by applying Lemma The Proof of the Main Theorem In this section we proceed to the proof of the partial regularity result and hence consider u W 1,m Ω,R N L Ω,R N 1 <m< to be a weak solution of 1.1. Then we have the following. Lemma 4.1. Consider ρ<1and ϕ C 0 B ρ x 0,R N with sup Bρ x 0 Dϕ 1. Furthermore fixed p 0 in R nn and set u 0 u x0,ρ udx and Φ x 0,ρ,p 0 1. Then for the weak solution u W 1,m Ω,R N L Ω,R N 1 <m< to systems 1.1 with sup Ω u M and am < λ being hold, there holds Aα i p j β ) ) x0,u 0,p 0 Du p0 D α ϕ i dx C e α n ρ n ω 1/ p 0, Φ 1/ x 0,ρ,p 0 ) ) Φ 1/ x 0,ρ,p 0 ) Φ x0,ρ,p 0 ) ρ β H p 0 ) ] 4.1 for C e C e C p,n,n,l and where one defines Φ x 0,ρ,p 0 ) V Du V p0 ) dx, 4. H t K M t 1 M t m/] σ, for σ max{σ, m/ m 1 } and K M t max{k M t,a,b,a,b,a m/ m 1,b m/ m 1 }.

14 14 Journal of Inequalities and Applications Proof. We assume initially that sup Bρ x 0 Dϕ 1. Applying Lemma.7 and noting that the definition of the weak solution of 1.1, for0 t 1, we deduce 1 A α i 0 p j β x0,u 0,p 0 t )) ) Du p 0 dt Du p0 D α ϕ i dx A α i x 0,u 0,Du A α )) i x0,u 0,p 0 Dα ϕ i dx A α i x 0,u 0,Du A α i x, u, Du ) D α ϕ i dx B i,u,du ϕ i dx. 4.3 Rearranging this, we find Aα i p j β ) ) x0,u 0,p 0 Du p0 D α ϕ i dx 1 A α i 0 p j β 1 0 ) ) x0,u 0,p 0 dt Du p0 D α ϕ i dx Aα i ) A α x0,u p j 0,p 0 i p j β β x0,u 0,p 0 t )) Du p 0 dt ) Du p 0 D α ϕ i dx A α i x 0,u 0,Du A α i x, u0 p 0 x x 0,Du )] D α ϕ i dx A α i x, u0 p 0 x x 0,Du ) A α i x, u, Du ] D α ϕ i dx B i,u,du ϕ i dx I II III IV. 4.4 Using the structure condition E1 and the estimate 1.5 for the modulus of continuity of A α i / pj,bylemma.7 and let β s 1 { Du p0 1 }, s { Du p0 > 1 }, 4.5

15 Journal of Inequalities and Applications 15 we can derive I 1 0 L 1 p0 m / L 1 p0 t Du p 0 ) ] m / 1/ L 1 p 0 p 0 t ) ) Du p 0 m / ω p 0, t )) ] 1/ Du p 0 dt Du p 0 dx C ω 1/ ) p0, Du p0 Du p0dx C ω 1/ ) p0, Du p0 Du p0 m/ dx. s 1 s 4.6 Noting that the estimates.6 and.7, using first Hölder s inequality and then Jensen s inequality: I Cα n ρ n Φ 1/ x 0,ρ,p 0 ) ω 1/ p 0, Φ 1/ x 0,ρ,p 0 ) ), 4.7 here we have used Φ 1/m x 0,ρ,p 0 Φ 1/ x 0,ρ,p 0 for Φ x 0,ρ,p 0 1. By E3, Young inequality,.6,.7, and noting the function K monotone nondecreasing and K M p 0 1andthatρ 1, we can estimate II as follows II K u 0 p0 ) ρ β 1 p0 ) β 1 Du m/ dx K M ) p0 ρ β 1 ) p0 β m/αn ρ n β K M ) ) ] p0 1 p0 β αn ρ n β Φ ) x 0,ρ,p 0 αn ρ n K M ) ) ] p0 1 p0 β 4/ 4 m αn ρ n 4β/ 4 m Φ ) x 0,ρ,p 0 αn ρ n K M ) ) ] p0 1 p0 β m/ σαn ρ n β, 4.8 for 1 < 4/ 4 m <σ. Similar to 3.11, to estimate III, one can divide the domain as previously mentioned. On the set { v/ρ > 1} { Du p 0 1},form/ m β < m/ m β σ, K u 0 p0 ) 1 Du m/ v β K M ) ) p0 1 p0 m/ v β K M ) p0 v β ε v m ρ C ε K M ) ) ] p0 1 p0 m/ m/ m β ρ mβ/ m β 4.9 C ε K M p 0 )] m/ m β ρ mβ/ m β ) εc v V ρ C ε K M ) ) ] p0 1 p0 m/ σρ β,

16 16 Journal of Inequalities and Applications while on the part { v/ρ 1} { Du p 0 > 1} and noting that 1 < / β < <σ, K u 0 p0 ) 1 Du m/ v β K M ) ) p0 1 p0 m/ρ β v ρ ε v ρ β K M p0 ) ρ β Du p0 m/ C ε K M ) ) ] p0 1 p0 m/ / β ρ β/ β 4.10 ε Du p 0 m C ε K M p 0 )] ρ β ) εc v V ρ εc V Dv C ε K M ) ) ] p0 1 p0 m/ σρ β. On { v/ρ 1} { Du p 0 1} K u 0 p 0 ) 1 Du m/ v β K M p 0 ) 1 p 0 ) m/ v β K M p 0 ) v β ε v ρ ) εc v V ρ C ε K M ) ) ] p0 1 p0 m/ / β ρ β/ β C ε K M ) ) ] p 0 1 p 0 m/ σρ β Finally, on the case { v/ρ > 1} { Du p 0 > 1}, there exists a constant 0 <m/ m β < 1 such that K u 0 p0 ) 1 Du m/ v β K M p0 ) 1 p0 ) m/ v β) m/ m β M p0 ρ ) β / m β K M p0 )Du p0 m/ v β) m/ m β M p0 ρ ) β / m β C ε v ρ m C ε Du p 0 m C ε K M ) ) ] p 0 1 p 0 m/ m β /mm p0 ρ ) β /m ρ β C ε K M p 0 )] m β / m β M p0 ρ ) β / m β ρ mβ/ m β ) C ε v V ρ C ε V Dv C ε, n, N K M ) ) ] p0 1 M p0 m/ σρ β, 4.1 for 1 < m β / m β β / m β < m/ m β σ.

17 Journal of Inequalities and Applications 17 Whereas, Lemma.1 yields III C ε V Du V p0 dx C ε C ε, n, N B S x 0 ) v V dx ρ K M p0 ) 1 M p0 ) m/ ] σαn ρ n β, 4.13 where σ is defined in Lemma 3.1. Noting that sup Bρ x 0 ϕ ρ 1, and by Young s inequality, we see IV C a Du m b )ϕ dx C a Du p0 m ϕdx C b p0 m) ρdx On D 1 { { Du p 0 > 1}}, by.7 and Young inequality, we have 4.14 C a V Du V ) p 0 dx Cαn ρ n 1 b p 0 m). D On the other hand, on D { { Du p 0 1}}, using.6 and Young inequality, we have Du p0 m Du p0 1 V Du V p Thus 4.14 C a ) V Du V p0 dx Cαn ρ n 1 a b p0 m). D 4.17 Combining these estimates and noting that definition of H t, we derive IV C V Du V p0 dx Cα n ρ n 1 H ) p By Lemma.6, there is III C ε, C P,n,N V Du V ) p 0 ) dx C ε H p 0 αn ρ n β Combining the above of I,II,III,IV with 4.4 and noting the definition of H t, we can get the lemma immediately.

18 18 Journal of Inequalities and Applications We next establish an initial excess-improvement estimate, assuming that the excess Φ ρ is initially sufficient small. We also define Γ ρ Φ ρ 4δ H ρ β,w x u x u x0,ρ γh x 0 Du x0,ρ x x 0,andγ C 6 C e Γ ρ, where C 6 stands for the constants C m, M form Lemma.1 vi. The precise statement is the following. Lemma 4. excess-improvement. Consider weak solution u W 1,m Ω,R N L Ω,R N 1 < m < satisfying the conditions of Theorem 1. and β fixed in E3). Then we can find positive constants C i,c k, and δ, and θ 0, 1/4 with C i depends only on n, N, m, λ, and L and with C k, δ and θ depending only on these quantities as well as β) such that the smallness condition ρ 0,ρ : C a γ 1, ω 1/ Du x0,ρ, Φ 1/ ρ )) Φ 1/ ρ ) δ, Du x0,ρ M 1, for given constant 0 M 1 < ) C i ρ β H 1 Du x0,ρ δ, 4.0 C e C a Φ ρ ) 1, together imply the growth condition Φ θρ ) θ β Φ ρ ) C k ρ β H )] 1 Du x0,ρ. 4.1 Here one uses the abbreviate Φ ρ Φ x 0,ρ, Du x0,ρ. Proof. For ε>0 to be determined later, we take δ δ n, N, λ, Λ,ε 0, 1 to be corresponding constant from the A-harmonic approximation lemma, that is, Lemma., andset w x u x u x0,ρ γh x 0 ) Du x0,ρ x x 0, Γ ρ ) Φ ρ ) 4δ H ρ β, γ C 6 C e Γ ρ ). 4. where C 6 stands for the constant C m, M from Lemma.1 vi. Then, from.4 and Lemma.1 vi, we have W Dw dx V Dw dx C 6 Φ ρ ) γ. 4.3 And by Lemma 4.1 and the smallness condition ω 1/ Du x0,ρ, Φ 1/ ρ )) Φ 1/ ρ ) δ, 4.4

19 Journal of Inequalities and Applications 19 we can deduce Aα i p j β ) x 0,u x0,ρ, Du x0,ρ Dw D α ϕ i dx ω 1/ Du x0,ρ, Φ 1/ ρ )) Φ 1/ ρ ) Φ ρ ) ) ρ H β Du x0,ρ γ C 1 Γ ρ ) γ ω 1/ Du x0,ρ, Φ 1/ ρ )) Φ 1/ ρ ) δ ] sup Dϕ γδ sup Dϕ. sup B ρ x 0 Dϕ 4.5 Inequalities 4.3 and 4.5 fulfill the condition of A-harmonic approximation lemma, which allow us to apply Lemma.. Therefore we can find a function h W 1,m,R N which is A α i / pj β x 0,u x0,ρ, Du x0,ρ -harmonic such that W Dh dx 1, ) w γh W dx γ ε. ρ 4.6 With the help of Lemma.1 iii and v, we have Φ θρ ) C C B θρ x 0 V Du V Du x0,θρ) dx B θρ x 0 B θρ x 0 V Du Du x0,θρ) dx V Du Du x0,ρ 0 ) γdh x dx CV Du x0,θρ Du x 0,ρ 0 ) γdh x, 4.7 where the constant C depends only on n, N, andm. We proceed to estimate the right-hand side of 4.7. Decomposing B θρ x 0 into the set with Du Du x0,ρ γdh x 0 1 and that with Du Du x0,ρ γdh x 0 > 1, that using Lemma.1 i and Hölder inequality, we obtain Du x0,θρ Du x 0,ρ γdh x 0 B θρ x 0 Du Du x0,ρ γdh x 0 dx I 1/ I 1/m), 4.8

20 0 Journal of Inequalities and Applications where we have abbreviated I B θρ x 0 V Du Du x0,ρ 0 ) γdh x dx. 4.9 Now, since V A V A and t V t is monotone increasing, we deduce from 4.7, alsobyusinglemma.1 i and ii, that there holds Φ θρ ) C I V I 1/ I 1/m)) C I I /m), 4.30 where C depends only on n, N, andm. Therefore it remains for us to estimate the quantity I. By considering the cases Dh 1and Dh > 1 seperately and keeping in mind 4.6, we have using Lemma.1 i : Dh dx Using the assumption Du x0,ρ M 1 and Lemma.4, this shows Du x0,ρ γ Dh x 0 M 1 γ Dh x 0 M 1 γc a Dh dx 4.3 M 1 γc a M 1 1. Lemma 3.1 applied on B θρ x 0 with u x0,ρ, respectively Du x0,ρ γdh x 0, instead of u 0, respectively, p 0 ; note that the constant C c depends only on n, N, m, L, λ, M: I C c B θρ x 0 ) u u V x0,ρ Du x0,ρ γdh x 0 x x 0 dx G, 4.33 θρ for ) G K u x0,ρ Du x0,ρ γdh x 0 1 Du x0,ρ γdh x ) ] m/ σ 0 ) θρ β max{ a Du x0,ρ γdh x 0 m ], b a Du x0,ρ γdh x 0 m m/ m 1 } θρ ). b] 4.34

21 Journal of Inequalities and Applications 1 Lemma.1 iii yields B θρ x 0 ) u u V x0,ρ Du x0,ρ γdh x 0 x x 0 dx θρ B θρ x 0 u V ux0,ρ γh x 0 ) Du x0,ρ x x 0 γh x γh x θρ 4.35 C B θρ x 0 γh x ) 0 γdh x 0 x x 0 dx θρ ) w γh x V θρ V γ h h ) 0 Dh x 0 x x 0 ) dx], θρ where the constant C is given by C m C c. To estimate the right-hand side of 4.33 we use.4, Lemma.1 ii note that 1/θ 1 and 4.6 to infer B θρ x 0 ) w γh V dx C m θρ B θρ x 0 ) w γh W dx θρ C m θ n C m θ n ) w γh W dx θρ ) w γh W dx ρ 4.36 C m n θ n γ ε. Using Lemma.1 i, Taylor s theorem applied to h on B θρ x 0, Lemma.4 and 4.31, we obtain B θρ x 0 V γ h h ) 0 Dh x 0 x x 0 dx θρ γ B θρ x 0 h h 0 Dh x 0 x x 0 θρ dx 4.37 γ 4θ ρ sup h x h x 0 Dh x 0 x x 0 8C aθ γ. B θρ x 0

22 Journal of Inequalities and Applications Using the smallness condition C a γ 1and 4.3 together with the definition of H yields ) Kux0 Du x0,ρ,ρ γdh x 0 1 Du x0,ρ γdh x ) ] m/ σ 0 ) θρ β K M ) ) ] m/ σ Du x0,ρ 1 Du x0,ρ ) θρ β H 1 θρ ) β, Du x0,ρ) 4.38 max{ a Du x0,ρ γdh x 0 m ], b a Du x0,ρ γdh x 0 m ] m/ m 1 } θρ ) b H 1 θρ ) β. Du x0,ρ) Combining all the above estimates with 4.33,andletε θ n 4 for θ 0, 1/4, weget I C i θ γ H 1 ) θρ Du x0,ρ ) ] β, 4.39 where the constant C i depends only on n, N, L, m, λ, M,andθ the dependency from θ occurs due to the fact that δ depends on θ. Choose θ 0, 1/4 suitable such that C i θ θ β,and inserting this into 4.30 we easily find recalling also that Φ ρ 1 : Φ θρ ) θ β Φ ρ ) C k H ) 1 Du x0,ρ ρ β], 4.40 where the constant C k has the same dependencies as C i. The regularity result then follows from the fact that this excess-decay estimate for any x in a neighborhood of x 0. From this estimate we conclude by Campanato s characterization of Hölder continuous functions 19, 0 that V Du has the modulus of continuity ρ Φ x 0,ρ by a constant times ρ β.bylemma.1 iv this modulus of continuity carries over to Du. Acknowledgments This work was supported by NCETXMU and the National Natural Science Foundation of China-NSAF no: References 1 E. De Giorgi, Frontiere orientate dimisura minima, in Seminario di Matematica della. Scuola Normale Superiore di Pisa, pp. 1 65, Pisa, Italy, E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Bollettino della Unione Matematica Italiana, vol. 1, pp , 1968.

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