Regularity Result for Quasilinear Elliptic Systems with Super Quadratic Natural Growth Condition

Abstract

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

1. Introduction

This paper considers boundary regularity for weak solutions of quasilinear elliptic systems −𝐷_𝛼𝐴^𝛼𝛽_𝑖𝑗(𝑥,𝑢)𝐷_𝛽𝑢^𝑗=𝐵_𝑖(𝑥,𝑢,𝐷𝑢),𝑥∈Ω, (1) where Ω is a bounded domain in 𝑅^𝑛 with boundary of class 𝐶¹,𝑛≥2 and 𝑢 takes value in 𝑅^𝑁,𝑁>1. Each 𝐴^𝛼𝛽_𝑖𝑗 maps Ω×𝑅^𝑁 into 𝑅, and each 𝐵_𝑖 maps Ω×𝑅^𝑁×𝑅^𝑛𝑁 into 𝑅. A partial regularity theory of (1) must have a priori existence weak solutions. Here we assume that weak solutions exist and consider partial regularity of weak solutions directly. We further impose certain structural conditions on 𝐴^𝛼𝛽_𝑖𝑗 and 𝐵_𝑖 with 𝑚>2 as follows.(H1) There exists 𝐿>0 such that 𝐴^𝛼𝛽_𝑖𝑗(𝑥,𝜉)(𝜈,𝜈)≤𝐿1+𝜉^{2(𝑚−2)/2}|𝜈𝜈|()∈||forall𝑥,𝜉Ω×𝑅^𝑁,𝜈,𝜈∈𝑅^𝑛𝑁. (2)(H2)𝐴^𝛼𝛽_𝑖𝑗(𝑥,𝜉) is uniformly strongly elliptic; that is, for some 𝜆>0 we have 𝐴^𝛼𝛽_𝑖𝑗(𝑥,𝜉)(𝜈,𝜈)≥𝜆1+𝜉^{2(𝑚−2)/2}|𝜈|²forall(𝑥,𝜉)∈Ω×𝑅^𝑁,𝜈∈𝑅^𝑛𝑁. (3)(H3) Assume that 𝐴^𝛼𝛽_𝑖𝑗∈𝐶⁰(Ω×𝑅^𝑁,𝑅^𝑛𝑁) and further that 𝐴^𝛼𝛽_𝑖𝑗 is uniformly continuous on sets of the form Ω×{𝜉∶|𝜉|≤𝑀}, for any fixed 𝑀,0<𝑀<∞.(H4) (Natural growth condition). There exist constants 𝑎 and 𝑏, with 𝑎 possibly depending on 𝑀>0, such that 𝐵_𝑖(𝑥,𝜉,𝜈)≤𝑎(𝑀)|𝜈|^𝑚+𝑏 (4) for all 𝑥∈Ω,𝜉∈𝑅^𝑁 with |𝜉|≤𝑀 and 𝜈∈𝑅^𝑛𝑁.

Further hypothesis (H3) deduces, writing 𝜔(⋅) for 𝜔(𝑀,⋅), the existence of a monotone nondecreasing concave function 𝜔∶[0,∞)→[0,∞) with 𝜔(0)=0, continuous at 0, such that 𝐴^𝛼𝛽_𝑖𝑗(𝑥,𝑢)−𝐴^𝛼𝛽_𝑖𝑗𝑦,𝑣≤𝜔𝑥−𝑦^𝑚+|𝑢−𝑣|^𝑚, (5) for all 𝑥,𝑦∈Ω,𝑢,𝑣∈𝑅^𝑁 with |𝑢|,|𝑣|≤𝑀 [1].(H5) There exist 𝑠 with 𝑠>𝑛 and a function 𝑔∈𝐻^1,𝑠(Ω,𝑅^𝑁), such that 𝑢|_𝜕Ω=𝑔|_𝜕Ω. (6)

Note that we trivially have 𝑔∈𝐻^1,2(Ω,𝑅^𝑁). Further, by the Sobolev 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 becomes as follows. (H5)′ There exist 𝑠 with 𝑠>𝑛 and a function 𝑔∈𝐻^1,𝑠(𝐵⁺,𝑅^𝑁), such that 𝑢|_𝐷=𝑔|_𝐷. (7)

Here we write 𝐵_𝜌(𝑥₀)={𝑥∈𝑅^𝑛∶|𝑥−𝑥₀|<𝜌}, and further 𝐵_𝜌=𝐵_𝜌(0), 𝐵=𝐵₁. Similarly we denote upper half balls as follows: for 𝑥₀∈𝑅^𝑛−1×{0}, we write 𝐵⁺_𝜌(𝑥₀) for {𝑥∈𝑅^𝑛∶𝑥_𝑛>0,|𝑥−𝑥₀|<𝜌} and set 𝐵⁺_𝜌=𝐵⁺_𝜌(0), 𝐵⁺=𝐵⁺₁. For 𝑥₀∈𝑅^𝑛−1×{0} we further write 𝐷_𝜌(𝑥₀) for {𝑥∈𝑅^𝑛∶𝑥_𝑛=0,|𝑥−𝑥₀|<𝜌} and set 𝐷_𝜌=𝐷_𝜌(0),𝐷=𝐷₁.

Definition 1. By a weak solution of (1) one means a vector valued function 𝑢∈𝑊^1,𝑚(Ω,𝑅^𝑁)∩𝐿^∞(Ω,𝑅^𝑁) such that _Ω𝐴^𝛼𝛽_𝑖𝑗(𝑥,𝑢)𝐷_𝛽𝑢^𝑗,𝐷_𝛼𝜑^𝑖𝑑𝑥=_Ω𝐵_𝑖(𝑥,𝑢,𝐷𝑢)⋅𝜑^𝑖𝑑𝑥 (8) holds for all test-functions 𝜑∈𝐶^∞₀(Ω,𝑅^𝑁) and, by approximation, for all 𝜑∈𝑊₀^1,𝑚(Ω,𝑅^𝑁)∩𝐿^∞(Ω,𝑅^𝑁).

Under such assumptions, even the boundary data is smooth, one cannot expect full regularity of (1) at the boundary [2]. Then, our goal is to establish partial boundary regularity.

After the partial regularity results of the type in this paper were proved by Giusti and Miranda in [3], there are some previous partial regularity results for quasilinear systems. For example, regularity up to boundary for nonlinear and quasilinear systems [4–6] has been studied by Arkhipova. Wiegner [7] established boundary regularity for systems in diagonal form first, and the proof was generalized and extended by Hildebrandt and Widman [8]. Jost and Meier [9] deduced full regularity in a neighborhood of the boundary for minima of functionals with the form ∫_Ω𝐴(𝑥,𝑢)|𝐷𝑢|²𝑑𝑥. Furthermore, Duzaar et al. obtained the boundary Hausdorff dimension on the singular sets of solutions to even more general systems in [10, 11] recently. Further discussion for regularity theory can be seen in [12, 13] and their references.

Inspired by [14], in this paper, we would establish boundary regularity for quasilinear systems under natural growth condition by the method of A-harmonic approximation.

The technique of A-harmonic approximation [15–17] is a natural extension of the harmonic approximation technique, which originated from Simon's proof of Allard's [18] 𝜀-regularity theorem. In this context, using the A-harmonic approximation technique, we obtain the following regularity results.

Theorem 2. Consider a bounded domain Ω in 𝑅^𝑁, with boundary of class 𝐶¹. Let 𝑢 be a bounded weak solution of (1) satisfying the boundary condition (H5), and ‖𝑢‖_𝐿^∞≤𝑀<∞ with 2𝑎(𝑀)𝑀<𝜆, where the structure conditions (H1)–(H3) hold for 𝐴^𝛼𝛽_𝑖𝑗 and (H4) holds for 𝐵_𝑖. Consider a fixed 𝛾∈(0,𝜎]. Then there exist positive 𝑅₁ and 𝜀₀ (depending only on 𝑛,𝑁,𝜆,𝐿,𝑏,𝑀,𝑎(𝑀),𝜔(⋅),𝑚, and 𝛾) with the property that _{𝐵𝑅𝑥0∩Ω}𝑢−𝑢^′_𝑥0,𝑅²𝑑𝑥+𝑔²_H1,𝑠𝑅^{2(1−(𝑛/𝑠))}+𝑅²≤𝜀²₀ (9) for some 𝑅∈(0,𝑅₁] for a given 𝑥₀∈𝜕Ω implies 𝑢∈𝐶^0,𝛾(𝐵_𝑅/2(𝑥₀)∩Ω,𝑅^𝑁).

Note in particular that the boundary condition (H5) means that 𝑢^′_𝑥0,𝑅 makes sense: in fact, we have 𝑢^′_𝑥0,𝑅=𝑔^′_𝑥0,𝑅. For 𝜈∈𝐿¹(𝜕Ω), 𝑥₀∈𝜕Ω, we set 𝜈^′_𝑥0,𝑅=⨍_{𝜕Ω∩𝐵𝑅(𝑥0)}𝜈𝑑𝐻^𝑛−1. In particular, for 𝜈∈𝐿¹(𝐷_𝜌(𝑥₀)), 𝑥₀∈𝐷, we write 𝜈^′_𝑥0,𝜌=⨍_{𝐷𝜌(𝑥0)}𝜈𝑑𝐻^𝑛−1.

Combining this result with the analogous interior [19] and a standard covering argument allows us to obtain the following bound on the size of the singular set.

Corollary 3. Under the assumptions of Theorem 2 the singular set of the weak solution 𝑢 has (𝑛−2)-dimensional Hausdorff measure zero in Ω.

If the domain of the main step in proving Theorem 2 is a half ball, the result then is the following.

Theorem 4. Consider a bounded weak solution of (1) on the upper half unit ball 𝐵⁺ which satisfies the boundary condition (𝐻5)^′ and ‖𝑢‖_𝐿^∞≤𝑀<∞ with 2𝑎(𝑀)𝑀<𝜆, where the structure conditions (H1)–(H3) hold for 𝐴^𝛼𝛽_𝑖𝑗 and (H4) holds for 𝐵_𝑖. Then there exist positive 𝑅₀ and 𝜀₀ (depending only on 𝑛,𝑁,𝜆,𝐿,𝑏,𝑀,𝑎(𝑀),𝑀,𝜔(⋅),𝑚, and 𝛾) with the property that_{𝐵+𝑅(𝑥0)}𝑢−𝑢^′_𝑥0,𝑅²𝑑𝑥+𝑔²_𝐻1,𝑠𝑅^{2(1−(𝑛/𝑠))}+𝑅²≤𝜀²₀, (10) for some 𝑅∈(0,𝑅₀] for a given 𝑥₀∈𝐷, implies that there holds: 𝑢∈𝐶^0,𝜎(𝐵⁺_𝑅/2(𝑥₀),𝑅^𝑁).

Note that analogous to the above, the boundary condition (H5)^′ ensures that 𝑢^′_𝑥0,𝑅 exists, and we have indeed 𝑢^′_𝑥0,𝑅=𝑔^′_𝑥0,𝑅.

2. The A-Harmonic Approximation Technique

In this section we present the A-harmonic approximation lemma [14] and some standard results due to Companato [20].

Lemma 5 (A-harmonic approximation lemma). Consider fixed positive 𝜆 and 𝐿, 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^𝑛𝑁) satisfying 𝐴(𝜈,𝜈)≥𝜆|𝜈|²𝑓𝑜𝑟𝑎𝑙𝑙𝜈∈𝑅^𝑛𝑁,|𝐴(𝜈,𝜈)|≤𝐿|𝜈||𝜈|𝑓𝑜𝑟𝑎𝑙𝑙𝜈,𝜈∈𝑅^𝑛𝑁 (11) for any 𝑤∈𝐻^1,2(𝐵⁺_𝜌(𝑥₀),𝑅^𝑁)   (for some 𝜌>0,𝑥₀∈𝑅^𝑛) satisfying 𝜌^2−𝑛_{𝐵+𝜌𝑥0}|𝐷𝑤|²𝜌𝑑𝑥≤1,_{2−𝑛𝐵+𝜌(𝑥0)}𝐴𝐷𝑤,𝐷𝜑𝑑𝑥≤𝛿𝜌sup_{𝐵+𝜌𝑥0},𝑤|𝐷𝜑_{𝐷𝜌(𝑥0)}=0 (12) for all 𝜑∈𝐶¹₀(𝐵⁺_𝜌(𝑥₀),𝑅^𝑁), there exists an A-harmonic function 𝑣∈𝐻=𝑤∈𝐻^1,2𝐵⁺_𝜌𝑥₀,𝑅^𝑁⋅𝜌^2−𝑛_{𝐵+𝜌(𝑥0)}|𝐷𝑤|²𝑑𝑥≤1,𝑤_{𝐷𝜌(𝑥0)}≡0 (13) with 𝜌^−𝑛_{𝐵+𝜌(𝑥0)}|𝑣−𝑤|²𝑑𝑥≤𝜀. (14)

Next we recall a slight modification of a characterization of Hölder continuous functions originally due to Campanato [21].

Lemma 6. Consider 𝑛∈𝑁,𝑛≥2, and 𝑥₀∈𝑅^𝑛−1×{0}. Suppose that there are positive constants 𝜅 and 𝛼, with 𝛼∈(0,1] such that, for some 𝜈∈𝐿²(B⁺_6𝑅(𝑥₀)), there holds the following: inf_𝜇∈𝑅𝜌^−𝑛_{𝐵+𝜌(𝑦)}𝜈−𝜇²𝑑𝑥≤𝜅²𝜌𝑅^2𝛼, (15) for all 𝑦∈𝐷_2𝑅(𝑥₀) and 𝜌≤4𝑅; and inf_𝜇∈𝑅𝜌^−𝑛_{𝐵𝜌(𝑦)}𝜈−𝜇²𝑑𝑥≤𝜅²𝜌𝑅^2𝛼, (16) for all 𝑦∈𝐵⁺_2𝑅(𝑥₀) and 𝐵_𝜌(𝑦)⊂𝐵⁺_2𝑅(𝑥₀).
Then there exists a Hölder continuous representative of the 𝐿²-class of 𝜈 on 𝐵⁺_𝑅(𝑥₀), and for this representative 𝜈 there holds |𝜈(𝑥)−𝜈(𝑧)|≤𝐶_𝜅|𝑥−𝑧|𝑅^𝛼, (17) for all 𝑥,𝑧∈𝐵⁺_𝑅(𝑥₀), for a constant 𝐶_𝜅 depending only on 𝑛 and 𝛼.

We close this section by a standard estimate for the solutions to homogeneous second-order elliptic systems with constant coefficients [20].

Lemma 7. Consider fixed positive 𝜆 and 𝐿, and 𝑛,𝑁∈𝑁 with 𝑛≥2. Then there exists 𝐶₀ depending only on 𝑛, 𝑁, 𝜆, and 𝐿   (without loss of generality we take 𝐶₀≥1) such that, for 𝐴∈Bil(𝑅^𝑛𝑁) satisfying (11), any A-harmonic function ℎ on 𝐵⁺_𝜌(𝑥₀) with ℎ|_{𝐷𝜌(𝑥0)}≡0 satisfies 𝜌²sup_{𝐵+𝜌/2𝑥0}|𝐷ℎ|²≤𝐶₀𝜌^2−𝑛_{𝐵+𝜌(𝑥0)}|𝐷ℎ|²𝑑𝑥. (18)

3. The Caccioppoli Inequality

In this section we would prove a suitable Caccioppoli inequality. First of all we recall two useful inequalities. The first is the Sobolev embedding theorem which yields the existence of a constant 𝐶_𝑠 depending only on 𝑠, 𝑛, and 𝑁 such that for 𝑥₀∈𝐷,𝜌≤1−|𝑥₀| there holds sup_{𝐵+𝜌𝑥0}𝑔−𝑔^′_𝑥0,𝜌≤𝐶_𝑠𝜌^{1−(𝑛/𝑠)}𝑔_{𝐻1,𝑠(𝐵+𝜌(𝑥0),𝑅𝑁)}. (19) Obviously, the inequality remains true if we replace ‖𝑔‖_{𝐻1,𝑠(𝐵+𝜌(𝑥0),𝑅𝑁)} by ‖𝑔‖_{𝐻1,𝑠(𝐵+,𝑅𝑁)}, which we will henceforth abbreviate simply as ‖𝑔‖_𝐻1,𝑠.

Next we note that the Poincaré inequality in this setting for 𝑥₀∈𝐷,𝜌≤1−|𝑥₀| yields _{𝐵+𝜌(𝑥0)}𝑔−𝑔_𝑥0,𝜌^𝑚𝑑𝑥≤𝐶_𝑝𝜌^𝑚_{𝐵+𝜌(𝑥0)}𝐷𝑔^𝑚𝑑𝑥, (20) for a constant 𝐶_𝑝 which depends only on 𝑛.

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

Then we establish an appropriate inequality for Caccioppoli.

Theorem 8 (Caccioppoli’s inequality). Let 𝑢∈𝑊^1,𝑚(Ω,𝑅^𝑁)∩𝐿^∞(Ω,𝑅^𝑁) with ‖𝑢‖_𝐿^∞≤𝑀<∞ and 2𝑎(𝑀)𝑀<𝜆 be a weak solution of systems (1) under assumption conditions (H1)–(H5). Then there exists 𝜌₀(𝐿,𝑀,𝑎(𝑀),𝑠,‖𝑔‖_𝐻1,𝑠)>0 such that, for all 𝐵⁺_𝜌(𝑥₀)⊂𝐵⁺, with 𝑥₀∈𝐷⁺, 0<𝜌<𝑅<𝜌₀, there holds _{𝐵+𝜌/2𝑥0}|𝐷𝑢|²𝑑𝑥≤𝐶_{1𝐵+𝜌𝑥0}𝑢(𝑥)−𝑢^′_𝑥0,𝑅²𝜌²𝑑𝑥+𝐶₂𝛼_𝑛𝜌^𝑛+𝐶₃𝛼_𝑛𝜌^{𝑛1−(2/𝑠)}𝑔²_𝐻1,𝑠, (21) where 𝐶₁ depends only on 𝜆, 𝐿, and 𝑀 and 𝐶₃ depends on these quantities, and in addition to 𝐶_𝑝, 𝐶₂ depends on 𝜆, 𝐿, 𝑀, 𝑎, 𝑏, and ‖𝑔‖_{𝐿∞(𝐵,𝑅𝑁)}.

Proof. Consider a cutoff function 𝜂∈𝐶^∞₀(𝐵⁺_𝜌/2(𝑥₀)), satisfying 0≤𝜂≤1,𝜂≡0 on 𝐵⁺_𝜌/2(𝑥₀) and |∇𝜂|<4/𝜌. Then the function (𝑢−𝑔)𝜂² is in 𝑊₀^1,𝑚(𝐵⁺_𝜌/2(𝑥₀,𝑅^𝑁)) and thus can be taken as a test-function.
Using (H1), (H4), (H5), and Young's inequality and noting that 2𝑎(𝑀)𝑀<𝜆, we can get from (8) with 𝜀 positive but arbitrary (to be fixed later) _{𝐵+𝜌(𝑥0)}𝐴^𝛼𝛽_𝑖𝑗(⋅,𝑢)𝐷_𝛽𝑢^𝑗,𝐷_𝛼𝑢^𝑖𝜂²𝑑𝑥≤𝐿_{𝐵+𝜌(𝑥0)}1+|𝑢|^{2(𝑚−2)/2}𝐷𝑔|𝐷𝑢|𝜂²𝑑𝑥+2𝐿_{𝐵+𝜌(𝑥0)}1+|𝑢|^{2(𝑚−2)/2}𝐷𝜂|𝐷𝑢|𝜂𝑢−𝑔𝑑𝑥+𝑎_{𝐵+𝜌(𝑥0)}|𝐷𝑢|^𝑚𝑢−𝑔𝜂²𝑑𝑥+𝑏_{𝐵+𝜌(𝑥0)}𝑢−𝑔𝜂²𝑑𝑥≤𝜀_{𝐵+𝜌(𝑥0)}1+|𝑢|^{2(𝑚−2)/2}|𝐷𝑢|²𝜂²𝑑𝑥+𝑎sup_{𝐵+𝜌𝑥0}𝑢−𝑢^′_{𝑥0,𝜌𝐵+𝜌(𝑥0)}|𝐷𝑢|^𝑚𝜂²𝑑𝑥+𝑎sup_{𝐵+𝜌𝑥0}𝑔−𝑔^′_{𝑥0,𝜌𝐵+𝜌(𝑥0)}|𝐷𝑢|^𝑚𝜂²+𝐿𝑑𝑥₂2𝜀_𝐵⁺_𝜌(𝑥0)1+|𝑢|^{2(𝑚−2)/2}𝐷𝑔²𝜂²+𝑑𝑥4𝐿²𝜀_𝐵⁺_𝜌(𝑥0)1+|𝑢|^{2(𝑚−2)/2}𝐷𝜂²𝑢−𝑢^′_𝑥0,𝜌²+𝑑𝑥4𝐿²𝜀_𝐵⁺_𝜌(𝑥0)1+|𝑢|^{2(𝑚−2)/2}𝐷𝜂²𝑔−𝑔^′_𝑥0,𝜌²+𝜀𝑑𝑥2𝑏²_{𝐵+𝜌(𝑥0)}𝜌²𝜂²1𝑑𝑥+𝜀𝜌²_𝐵⁺_𝜌(𝑥0)𝑢−𝑢^′_𝑥0,𝜌²+1𝑑𝑥𝜀𝜌²_𝐵⁺_𝜌(𝑥0)𝑔−𝑔^′_𝑥0,𝜌²𝑑𝑥≤𝜀_{𝐵+𝜌(𝑥0)}1+|𝑢|^{2(𝑚−2)/2}|𝐷𝑢|²𝜂²𝑔𝑑𝑥+𝑎𝑀+_{𝐿∞(𝐵+,𝑅𝑁)𝐵+𝜌(𝑥0)}|𝐷𝑢|^𝑚𝜂²+𝑑𝑥64𝐿²+1𝜀_𝐵⁺_𝜌(𝑥0)1+|𝑢|^{2(𝑚−2)/2}1𝜌²𝑢−𝑢^′_𝑥0,𝜌²+𝜀𝑑𝑥4𝑏²𝜂²𝛼_𝑛𝜌^𝑛+2+𝐿²+2𝜀64𝐿²𝐶_𝑝+2𝜀4𝐶_𝑝𝜀⋅_{𝐵+𝜌(𝑥0)}1+|𝑢|^{2(𝑚−2)/2}𝐷𝑔²𝜂²𝑑𝑥≤𝜀_{𝐵+𝜌(𝑥0)}1+|𝑢|^{2(𝑚−2)/2}𝐷𝑔²𝜂²𝑔𝑑𝑥+𝑎𝑀+_{𝐿∞(𝐵+,𝑅𝑁)}𝐶‖𝑢‖_𝑊^1,𝑚_(𝐵⁺_𝜌(𝑥0))𝛼_𝑛𝜌^𝑛+64𝐿²+1𝜀_𝐵⁺_𝜌(𝑥0)1+|𝑢|^{2(𝑚−2)/2}𝑢−𝑢^′_𝑥0,𝜌𝜌²+𝜀𝑑𝑥4𝑏²𝜂²𝛼_𝑛𝜌^𝑛+2+1+𝑀^{2(𝑚−2)/2}𝐿²+2𝜀64𝐿²𝐶_𝑝+2𝜀4𝐶_𝑝𝜀⋅_{𝐵+𝜌(𝑥0)}𝐷𝑔²𝜂²𝑑𝑥. (22) Using (H2), (19), and (20), we thus have (𝜆−𝜀)_{𝐵+𝜌𝑥0}|𝐷𝑢|²𝜂²≤()𝑑𝑥𝜆−𝜀_{𝐵+𝜌𝑥0}1+|𝑢|^{2(𝑚−2)/2}|𝐷𝑢|²𝜂²≤𝑑𝑥64𝐿²+1𝜀_𝐵⁺_𝜌𝑥01+|𝑢|^{2(𝑚−2)/2}1𝜌²𝑢−𝑢^′_𝑥0²𝑔𝑑𝑥+𝐶𝑎,𝑀,_{𝐿∞(𝐵+,𝑅𝑁)},‖𝑢‖_𝑊^1,𝑚_(𝐵⁺_𝜌(𝑥0)),𝑏𝛼_𝑛𝜌^𝑛+𝐿,𝐶_𝑝,𝑀_{𝐵+𝜌𝑥0}𝐷𝑔²≤𝑑𝑥64𝐿²+1𝜀1+𝑀^{2(𝑚−2)/2}_{𝐵+𝜌𝑥0}1𝜌²𝑢−𝑢^′_𝑥0²𝑔𝑑𝑥+𝐶𝑎,𝑀,_{𝐿∞(𝐵+,𝑅𝑁)},‖𝑢‖_𝑊^1,𝑚_(𝐵⁺_𝜌(𝑥0)),𝑏𝛼_𝑛𝜌^𝑛+𝐿,𝐶_𝑝,𝑀𝛼_𝑛𝜌^{𝑛1−(2/𝑠)}𝑔²_𝐻1,𝑠. (23) Thus, we fix 𝜀 small enough to yield the desired inequality.

4. The Proof of the Main Theorem

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

Lemma 9. Consider 𝑢∈𝑊^1,𝑚(Ω,𝑅^𝑁)∩𝐿^∞(Ω,𝑅^𝑁) to be a weak solution of (1), 𝑥₀∈𝐷 and 𝑦∈𝐷_𝑅(𝑥₀), 𝐷_𝜌(𝑦)⊂⊂𝐷_𝑅(𝑥₀), for 𝑅<1−|𝑥₀|, and 𝜑∈𝐶^∞₀(𝐵⁺_𝜌/2(𝑦),𝑅^𝑁) with sup_{𝐵+𝜌(𝑦)}|𝐷𝜑|≤1. We have 𝜌2^2−𝑛_𝐵⁺_𝜌/2(𝑦)𝐴^𝛼𝛽_𝑖𝑗𝑦,𝑢⁺_𝑦,𝜌𝐷_𝛽𝑢^𝑗,𝐷_𝛼𝜑^𝑖𝑑𝑥≤𝐶₄√𝐼√𝐼+𝜔(𝐼)𝜌sup_{𝐵+𝜌/2𝑥0}𝐷𝜑. (24) Here and hereafter, we define 𝐼(𝑧,𝑟)=_{𝐵+𝑟(𝑧)}𝑢−𝑢^′_𝑧,𝑟²𝑑𝑥+𝑔²_𝐻1,𝑠𝑟^{2(1−(𝑛/𝑠))}+𝑟², (25) for 𝑧∈𝐷,𝑟∈(0,1−|𝑧|).

Proof. Using (8) we have _{𝐵+𝜌/2𝑦}𝐴^𝛼𝛽_𝑖𝑗𝑦,𝑢^′_𝑦,𝜌𝐷_𝛽𝑢^𝑗,𝐷_𝛼𝜑^𝑖≤𝑑𝑥𝑎_{𝐵+𝜌/2𝑦}|𝐷𝑢|^𝑚𝑑𝑥+2^−𝑛−1𝛼_𝑛𝑏𝜌^𝑛⋅𝜌sup_{𝐵+𝜌/2𝑦}+𝐷𝜑_{𝐵+𝜌/2𝑦}𝐴^𝛼𝛽_𝑖𝑗𝑦,𝑢^′_𝑦,𝜌−𝐴^𝛼𝛽_𝑖𝑗(⋅||𝑥,𝑢)𝐷𝑢𝑑𝑥sup_{𝐵+𝜌/2𝑦}𝐷𝜑. (26)
Applying in turn Young's inequality, (H3), the Caccioppoli inequality (Theorem 8), and Jensen's inequality, we calculate from (26) _{𝐵+𝜌/2𝑦}𝐴^𝛼𝛽_𝑖𝑗𝑦,𝑢^′_𝑦,𝜌𝐷_𝛽𝑢^𝑗,𝐷_𝛼𝜑^𝑖≤𝑑𝑥𝑎_{𝐵+𝜌/2𝑦}|𝐷𝑢|^𝑚𝑑𝑥+2^−𝑛−1𝛼_𝑛𝑏𝜌^𝑛+⋅𝜌_{𝐵+𝜌/2𝑦}𝐴^𝛼𝛽_𝑖𝑗𝑦,𝑢^′_𝑦,𝜌−𝐴^𝛼𝛽_𝑖𝑗(𝑥,𝑢)^1/2𝑑𝑥^1/2⋅_{𝐵+𝜌/2(𝑦)}|𝐷𝑢|²𝑑𝑥^1/2≤𝛼_𝑛𝜌^𝑛−12𝑎_{𝐵+𝜌(𝑦)}|𝐷𝑢|^𝑚𝑥+2^−𝑛𝑏𝜌²+𝛼_𝑛𝜌^𝑛−1𝜔𝜌^𝑚+𝑀^𝑚−2_{𝐵+𝜌(𝑦)}𝑢−𝑢^′_𝑦,𝜌²⋅𝐶𝑑𝑥_{1𝐵+𝜌(𝑦)}𝑢−𝑢^′_𝑦,𝜌²𝑑𝑥+𝐶₃𝑔²_𝐻1,𝑠𝜌^{2(1−(𝑛/𝑠))}+𝐶₂𝜌^21/2≤𝛼_𝑛𝜌^𝑛−12𝐶₅𝛼𝐼+^𝑛𝜌^𝑛−12𝐶₆𝜔(𝐼)√𝐼≤𝐶₇𝛼_𝑛𝜌^𝑛−1𝐼+𝜔(𝐼)√𝐼, (27) where 𝐶₅=𝑎‖𝑢‖_𝑊^1,𝑚+𝑏,𝐶₆=max{𝐶₁,𝐶₂,𝐶₃}, and 𝐶₇=(1/2)(𝐶₅+𝐶₆), for 𝑧∈𝐷, 𝑟∈(0,1−|𝑧|). We introduce the notation 𝐼(𝑧,𝑟)=_{𝐵+𝑟(𝑧)}𝑢−𝑢^′_𝑧,𝑟²𝑑𝑧+𝑔²_𝐻1,𝑠𝑟^{2(1−(𝑛/𝑠))}+𝑟² (28) and further write 𝐼 for 𝐼(𝑦,𝜌). For arbitrary 𝜑∈𝐶^∞₀(Ω,𝑅^𝑁) we thus have, by rescalling, _{𝐵+𝜌/2(𝑦)}𝐴^𝛼𝛽_𝑖𝑗𝑦,𝑢^′_𝑦,𝜌𝐷_𝛽𝑢^𝑗,𝐷_𝛼𝜑^𝑖𝑑𝑥≤𝐶₇𝛼_𝑛𝜌^𝑛−1√𝐼√𝐼+𝜔(𝐼). (29)
Multiplying (29) through by (𝜌/2)^2−𝑛 yields𝜌2^2−𝑛_𝐵⁺_𝜌/2(𝑦)𝐴^𝛼𝛽_𝑖𝑗𝑦,𝑢^′_𝑦,𝜌𝐷_𝛽𝑢^𝑗,𝐷_𝛼𝜑^𝑖𝑑𝑥≤𝐶₄√𝐼√𝐼+𝜔(𝐼)𝜌sup_{𝐵+𝜌/2𝑥0}𝐷𝜑, (30) for 𝐶₄ defined by 𝐶₄=2^𝑛−3𝛼_𝑛𝐶₇.

Lemma 10. Consider 𝑢 satisfying the conditions of Theorem 2 and 𝜎 fixed; then we can find 𝛿 and 𝑠₀ together, with positive constants 𝐶₈ such that the smallness conditions: 0<𝜔(𝑠₀)≤𝛿/2 and 𝐼(𝑥₀,𝑅)≤𝐶₈⁻¹min{𝛿²/4,𝑠₀} together, imply the growth condition 𝐼𝑦,𝜃𝜌≤𝜃^2𝜎𝐼𝑦,𝜌. (31)

Proof. We now set 𝑤=𝑢−𝑔, using in turn (H1), Young's inequality, and Hölder's inequality. We have from (30) 𝜌2^2−𝑛_𝐵⁺_𝜌/2(𝑦)𝐴^𝛼𝛽_𝑖𝑗𝑦,𝑢^′_𝑦,𝜌𝐷_𝛽𝑤^𝑗,𝐷_𝛼𝜑^𝑖≤𝜌𝑑𝑥2^2−𝑛_𝐵⁺_𝜌/2(𝑦)𝐴^𝛼𝛽_𝑖𝑗𝑦,𝑢^′_𝑦,𝜌𝐷_𝛽𝑢^𝑗,𝐷_𝛼𝜑^𝑖+𝜌𝑑𝑥2^2−𝑛_𝐵⁺_𝜌/2(𝑦)𝐴^𝛼𝛽_𝑖𝑗𝑦,𝑢^′_𝑦,𝜌𝐷_𝛽𝑔^𝑗,𝐷_𝛼𝜑^𝑖𝑑𝑥≤𝐶₉√𝐼√𝐼+𝜔(𝐼)𝜌sup_{𝐵+𝜌/2𝑥0}𝐷𝜑, (32) for 𝐶₉=max{𝐶₄,(𝛼_𝑛/2)^{1−(𝑛/𝑠)}}.
We now set 𝑣=𝑤/𝛾, for 𝛾=𝐶₉√𝐼. From (32) we then have 𝜌2^2−𝑛_𝐵⁺_𝜌/2(𝑦)𝐴^𝛼𝛽_𝑖𝑗𝑦,𝑢^′_𝑦,𝜌𝐷_𝛽𝑣^𝑗,𝐷_𝛼𝜑^𝑖≤√𝑑𝑥𝐼+𝜔(𝐼)𝜌sup_{𝐵+𝜌/2𝑥0}𝐷𝜑, (33) and from (32) we observe from the definition of 𝐶₉ (recalling also the definition of 𝛾) 𝜌2^2−𝑛_𝐵⁺_𝜌/2(𝑦)|𝐷𝑣|²𝑑𝑥<1. (34) Further we note 𝑣|_{𝐷𝜌(𝑦)}=1𝛾𝑤|_{𝐷𝜌(𝑦)}=1𝛾𝑢−𝑔|_{𝐷𝜌(𝑦)}≡0. (35)
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 (36) holds. Then in view of (33), (34), and (35) we would be able to apply Lemma 5 to conclude the existence of a function ℎ∈𝐻^1,2(𝐵⁺_𝜌/2(𝑦),𝑅^𝑁) which is 𝐴^𝛼𝛽_𝑖𝑗(𝑦,𝑢^′_𝑦,𝜌)-harmonic, with ℎ|_{𝐷𝜌/2(𝑦)}≡0 such that 𝜌2^2−𝑛_𝐵⁺_𝜌/2(𝑦)|𝐷ℎ|²𝜌𝑑𝑥≤1,2^−𝑛_𝐵⁺_𝜌/2(𝑦)|𝑣−ℎ|²𝑑𝑥≤𝜀. (37)(38)
For 𝜃∈(0,1/4] arbitrary (to be fixed later), we have from the Campanato theorem, noting (37) and recalling also that ℎ(𝑦)=0, sup_{𝐵+𝜃𝜌𝑦}|ℎ|²≤𝜃²𝜌²sup_{𝐵+𝜌/4𝑦}|𝐷ℎ|²≤4𝐶₀𝜃². (39)
Using (38) and (39) we observe 𝜃𝜌^−𝑛_{𝐵+𝜃𝜌𝑦}|𝑣|²𝑑𝑥≤2𝜃𝜌^−𝑛_{𝐵+𝜃𝜌𝑦}|𝑣−ℎ|²𝑑𝑥+_{𝐵+𝜃𝜌𝑦}|ℎ|²𝑑𝑥≤2𝜃𝜌^−𝑛𝜌2^𝑛1𝜀+2𝛼_𝑛𝜃𝜌^𝑛sup_{B+𝜃𝜌𝑦}|ℎ|²≤2^1−𝑛𝜃^−𝑛𝜀+4𝛼_𝑛𝐶₀𝜃², (40) and, hence, on multiplying this through by 𝛾², we obtain the estimate 𝜃𝜌^−𝑛_{𝐵+𝜃𝜌(𝑦)}|𝑤|²𝑑𝑥≤𝐶²₉2^1−𝑛𝜃^−𝑛𝜀+4𝛼_𝑛𝐶₀𝜃²𝐼. (41)
For the time being, we restrict to the case that 𝑔 does not vanish identically. Recalling that 𝑤=𝑢−𝑔, using in turn Poincaré’s, Sobolev’s, and then Hölder’s inequalities, and noting also that 𝑢^′_{𝑦,𝜃𝜌}=𝑔^′_{𝑦,𝜃𝜌}, thus from (41) we get 𝜃𝜌^−𝑛_{𝐵+𝜃𝜌𝑦}𝑢−u^′_{𝑦,𝜃𝜌}²𝑑𝑥≤2𝜃𝜌^−𝑛_{𝐵+𝜃𝜌𝑦}𝑢−𝑔²𝑑𝑥+_{𝐵+𝜃𝜌𝑦}𝑔−𝑔^′_{𝑦,𝜃𝜌}²𝑑𝑥≤2𝐶²₉2^1−𝑛𝜃^−𝑛𝜀+4𝛼_𝑛𝐶₀𝜃²𝐼+2𝐶_𝑝𝜃𝜌^2−𝑛12𝛼_𝑛𝜃𝜌^{𝑛1−(2/𝑠)}𝑔²_𝐻1,𝑠≤𝐶₁₀𝜃^−𝑛𝜀+𝜃²𝐼+𝐶₁₀𝜃^{2(1−(𝑛/𝑠))}𝐼, (42) for 𝐶₁₀=max{8𝛼_𝑛𝐶₀𝐶²₉,2^2/𝑠𝐶_𝑝𝛼_𝑛^1−(2/𝑠)}, and provided 𝜀=𝜃^𝑛+2, we have 𝜃𝜌^−𝑛_{𝐵+𝜃𝜌(𝑦)}𝑢−𝑢^′_{𝑦,𝜃𝜌}²𝑑𝑥≤3𝐶₁₀𝜃^{2(1−(𝑛/𝑠))}𝐼. (43)
Note that fix 𝜀=𝜃^𝑛+2, which is also fixed 𝛿. Since 𝜌≤1, we see from the definition of 𝐼𝑔²_𝐻1,𝑠𝜃𝜌^{2(1−(𝑛/𝑠))}≤𝜃^{2(1−(𝑛/𝑠))}𝐼, (44) and further 𝜃𝜌²≤𝜃²𝐼. (45)
Combining these estimates with (43), we can get 𝐼𝑦,𝜃𝜌≤3𝐶₁₀+1𝜃^{2(1−(𝑛/𝑠))}𝐼. (46)
Choose 𝜃∈(0,1/4] sufficiently small that there holds: 3(𝐶₁₀+1)𝜃^{2(1−(𝑛/𝑠))}≤𝜃^2𝜎.
We can see from (46) 𝐼𝑦,𝜃𝜌≤𝜃^2𝜎𝐼. (47)
We now choose 𝑠₀>0 such that 0<𝜔(𝑠₀)<(𝛿/2) and define 𝐶₈ by 𝐶₈=max2^𝑛−1,2𝐶²₉+1,2𝐶²_𝑠+1. (48) Suppose that we have 𝐼𝑥₀,𝑅≤𝐶₈⁻¹𝛿min²4,𝑠₀, (49) for some 𝑅∈(0,𝑅₀], where 𝑅₀=min{2𝑠₀,1−|𝑥₀|}.
For any 𝑦∈𝐷_𝑅/2(𝑥₀) we use the Sobolev inequality to calculate 𝛼_𝑛𝑅^𝑛2^𝑛+1𝑢^′_𝑥0,𝑅−𝑢^′_𝑦,𝑅/2²=_𝐵+𝑅/2𝑢^′_𝑥0,𝑅−𝑢^′_𝑦,𝑅/2²𝑑𝑥=_𝐵+𝑅/2𝑔^′_𝑥0,𝑅−𝑔^′_𝑦,𝑅/2²𝑑𝑥≤2_𝐵+𝑅/2𝑔−𝑔^′_𝑥0,𝑅²𝑑𝑥+2_𝐵+𝑅/2𝑔−𝑔^′_𝑦,𝑅/2²𝑑𝑥≤2𝛼_𝑛𝐶²_𝑠𝑔²_𝐻1,𝑠𝑅^{𝑛+2(1−(𝑛/𝑠))}. (50)
Then we can calculate 𝐼1𝑦,2𝑅≤2^𝑛−1_{𝐵+𝑅/2(𝑦)}𝑢−𝑢^′_𝑥0,𝑅²+𝑑𝑥2𝐶²_𝑠+1𝑔²_𝐻1,𝑠𝑅^{2(1−(𝑛/𝑠))}+14𝑅²≤𝐶₈𝐼𝑥₀,𝑅. (51)
Then we have 𝐼1𝑦,2𝑅𝐼1+𝜔𝑦,2𝑅≤𝐶₈𝐼𝑥₀,𝑅+𝜔𝐶₈𝐼𝑥₀≤1,𝑅2𝛿+𝜔𝑠₀≤𝛿, (52) which means that the condition (49) is sufficient to guarantee the smallness condition (37) for 𝜌=𝑅/2, for all 𝑦∈𝐷_𝑅/2(𝑥₀). We can thus conclude that (46) holds in this situation. From (46) we thus have 𝐼𝑦,𝜃𝜌2+𝜔𝐼𝑦,𝜃𝜌2≤𝐼1y,2𝑅+𝜔𝐼1𝑦,2𝑅≤𝛿, (53) meaning that we can apply (46) on 𝐵⁺_𝜃𝜌/2(𝑦) as well, yielding 𝐼𝜃𝑦,²𝑅2≤𝜃^4𝜎𝐼𝑅𝑦,2, (54) and inductively 𝐼𝜃𝑦,^𝑘𝑅2≤𝜃^2𝑘𝜎𝐼𝑅𝑦,2. (55)
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∈𝑁₀ such that 𝜃^𝑘+1𝑅/2<𝜌≤𝜃^𝑘𝑅/2. Firstly we use the Sobolev inequality, to see_{𝐵+𝜌(𝑦)}𝑢^′_𝑦,𝜌−𝑢^′_{𝑦,𝜃𝑘𝑅/2}²𝑑𝑥≤2𝛼_𝑛12𝜃^𝑘𝑅^𝑛𝐶²_𝑠𝑔²_𝐻1,𝑠12𝜃^𝑘𝑅^{2(1−(𝑛/𝑠))}, (56) which allows us to deduce _{𝐵+𝜌(𝑦)}𝑢−𝑢^′_𝑦,𝜌²𝑑𝑥≤2_{𝐵+𝜌(𝑦)}𝑢−𝑢_𝑦,𝜃^𝑘_𝑅/2′²𝑑𝑥+4𝛼_𝑛12𝜃^𝑘𝑅^𝑛𝐶²_𝑠𝑔²_𝐻1,𝑠12𝜃^𝑘𝑅^{2(1−(𝑛/𝑠))}, (57) and, hence, 𝐼𝑦,𝜌≤𝐶₁₁𝐼𝜃𝑦,^𝑘𝑅2, (58) for 𝐶₁₁=8𝜃^−𝑛𝐶²_𝑠+1. Combining this with (55) and (51), we have 𝐼𝑦,𝜌≤𝐶₁₁𝜃^2𝑘𝜎𝐼𝑅𝑦,2≤𝐶₈𝐶₁₁𝜃^−2𝜎2𝜌𝑅^2𝜎𝐼𝑥₀,𝑅≤𝐶₈𝐶₁₁2𝜃𝐼𝑥₀𝜌,𝑅𝑅^2𝜎, (59) and more particularly inf_{𝜇∈𝑅𝑁𝐵+𝜌(𝑦)}𝑢−𝜇²𝑑𝑥≤𝐶₁₂𝐼𝑥₀𝜌,𝑅𝑅^2𝜎, (60) for 𝐶₁₂=𝐶₈𝐶₁₁(2/𝜃)^2𝜎. Recall that this estimate is valid for all 𝑦∈𝐷 and 𝜌 with 𝐷_𝜌(𝑦)⊂𝐷_𝑅/2(𝑥₀); assume only the condition (49) on 𝐼(𝑥₀,𝑅). This yields after replacing 𝑅 with 6𝑅 the boundary estimate (15) which requires to apply Lemma 6.

Combining the boundary and interior estimates [19] we can derive the desired result. As the argument for combining the boundary and interior regularity results is relatively standard, we omit it. Hence we can apply Lemma 6 and conclude the desired Hölder continuity.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11201415, 11271305), Natural Science Foundation of Fujian Province (2012J01027), and Training Programme Foundation for Excellent Youth Researching Talents of Fujian's Universities (JA12205).