1.Introduction ShuhongChen andZhongTan RegularityResultforQuasilinearEllipticSystemswithSuperQuadraticNaturalGrowthCondition ResearchArticle

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Volume 2013, Article ID 676215,8pages http://dx.doi.org/10.1155/2013/676215

Research Article

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

Shuhong Chen

¹

and Zhong Tan

²

1Department of Mathematics and Information Science, Zhangzhou Normal University, Zhangzhou, Fujian 363000, China

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

Correspondence should be addressed to Shuhong Chen; [email protected] Received 31 December 2012; Accepted 2 April 2013

Academic Editor: Paul Eloe

Copyright © 2013 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 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_𝛼(𝐴^𝛼𝛽_𝑖𝑗 (𝑥, 𝑢) 𝐷_𝛽𝑢^𝑗) = 𝐵_𝑖(𝑥, 𝑢, 𝐷𝑢) , 𝑥 ∈ Ω, (1)

whereΩis a bounded domain in𝑅^𝑛with boundary of class 𝐶¹, 𝑛 ≥ 2and𝑢takes value in𝑅^𝑁, 𝑁 > 1. Each𝐴^𝛼𝛽_𝑖𝑗 maps Ω × 𝑅^𝑁into𝑅, and each𝐵_𝑖 mapsΩ × 𝑅^𝑁× 𝑅^𝑛𝑁into𝑅. 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 on𝐴^𝛼𝛽_𝑖𝑗 and𝐵_𝑖 with𝑚 > 2as follows.

(H1) There exists𝐿 > 0such that

𝐴^𝛼𝛽_𝑖𝑗 (𝑥, 𝜉) (], ̃]) ≤ 𝐿(1 + 󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨²)^(𝑚−2)/2|]| |̃]|

for all (𝑥, 𝜉) ∈ Ω × 𝑅^𝑁, ], ̃]∈ 𝑅^𝑛𝑁.

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(H2)𝐴^𝛼𝛽_𝑖𝑗(𝑥, 𝜉) is uniformly strongly elliptic; that is, for some𝜆 > 0we have

𝐴^𝛼𝛽_𝑖𝑗 (𝑥, 𝜉) (],]) ≥ 𝜆(1 + 󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨²)^(𝑚−2)/2|]|² for all (𝑥, 𝜉) ∈ Ω × 𝑅^𝑁, ]∈ 𝑅^𝑛𝑁.

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(H3) Assume that𝐴^𝛼𝛽_𝑖𝑗 ∈ 𝐶⁰(Ω × 𝑅^𝑁, 𝑅^𝑛𝑁)and further that 𝐴^𝛼𝛽_𝑖𝑗 is uniformly continuous on sets of the formΩ × {𝜉 : |𝜉| ≤ 𝑀}, for any fixed𝑀, 0 < 𝑀 < ∞.

(H4) (Natural growth condition). There exist constants 𝑎 and𝑏, with𝑎possibly depending on𝑀 > 0, such that

󵄨󵄨󵄨󵄨𝐵^𝑖(𝑥, 𝜉,])󵄨󵄨󵄨󵄨 ≤ 𝑎 (𝑀) |]|^𝑚+ 𝑏 (4) for all𝑥 ∈ Ω, 𝜉 ∈ 𝑅^𝑁with|𝜉| ≤ 𝑀and]∈ 𝑅^𝑛𝑁. Further hypothesis (H3) deduces, writing𝜔(⋅)for𝜔(𝑀, ⋅), the existence of a monotone nondecreasing concave function 𝜔 : [0, ∞) → [0, ∞)with𝜔(0) = 0, continuous at 0, such that󵄨󵄨󵄨󵄨󵄨󵄨𝐴^𝛼𝛽^𝑖𝑗 (𝑥, 𝑢) − 𝐴^𝛼𝛽_𝑖𝑗 (𝑦,V)󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 𝜔 (󵄨󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨󵄨^𝑚+ |𝑢 −V|^𝑚) , (5) for all𝑥, 𝑦 ∈ Ω, 𝑢,V∈ 𝑅^𝑁with|𝑢|, |V| ≤ 𝑀[1].

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(H5) There exist𝑠with𝑠 > 𝑛and a function𝑔 ∈ 𝐻^1,𝑠(Ω, 𝑅^𝑁), such that

𝑢|_𝜕Ω= 𝑔|_𝜕Ω. (6)

Note that we trivially have𝑔 ∈ 𝐻^1,2(Ω, 𝑅^𝑁). Further, by the Sobolev embedding theorem we have𝑔 ∈ 𝐶^0,𝜅(Ω, 𝑅^𝑁)for any𝜅 ∈ [0, 1 − (𝑛/𝑠)]. If𝑔|_𝜕Ω≡ 0, we will take𝑔 ≡ 0onΩ.

If the domain we consider is an upper half unit ball𝐵⁺, 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 write𝐵_𝜌(𝑥₀) = {𝑥 ∈ 𝑅^𝑛 : |𝑥 − 𝑥₀| < 𝜌}, and further𝐵_𝜌 = 𝐵_𝜌(0),𝐵 = 𝐵₁. Similarly we denote upper half balls as follows: for𝑥₀∈ 𝑅^𝑛−1× {0}, we write𝐵⁺_𝜌(𝑥₀)for{𝑥 ∈ 𝑅^𝑛 : 𝑥_𝑛 > 0, |𝑥 − 𝑥₀| < 𝜌}and set𝐵⁺_𝜌 = 𝐵⁺_𝜌(0),𝐵⁺ = 𝐵⁺₁. For 𝑥₀ ∈ 𝑅^𝑛−1× {0}we further write𝐷_𝜌(𝑥₀)for{𝑥 ∈ 𝑅^𝑛 : 𝑥_𝑛 = 0, |𝑥 − 𝑥₀| < 𝜌}and set𝐷_𝜌= 𝐷_𝜌(0), 𝐷 = 𝐷₁.

Definition 1. By a weak solution of (1) one means a vector valued function𝑢 ∈ 𝑊^1,𝑚(Ω, 𝑅^𝑁) ∩ 𝐿^∞(Ω, 𝑅^𝑁)such that

∫Ω𝐴^𝛼𝛽_𝑖𝑗 (𝑥, 𝑢) (𝐷_𝛽𝑢^𝑗, 𝐷_𝛼𝜑^𝑖) 𝑑𝑥 = ∫

Ω𝐵_𝑖(𝑥, 𝑢, 𝐷𝑢) ⋅ 𝜑^𝑖𝑑𝑥 (8) holds for all test-functions𝜑 ∈ 𝐶₀^∞(Ω, 𝑅^𝑁)and, by approximation, for all𝜑 ∈ 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 class𝐶¹. Let𝑢be a bounded weak solution of (1) satisfying the boundary condition (H5), and‖𝑢‖_𝐿^∞ ≤ 𝑀 <

∞with2𝑎(𝑀)𝑀 < 𝜆, where the structure conditions (H1)–

(H3) hold for 𝐴^𝛼𝛽_𝑖𝑗 and (H4) holds for𝐵_𝑖. Consider a fixed 𝛾 ∈ (0, 𝜎]. Then there exist positive𝑅₁and𝜀₀(depending only on𝑛, 𝑁, 𝜆, 𝐿, 𝑏, 𝑀, 𝑎(𝑀), 𝜔(⋅), 𝑚, and𝛾) with the property that

−∫𝐵𝑅(^𝑥0)^∩Ω󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^𝑥^󸀠⁰^,𝑅󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 + 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩²^H^1,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 that𝑢^󸀠_𝑥₀_,𝑅makes sense: in fact, we have𝑢^󸀠_𝑥₀_,𝑅 = 𝑔_𝑥^󸀠₀_,𝑅. For] ∈ 𝐿¹(𝜕Ω), 𝑥₀ ∈ 𝜕Ω, we set]^󸀠_𝑥₀_,𝑅 =∫⁻_{𝜕Ω∩𝐵}

𝑅(𝑥0)]𝑑𝐻^𝑛−1. In particular, for] ∈ 𝐿¹(𝐷_𝜌(𝑥₀)), 𝑥₀ ∈ 𝐷, we write]^󸀠_𝑥₀_,𝜌 =

∫−_𝐷

𝜌(𝑥0)]𝑑𝐻^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 ofTheorem 2the singular set of the weak solution𝑢has(𝑛 − 2)-dimensional Hausdorff measure zero inΩ.

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

Theorem 4. Consider a bounded weak solution of (1)on the upper half unit ball𝐵⁺which satisfies the boundary condition (H5)^󸀠and‖𝑢‖_𝐿^∞ ≤ 𝑀 < ∞with2𝑎(𝑀)𝑀 < 𝜆, where the structure conditions (H1)–(H3) hold for𝐴^𝛼𝛽_𝑖𝑗 and (H4) holds for 𝐵_𝑖. Then there exist positive 𝑅₀ and 𝜀₀ (depending only on𝑛, 𝑁, 𝜆, 𝐿, 𝑏, 𝑀, 𝑎(𝑀), 𝑀, 𝜔(⋅), 𝑚, and𝛾) with the property that

−∫𝐵⁺_𝑅(𝑥0)󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^󸀠^𝑥⁰^,𝑅󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 + 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩²𝐻^1,𝑠𝑅^{2(1−(𝑛/𝑠))}+ 𝑅²≤ 𝜀₀², (10) for some𝑅 ∈ (0, 𝑅₀]for a given𝑥₀ ∈ 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 that𝑢^󸀠_𝑥₀_,𝑅 exists, and we have indeed𝑢^󸀠_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].

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Lemma 5 (A-harmonic approximation lemma). Consider fixed positive𝜆and𝐿, and𝑛, 𝑁 ∈ 𝑁with𝑛 ≥ 2. Then for any given𝜀 > 0there exists𝛿 = 𝛿(𝑛, 𝑁, 𝜆, 𝐿, 𝜀) ∈ (0, 1]with the following property: for any𝐴 ∈Bil(𝑅^𝑛𝑁)satisfying

𝐴 (],]) ≥ 𝜆|]|² 𝑓𝑜𝑟 𝑎𝑙𝑙 ]∈ 𝑅^𝑛𝑁,

|𝐴 (],])| ≤ 𝐿 |]| |]| 𝑓𝑜𝑟 𝑎𝑙𝑙],]∈ 𝑅^𝑛𝑁 (11) for any𝑤 ∈ 𝐻^1,2(𝐵⁺_𝜌(𝑥₀), 𝑅^𝑁) (for some𝜌 > 0, 𝑥₀ ∈ 𝑅^𝑛) satisfying

𝜌^2−𝑛∫

𝐵⁺_𝜌(^𝑥0)|𝐷𝑤|²𝑑𝑥 ≤ 1,

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨𝜌^2−𝑛∫

𝐵⁺_𝜌(𝑥₀)𝐴 (𝐷𝑤, 𝐷𝜑) 𝑑𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨≤ 𝛿𝜌sup

𝐵⁺_𝜌(^𝑥0)󵄨󵄨󵄨󵄨𝐷𝜑󵄨󵄨󵄨󵄨, 𝑤|_𝐷_𝜌_(𝑥₀₎= 0

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for all 𝜑 ∈ 𝐶¹₀(𝐵⁺_𝜌(𝑥₀), 𝑅^𝑁), there exists an A-harmonic function

V∈ ̃𝐻 ={ {{

̃

𝑤 ∈ 𝐻^1,2(𝐵⁺_𝜌(𝑥₀) , 𝑅^𝑁)

⋅󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝜌^2−𝑛∫

𝐵⁺_𝜌(𝑥0)|𝐷̃𝑤|²𝑑𝑥 ≤ 1, ̃𝑤󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝐷𝜌(𝑥0)

≡ 0} }}

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with

𝜌^−𝑛∫

𝐵⁺𝜌(𝑥0)|V− 𝑤|²𝑑𝑥 ≤ 𝜀. (14) Next we recall a slight modification of a characterization of H¨older continuous functions originally due to Campanato [21].

Lemma 6. Consider 𝑛 ∈ 𝑁, 𝑛 ≥ 2, and𝑥₀ ∈ 𝑅^𝑛−1× {0}.

Suppose that there are positive constants𝜅and𝛼, with𝛼 ∈ (0, 1]such that, for some ] ∈ 𝐿²(B⁺_6𝑅(𝑥₀)), there holds the following:

inf𝜇∈𝑅{𝜌^−𝑛∫

𝐵⁺𝜌(𝑦)󵄨󵄨󵄨󵄨]− 𝜇󵄨󵄨󵄨󵄨²𝑑𝑥} ≤ 𝜅²(𝜌

𝑅)^2𝛼, (15) for all𝑦 ∈ 𝐷_2𝑅(𝑥₀)and𝜌 ≤ 4𝑅; and

inf𝜇∈𝑅𝜌^−𝑛{∫

𝐵_𝜌(𝑦)󵄨󵄨󵄨󵄨]− 𝜇󵄨󵄨󵄨󵄨²𝑑𝑥} ≤ 𝜅²(𝜌

𝑅)^2𝛼, (16) for all𝑦 ∈ 𝐵⁺_2𝑅(𝑥₀)and𝐵_𝜌(𝑦) ⊂ 𝐵⁺_2𝑅(𝑥₀).

Then there exists a H¨older continuous representative of the 𝐿²-class of]on𝐵⁺_𝑅(𝑥₀), and for this representative]there holds

|](𝑥) −](𝑧)| ≤ 𝐶_𝜅(|𝑥 − 𝑧|

𝑅 )^𝛼, (17)

for all𝑥, 𝑧 ∈ 𝐵⁺_𝑅(𝑥₀), for a constant𝐶_𝜅depending only on 𝑛 and𝛼.

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 positive𝜆and𝐿, and𝑛, 𝑁 ∈ 𝑁with 𝑛 ≥ 2. Then there exists𝐶₀ depending only on𝑛,𝑁,𝜆, and 𝐿 (without loss of generality we take 𝐶₀ ≥ 1)such that, for 𝐴 ∈ Bil(𝑅^𝑛𝑁)satisfying(11), any A-harmonic functionℎon 𝐵⁺_𝜌(𝑥₀)withℎ|_𝐷_𝜌_(𝑥₀₎≡ 0satisfies

𝜌² sup

𝐵⁺_𝜌/2(^𝑥0)|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 constant𝐶_𝑠depending only on𝑠, 𝑛, and𝑁such that for 𝑥₀∈ 𝐷, 𝜌 ≤ 1 − |𝑥₀|there holds

sup

𝐵_𝜌⁺(^𝑥0)󵄨󵄨󵄨󵄨󵄨𝑔 − 𝑔^𝑥^󸀠⁰^,𝜌󵄨󵄨󵄨󵄨󵄨 ≤ 𝐶^𝑠𝜌^{1−(𝑛/𝑠)}󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩𝐻^1,𝑠(𝐵⁺_𝜌(𝑥0),𝑅^𝑁). (19) Obviously, the inequality remains true if we replace

‖𝑔‖_𝐻1,𝑠(𝐵⁺_𝜌(𝑥0),𝑅^𝑁) by ‖𝑔‖_𝐻1,𝑠(𝐵⁺,𝑅^𝑁), which we will henceforth abbreviate simply as‖𝑔‖_𝐻1,𝑠.

Next we note that the Poincar´e inequality in this setting for𝑥₀∈ 𝐷, 𝜌 ≤ 1 − |𝑥₀|yields

∫𝐵⁺_𝜌(𝑥₀)󵄨󵄨󵄨󵄨󵄨𝑔 − 𝑔^𝑥⁰^,𝜌󵄨󵄨󵄨󵄨󵄨^𝑚𝑑𝑥 ≤ 𝐶_𝑝𝜌^𝑚∫

𝐵⁺_𝜌(𝑥₀)󵄨󵄨󵄨󵄨𝐷𝑔󵄨󵄨󵄨󵄨^𝑚𝑑𝑥, (20) for a constant𝐶_𝑝which depends only on𝑛.

Finally, we fix an exponent𝜎 ∈ (0, 1)as follows: if𝑔 ≡ 0, 𝜎can be chosen arbitrarily (but henceforth fixed); otherwise we take𝜎fixed in(0, 1 − (𝑛/𝑠)).

Then we establish an appropriate inequality for Cacciop- poli.

Theorem 8 (Caccioppoli’s inequality). Let 𝑢 ∈ 𝑊^1,𝑚(Ω, 𝑅^𝑁) ∩ 𝐿^∞(Ω, 𝑅^𝑁)with‖𝑢‖_𝐿^∞ ≤ 𝑀 < ∞and2𝑎(𝑀)𝑀 <

𝜆be a weak solution of systems(1)under assumption conditions (H1)–(H5). Then there exists𝜌₀(𝐿, 𝑀, 𝑎(𝑀), 𝑠, ‖𝑔‖_𝐻1,𝑠) > 0 such that, for all𝐵⁺_𝜌(𝑥₀) ⊂ 𝐵⁺, with𝑥₀ ∈ 𝐷⁺,0 < 𝜌 < 𝑅 < 𝜌₀, there holds

∫𝐵⁺_𝜌/2(^𝑥0)|𝐷𝑢|²𝑑𝑥 ≤ 𝐶₁∫

𝐵⁺_𝜌(^𝑥0)

󵄨󵄨󵄨󵄨󵄨𝑢 (𝑥) − 𝑢^𝑥^󸀠⁰^,𝑅󵄨󵄨󵄨󵄨󵄨²

𝜌² 𝑑𝑥 + 𝐶₂𝛼_𝑛𝜌^𝑛 + 𝐶₃(𝛼_𝑛𝜌^𝑛)^1−(2/𝑠)󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩²𝐻^1,𝑠,

(21) where𝐶₁depends only on𝜆,𝐿, and𝑀and𝐶₃depends on these quantities, and in addition to𝐶_𝑝,𝐶₂depends on𝜆,𝐿,𝑀,𝑎,𝑏, and‖𝑔‖_𝐿∞(𝐵,𝑅^𝑁).

(4)

Proof. Consider a cutoff function𝜂 ∈ 𝐶^∞₀ (𝐵⁺_𝜌/2(𝑥₀)), satisfy- ing0 ≤ 𝜂 ≤ 1, 𝜂 ≡ 0on𝐵_𝜌/2⁺ (𝑥₀)and|∇𝜂| < 4/𝜌. Then the function(𝑢 − 𝑔)𝜂²is in𝑊₀^1,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 that2𝑎(𝑀)𝑀 < 𝜆, we can get from (8) with𝜀positive but arbitrary (to be fixed later)

∫𝐵⁺_𝜌(𝑥0)𝐴^𝛼𝛽_𝑖𝑗 (⋅, 𝑢) (𝐷_𝛽𝑢^𝑗, 𝐷_𝛼𝑢^𝑖) 𝜂²𝑑𝑥

≤ 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|𝜂󵄨󵄨󵄨󵄨𝑢 − 𝑔󵄨󵄨󵄨󵄨 𝑑𝑥 + 𝑎 ∫

𝐵⁺_𝜌(𝑥0)|𝐷𝑢|^𝑚󵄨󵄨󵄨󵄨𝑢 − 𝑔󵄨󵄨󵄨󵄨𝜂²𝑑𝑥 + 𝑏 ∫

𝐵⁺_𝜌(𝑥0)󵄨󵄨󵄨󵄨𝑢 − 𝑔󵄨󵄨󵄨󵄨𝜂²𝑑𝑥

≤ 𝜀 ∫

𝐵⁺𝜌(𝑥0)(1 + |𝑢|²)^(𝑚−2)/2|𝐷𝑢|²𝜂²𝑑𝑥 + 𝑎sup

𝐵⁺_𝜌(^𝑥0)󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^󸀠^𝑥⁰^,𝜌󵄨󵄨󵄨󵄨󵄨 ∫_𝐵+

𝜌(𝑥₀)|𝐷𝑢|^𝑚𝜂²𝑑𝑥 + 𝑎sup

𝐵⁺_𝜌(^𝑥0)󵄨󵄨󵄨󵄨󵄨𝑔 − 𝑔^󸀠^𝑥⁰^,𝜌󵄨󵄨󵄨󵄨󵄨 ∫_𝐵+

𝜌(𝑥0)|𝐷𝑢|^𝑚𝜂²𝑑𝑥 +𝐿²

2𝜀∫

𝐵⁺_𝜌(𝑥0)(1 + |𝑢|²)^(𝑚−2)/2󵄨󵄨󵄨󵄨𝐷𝑔󵄨󵄨󵄨󵄨²𝜂²𝑑𝑥 +4𝐿²

𝜀 ∫

𝐵⁺_𝜌(𝑥₀)(1 + |𝑢|²)^(𝑚−2)/2󵄨󵄨󵄨󵄨𝐷𝜂󵄨󵄨󵄨󵄨²󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^󸀠^𝑥⁰^,𝜌󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 +4𝐿²

𝜀 ∫

𝐵⁺𝜌(𝑥0)(1 + |𝑢|²)^(𝑚−2)/2󵄨󵄨󵄨󵄨𝐷𝜂󵄨󵄨󵄨󵄨²󵄨󵄨󵄨󵄨󵄨𝑔 − 𝑔^󸀠^𝑥⁰^,𝜌󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 +𝜀

2𝑏²∫

𝐵_𝜌⁺(𝑥₀)𝜌²𝜂²𝑑𝑥 + 1 𝜀𝜌²∫

𝐵⁺_𝜌(𝑥₀)󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^󸀠^𝑥⁰^,𝜌󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 + 1

𝜀𝜌²∫

𝐵_𝜌⁺(𝑥0)󵄨󵄨󵄨󵄨󵄨𝑔 − 𝑔^󸀠^𝑥⁰^,𝜌󵄨󵄨󵄨󵄨󵄨²𝑑𝑥

≤ 𝜀 ∫

𝐵⁺𝜌(𝑥0)(1 + |𝑢|²)^(𝑚−2)/2|𝐷𝑢|²𝜂²𝑑𝑥 + 𝑎 (𝑀 + 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩𝐿^∞(𝐵⁺,𝑅^𝑁)) ∫

𝐵⁺_𝜌(𝑥₀)|𝐷𝑢|^𝑚𝜂²𝑑𝑥 +64𝐿²+ 1

𝜀 ∫

𝐵⁺𝜌(𝑥0)(1 + |𝑢|²)^(𝑚−2)/2 1

𝜌²󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^󸀠^𝑥⁰^,𝜌󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 +𝜀

4𝑏²𝜂²𝛼_𝑛𝜌^𝑛+2 + (𝐿²

2𝜀 +64𝐿²𝐶_𝑝 2𝜀 +4𝐶_𝑝

𝜀 )

⋅ ∫𝐵⁺_𝜌(𝑥0)(1 + |𝑢|²)^(𝑚−2)/2󵄨󵄨󵄨󵄨𝐷𝑔󵄨󵄨󵄨󵄨²𝜂²𝑑𝑥

≤ 𝜀 ∫

𝐵⁺𝜌(𝑥0)(1 + |𝑢|²)^(𝑚−2)/2󵄨󵄨󵄨󵄨𝐷𝑔󵄨󵄨󵄨󵄨²𝜂²𝑑𝑥

+ 𝑎 (𝑀 + 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩𝐿^∞(𝐵⁺,𝑅^𝑁)) 𝐶 (‖𝑢‖_𝑊^1,𝑚_(𝐵⁺_𝜌_(𝑥₀₎₎) 𝛼_𝑛𝜌^𝑛 +64𝐿²+ 1

𝜀 ∫

𝐵⁺_𝜌(𝑥0)(1 + |𝑢|²)^(𝑚−2)/2(𝑢 − 𝑢^󸀠_𝑥₀_,𝜌

𝜌 )

2

𝑑𝑥

+𝜀

4𝑏²𝜂²𝛼_𝑛𝜌^𝑛+2 + (1 + 𝑀²)^(𝑚−2)/2(𝐿²

2𝜀 +64𝐿²𝐶_𝑝 2𝜀 +4𝐶_𝑝

𝜀 )

⋅ ∫𝐵⁺_𝜌(𝑥₀)󵄨󵄨󵄨󵄨𝐷𝑔󵄨󵄨󵄨󵄨²𝜂²𝑑𝑥.

(22)

Using (H2), (19), and (20), we thus have

(𝜆 − 𝜀) ∫

𝐵⁺_𝜌(^𝑥0)|𝐷𝑢|²𝜂²𝑑𝑥

≤ (𝜆 − 𝜀) ∫

𝐵⁺_𝜌(^𝑥0)(1 + |𝑢|²)^(𝑚−2)/2|𝐷𝑢|²𝜂²𝑑𝑥

≤ 64𝐿²+ 1

𝜀 ∫

𝐵⁺_𝜌(^𝑥0)(1 + |𝑢|²)^(𝑚−2)/2 1

𝜌²󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^󸀠^𝑥⁰󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 + 𝐶 (𝑎, 𝑀, 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩𝐿^∞(𝐵⁺,𝑅^𝑁), ‖𝑢‖_𝑊^1,𝑚_(𝐵⁺_𝜌_(𝑥₀₎₎, 𝑏) 𝛼_𝑛𝜌^𝑛 + (𝐿, 𝐶_𝑝, 𝑀) ∫

𝐵⁺𝜌(^𝑥0)󵄨󵄨󵄨󵄨𝐷𝑔󵄨󵄨󵄨󵄨²𝑑𝑥

≤ 64𝐿²+ 1

𝜀 (1 + 𝑀²)^(𝑚−2)/2∫

𝐵⁺_𝜌(^𝑥0) 1

𝜌²󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^󸀠^𝑥⁰󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 + 𝐶 (𝑎, 𝑀, 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩𝐿^∞(𝐵⁺,𝑅^𝑁), ‖𝑢‖_𝑊^1,𝑚_(𝐵⁺_𝜌_(𝑥₀₎₎, 𝑏) 𝛼_𝑛𝜌^𝑛 + (𝐿, 𝐶_𝑝, 𝑀) (𝛼_𝑛𝜌^𝑛)^1−(2/𝑠)󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩²𝐻^1,𝑠.

(23)

Thus, we fix𝜀small 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.

(5)

Lemma 9. Consider𝑢 ∈ 𝑊^1,𝑚(Ω, 𝑅^𝑁) ∩ 𝐿^∞(Ω, 𝑅^𝑁)to be a weak solution of (1),𝑥₀ ∈ 𝐷and𝑦 ∈ 𝐷_𝑅(𝑥₀), 𝐷_𝜌(𝑦) ⊂⊂

𝐷_𝑅(𝑥₀), for𝑅 < 1 − |𝑥₀|, and 𝜑 ∈ 𝐶^∞₀ (𝐵⁺_𝜌/2(𝑦), 𝑅^𝑁) with sup

𝐵⁺_𝜌(𝑦)|𝐷𝜑| ≤ 1. We have

(𝜌 2)^2−𝑛∫

𝐵⁺_𝜌/2(𝑦)𝐴^𝛼𝛽_𝑖𝑗 (𝑦, 𝑢⁺_𝑦,𝜌) (𝐷_𝛽𝑢^𝑗, 𝐷_𝛼𝜑^𝑖) 𝑑𝑥

≤ 𝐶₄√𝐼 (√𝐼 + 𝜔 (𝐼)) 𝜌 sup

𝐵⁺_𝜌/2(^𝑥0)󵄨󵄨󵄨󵄨𝐷𝜑󵄨󵄨󵄨󵄨. (24) Here and hereafter, we define

𝐼 (𝑧, 𝑟) = −∫

𝐵⁺𝑟(𝑧)󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^󸀠^𝑧,𝑟󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 + 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩²^𝐻^1,𝑠𝑟^{2(1−(𝑛/𝑠))}+ 𝑟², (25) for𝑧 ∈ 𝐷, 𝑟 ∈ (0, 1 − |𝑧|).

Proof. Using (8) we have

∫𝐵⁺_𝜌/2(^𝑦)𝐴^𝛼𝛽_𝑖𝑗 (𝑦, 𝑢^󸀠_𝑦,𝜌) (𝐷_𝛽𝑢^𝑗, 𝐷_𝛼𝜑^𝑖) 𝑑𝑥

≤ [𝑎 ∫

𝐵⁺_𝜌/2(^𝑦)|𝐷𝑢|^𝑚𝑑𝑥 + 2^−𝑛−1𝛼_𝑛𝑏𝜌^𝑛] ⋅ 𝜌 sup

𝐵⁺_𝜌/2(^𝑦)󵄨󵄨󵄨󵄨𝐷𝜑󵄨󵄨󵄨󵄨

+ ∫𝐵⁺_𝜌/2(^𝑦)󵄨󵄨󵄨󵄨󵄨󵄨𝐴^𝛼𝛽^𝑖𝑗 (𝑦, 𝑢^󸀠_𝑦,𝜌) − 𝐴^𝛼𝛽_𝑖𝑗 (𝑥, 𝑢)󵄨󵄨󵄨󵄨󵄨󵄨

⋅ |𝐷𝑢| 𝑑𝑥 sup

𝐵⁺_𝜌/2(^𝑦)󵄨󵄨󵄨󵄨𝐷𝜑󵄨󵄨󵄨󵄨.

(26)

Applying in turn Young’s inequality, (H3), the Cac- cioppoli inequality (Theorem 8), and Jensen’s inequality, we calculate from (26)

∫𝐵_𝜌/2⁺ (^𝑦)𝐴^𝛼𝛽_𝑖𝑗 (𝑦, 𝑢^󸀠_𝑦,𝜌) (𝐷_𝛽𝑢^𝑗, 𝐷_𝛼𝜑^𝑖) 𝑑𝑥

≤ [𝑎 ∫

𝐵⁺_𝜌/2(^𝑦)|𝐷𝑢|^𝑚𝑑𝑥 + 2^−𝑛−1𝛼_𝑛𝑏𝜌^𝑛] ⋅ 𝜌 + [∫𝐵⁺_𝜌/2(^𝑦)󵄨󵄨󵄨󵄨󵄨󵄨𝐴^𝛼𝛽^𝑖𝑗 (𝑦, 𝑢^󸀠_𝑦,𝜌) − 𝐴^𝛼𝛽_𝑖𝑗 (𝑥, 𝑢)󵄨󵄨󵄨󵄨󵄨󵄨^1/2𝑑𝑥]

1/2

⋅ [∫𝐵⁺_𝜌/2(𝑦)|𝐷𝑢|²𝑑𝑥]

1/2

≤𝛼_𝑛𝜌^𝑛−1

2 {(𝑎−∫

𝐵⁺_𝜌(𝑦)|𝐷𝑢|^𝑚𝑥 + 2^−𝑛𝑏) 𝜌²} + 𝛼_𝑛𝜌^𝑛−1𝜔 (𝜌^𝑚+ 𝑀^𝑚−2−∫

𝐵⁺𝜌(𝑦)󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^𝑦,𝜌^󸀠 󵄨󵄨󵄨󵄨󵄨²𝑑𝑥)

⋅ {𝐶₁−∫

𝐵⁺_𝜌(𝑦)󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^󸀠^𝑦,𝜌󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 + 𝐶₃󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩²𝐻^1,𝑠𝜌^{2(1−(𝑛/𝑠))} + 𝐶₂𝜌²}

1/2

≤𝛼_𝑛𝜌^𝑛−1

2 𝐶₅𝐼 +𝛼_𝑛𝜌^𝑛−1

2 𝐶₆𝜔 (𝐼) √𝐼

≤ 𝐶₇𝛼_𝑛𝜌^𝑛−1(𝐼 + 𝜔 (𝐼) √𝐼) ,

(27)

where𝐶₅ = 𝑎‖𝑢‖_𝑊^1,𝑚 + 𝑏, 𝐶₆ = max{√𝐶₁, √𝐶₂, √𝐶₃}, and 𝐶₇= (1/2)(𝐶₅+ 𝐶₆), for𝑧 ∈ 𝐷,𝑟 ∈ (0, 1 − |𝑧|). We introduce the notation

𝐼 (𝑧, 𝑟) = −∫

𝐵⁺_𝑟(𝑧)󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^󸀠^𝑧,𝑟󵄨󵄨󵄨󵄨󵄨²𝑑𝑧 + 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩²𝐻^1,𝑠𝑟^{2(1−(𝑛/𝑠))}+ 𝑟² (28) and further write𝐼for𝐼(𝑦, 𝜌). For arbitrary𝜑 ∈ 𝐶^∞₀ (Ω, 𝑅^𝑁) we thus have, by rescalling,

∫𝐵⁺_𝜌/2(𝑦)𝐴^𝛼𝛽_𝑖𝑗 (𝑦, 𝑢^󸀠_𝑦,𝜌) (𝐷_𝛽𝑢^𝑗, 𝐷_𝛼𝜑^𝑖) 𝑑𝑥

≤ 𝐶₇𝛼_𝑛𝜌^𝑛−1√𝐼 (√𝐼 + 𝜔 (𝐼)) .

(29)

Multiplying (29) through by(𝜌/2)^2−𝑛yields

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨(𝜌 2)^2−𝑛∫

𝐵_𝜌/2⁺ (𝑦)𝐴^𝛼𝛽_𝑖𝑗 (𝑦, 𝑢^󸀠_𝑦,𝜌) (𝐷_𝛽𝑢^𝑗, 𝐷_𝛼𝜑^𝑖) 𝑑𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

≤ 𝐶₄√𝐼 (√𝐼 + 𝜔 (𝐼)) 𝜌 sup

𝐵⁺_𝜌/2(^𝑥0)󵄨󵄨󵄨󵄨𝐷𝜑󵄨󵄨󵄨󵄨, (30) for𝐶₄defined by𝐶₄= 2^𝑛−3𝛼_𝑛𝐶₇.

Lemma 10. Consider𝑢satisfying the conditions ofTheorem 2 and𝜎fixed; then we can find𝛿and𝑠₀together, with positive constants𝐶₈such that the smallness conditions:0 < 𝜔(𝑠₀) ≤ 𝛿/2 and 𝐼(𝑥₀, 𝑅) ≤ 𝐶⁻¹₈ min{𝛿²/4, 𝑠₀} together, imply the growth condition

𝐼 (𝑦, 𝜃𝜌) ≤ 𝜃^2𝜎𝐼 (𝑦, 𝜌) . (31)

(6)

Proof. We now set𝑤 = 𝑢 − 𝑔, using in turn (H1), Young’s inequality, and H¨older’s inequality. We have from (30)

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨(𝜌 2)^2−𝑛∫

𝐵⁺_𝜌/2(𝑦)𝐴^𝛼𝛽_𝑖𝑗 (𝑦, 𝑢^󸀠_𝑦,𝜌) (𝐷_𝛽𝑤^𝑗, 𝐷_𝛼𝜑^𝑖) 𝑑𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

≤󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨(𝜌 2)^2−𝑛∫

𝐵⁺_𝜌/2(𝑦)𝐴^𝛼𝛽_𝑖𝑗 (𝑦, 𝑢^󸀠_𝑦,𝜌) (𝐷_𝛽𝑢^𝑗, 𝐷_𝛼𝜑^𝑖) 𝑑𝑥󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨

+󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨(𝜌 2)^2−𝑛∫

𝐵⁺_𝜌/2(𝑦)𝐴^𝛼𝛽_𝑖𝑗 (𝑦, 𝑢^󸀠_𝑦,𝜌) (𝐷_𝛽𝑔^𝑗, 𝐷_𝛼𝜑^𝑖) 𝑑𝑥󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨

≤ 𝐶₉√𝐼 [√𝐼 + 𝜔 (𝐼)] 𝜌 sup

𝐵⁺_𝜌/2(^𝑥0)󵄨󵄨󵄨󵄨𝐷𝜑󵄨󵄨󵄨󵄨,

(32) for𝐶₉=max{𝐶₄, (𝛼_𝑛/2)^{1−(𝑛/𝑠)}}.

We now setV= 𝑤/𝛾, for𝛾 = 𝐶₉√𝐼. From (32) we then have

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨(𝜌 2)^2−𝑛∫

𝐵⁺_𝜌/2(𝑦)𝐴^𝛼𝛽_𝑖𝑗 (𝑦, 𝑢_𝑦,𝜌^󸀠 ) (𝐷_𝛽V^𝑗, 𝐷_𝛼𝜑^𝑖) 𝑑𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

≤ (√𝐼 + 𝜔 (𝐼)) 𝜌 sup

𝐵⁺_𝜌/2(^𝑥0)󵄨󵄨󵄨󵄨𝐷𝜑󵄨󵄨󵄨󵄨, (33) and from (32) we observe from the definition of𝐶₉(recalling also the definition of𝛾)

(𝜌 2)^2−𝑛∫

𝐵⁺_𝜌/2(𝑦)|𝐷V|²𝑑𝑥 < 1. (34) Further we note

V|_𝐷_𝜌_(𝑦)=1

𝛾𝑤|_𝐷_𝜌_(𝑦)=1

𝛾(𝑢 − 𝑔)|_𝐷_𝜌_(𝑦)≡ 0. (35) For𝜀 > 0 we take 𝛿 = 𝛿(𝑛, 𝑁, 𝜆, 𝐿, 𝜀) to be the cor- responding𝛿from 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 5to conclude the existence of a function ℎ ∈ 𝐻^1,2(𝐵⁺_𝜌/2(𝑦), 𝑅^𝑁)which is𝐴^𝛼𝛽_𝑖𝑗(𝑦, 𝑢^󸀠_𝑦,𝜌)-harmonic, with ℎ|_𝐷_𝜌/2_(𝑦)≡ 0such that

(𝜌 2)^2−𝑛∫

𝐵⁺_𝜌/2(𝑦)|𝐷ℎ|²𝑑𝑥 ≤ 1, (37) (𝜌

2)^−𝑛∫

𝐵⁺_𝜌/2(𝑦)|V− ℎ|²𝑑𝑥 ≤ 𝜀. (38) For𝜃 ∈ (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⁺_𝜃𝜌(^𝑦)|ℎ|²≤ 𝜃²𝜌² sup

𝐵⁺_𝜌/4(^𝑦)|𝐷ℎ|²≤ 4𝐶₀𝜃². (39)

Using (38) and (39) we observe (𝜃𝜌)^−𝑛∫

𝐵⁺_𝜃𝜌(^𝑦)|V|²𝑑𝑥

≤ 2(𝜃𝜌)^−𝑛[∫

𝐵⁺_𝜃𝜌(^𝑦)|V− ℎ|²𝑑𝑥 + ∫

𝐵⁺_𝜃𝜌(^𝑦)|ℎ|²𝑑𝑥]

≤ 2(𝜃𝜌)^−𝑛[ [

(𝜌 2)^𝑛𝜀 +1

2𝛼_𝑛(𝜃𝜌)^𝑛 sup

B⁺_𝜃𝜌(^𝑦)|ℎ|²] ]

≤ 2^1−𝑛𝜃^−𝑛𝜀 + 4𝛼_𝑛𝐶₀𝜃²,

(40)

and, hence, on multiplying this through by𝛾², we obtain the estimate

(𝜃𝜌)^−𝑛∫

𝐵⁺_𝜃𝜌(𝑦)|𝑤|²𝑑𝑥 ≤ 𝐶²₉(2^1−𝑛𝜃^−𝑛𝜀 + 4𝛼_𝑛𝐶₀𝜃²) 𝐼. (41) For the time being, we restrict to the case that 𝑔does not vanish identically. Recalling that 𝑤 = 𝑢 − 𝑔, using in turn Poincar´e’s, Sobolev’s, and then H¨older’s inequalities, and noting also that𝑢^󸀠_{𝑦,𝜃𝜌}= 𝑔_{𝑦,𝜃𝜌}^󸀠 , thus from (41) we get

(𝜃𝜌)^−𝑛∫

𝐵⁺_𝜃𝜌(^𝑦)󵄨󵄨󵄨󵄨󵄨𝑢 −u^󸀠_{𝑦,𝜃𝜌}󵄨󵄨󵄨󵄨󵄨²𝑑𝑥

≤ 2(𝜃𝜌)^−𝑛[∫

𝐵⁺_𝜃𝜌(^𝑦)󵄨󵄨󵄨󵄨𝑢 − 𝑔󵄨󵄨󵄨󵄨²𝑑𝑥 + ∫

𝐵⁺_𝜃𝜌(^𝑦)󵄨󵄨󵄨󵄨󵄨𝑔 − 𝑔^{𝑦,𝜃𝜌}^󸀠 󵄨󵄨󵄨󵄨󵄨

2𝑑𝑥]

≤ 2𝐶²₉(2^1−𝑛𝜃^−𝑛𝜀 + 4𝛼_𝑛𝐶₀𝜃²) 𝐼 + 2𝐶_𝑝(𝜃𝜌)^2−𝑛[1

2𝛼_𝑛(𝜃𝜌)^𝑛]^1−(2/𝑠)󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩²𝐻^1,𝑠

≤ 𝐶₁₀(𝜃^−𝑛𝜀 + 𝜃²) 𝐼 + 𝐶₁₀𝜃^{2(1−(𝑛/𝑠))}𝐼,

(42) for𝐶₁₀ = max{8𝛼_𝑛𝐶₀𝐶²₉, 2^2/𝑠𝐶_𝑝𝛼_𝑛^1−(2/𝑠)}, and provided𝜀 = 𝜃^𝑛+2, we have

(𝜃𝜌)^−𝑛∫

𝐵⁺_𝜃𝜌(𝑦)󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^󸀠^{𝑦,𝜃𝜌}󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 ≤ 3𝐶₁₀𝜃^{2(1−(𝑛/𝑠))}𝐼. (43) Note that fix𝜀 = 𝜃^𝑛+2, which is also fixed𝛿. Since𝜌 ≤ 1, we see from the definition of𝐼

󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩²𝐻^1,𝑠(𝜃𝜌)^{2(1−(𝑛/𝑠))}≤ 𝜃^{2(1−(𝑛/𝑠))}𝐼, (44) and further

(𝜃𝜌)²≤ 𝜃²𝐼. (45)

Combining these estimates with (43), we can get 𝐼 (𝑦, 𝜃𝜌) ≤ 3 (𝐶₁₀+ 1) 𝜃^{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.

(7)

We can see from (46)

𝐼 (𝑦, 𝜃𝜌) ≤ 𝜃^2𝜎𝐼. (47)

We now choose𝑠₀ > 0such that0 < 𝜔(𝑠₀) < (𝛿/2)and define𝐶₈by

𝐶₈=max{2^𝑛−1, 2𝐶²₉+ 1, 2𝐶²_𝑠+ 1} . (48) Suppose that we have

𝐼 (𝑥₀, 𝑅) ≤ 𝐶⁻¹₈ min{𝛿²

4, 𝑠₀} , (49) for some𝑅 ∈ (0, 𝑅₀], where𝑅₀=min{√2𝑠₀, 1 − |𝑥₀|}.

For any𝑦 ∈ 𝐷_𝑅/2(𝑥₀)we use the Sobolev inequality to calculate

𝛼_𝑛𝑅^𝑛

2^𝑛+1󵄨󵄨󵄨󵄨󵄨𝑢^󸀠^𝑥⁰^,𝑅− 𝑢^󸀠_𝑦,𝑅/2󵄨󵄨󵄨󵄨󵄨²

= ∫𝐵⁺_𝑅/2󵄨󵄨󵄨󵄨󵄨𝑢^󸀠^𝑥⁰^,𝑅− 𝑢^󸀠_𝑦,𝑅/2󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 = ∫

𝐵⁺_𝑅/2󵄨󵄨󵄨󵄨󵄨𝑔^󸀠^𝑥⁰^,𝑅− 𝑔_𝑦,𝑅/2^󸀠 󵄨󵄨󵄨󵄨󵄨²𝑑𝑥

≤ 2 ∫

𝐵⁺_𝑅/2󵄨󵄨󵄨󵄨󵄨𝑔 − 𝑔^󸀠^𝑥⁰^,𝑅󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 + 2 ∫

𝐵⁺_𝑅/2󵄨󵄨󵄨󵄨󵄨𝑔 − 𝑔^𝑦,𝑅/2^󸀠 󵄨󵄨󵄨󵄨󵄨²𝑑𝑥

≤ 2𝛼_𝑛𝐶²_𝑠󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩²𝐻^1,𝑠𝑅𝑛+2(1−(𝑛/𝑠)).

(50) Then we can calculate

𝐼 (𝑦,1 2𝑅)

≤ 2^𝑛−1−∫

𝐵⁺_𝑅/2(𝑦)󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^󸀠^𝑥⁰^,𝑅󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 + (2𝐶²_𝑠 + 1) 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩²𝐻^1,𝑠𝑅^{2(1−(𝑛/𝑠))}+1

4𝑅²

≤ 𝐶₈𝐼 (𝑥₀, 𝑅) .

(51)

Then we have

√𝐼 (𝑦, 12𝑅) + 𝜔 (𝐼 (𝑦,1 2𝑅))

≤ √𝐶₈𝐼 (𝑥₀, 𝑅) + √𝜔 (𝐶₈𝐼 (𝑥₀, 𝑅) )

≤ 1

2𝛿 + 𝜔 (𝑠₀) ≤ 𝛿,

(52)

which means that the condition (49) is sufficient to guarantee the smallness condition (37) for𝜌 = 𝑅/2, for all𝑦 ∈ 𝐷_𝑅/2(𝑥₀).

We can thus conclude that (46) holds in this situation. From (46) we thus have

√𝐼 (𝑦, 𝜃𝜌2) + √𝜔 (𝐼 (𝑦,𝜃𝜌 2))

≤ √𝐼 (y,1

2𝑅) + √𝜔 (𝐼 (𝑦,1

2𝑅)) ≤ 𝛿,

(53)

meaning that we can apply (46) on𝐵⁺_𝜃𝜌/2(𝑦)as well, yielding 𝐼 (𝑦,𝜃²𝑅

2 ) ≤ 𝜃^4𝜎𝐼 (𝑦,𝑅

2) , (54)

and inductively

𝐼 (𝑦,𝜃^𝑘𝑅

2 ) ≤ 𝜃^2𝑘𝜎𝐼 (𝑦,𝑅

2) . (55)

The next step is to go from a discrete to a continuous version of the decay estimate. Given𝜌 ∈ (0, 𝑅/2], we can find 𝑘 ∈ 𝑁₀such that𝜃^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_𝑛( 1

2𝜃^𝑘𝑅)^𝑛𝐶²_𝑠󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩²𝐻^1,𝑠( 1

2𝜃^𝑘𝑅)^{2(1−(𝑛/𝑠))},

(56)

which allows us to deduce

∫𝐵⁺_𝜌(𝑦)󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^𝑦,𝜌^󸀠 󵄨󵄨󵄨󵄨󵄨²𝑑𝑥

≤ 2 ∫

𝐵⁺𝜌(𝑦)󵄨󵄨󵄨󵄨󵄨𝑢 − 𝑢^𝑦,𝜃^𝑘^𝑅/2^󸀠󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 + 4𝛼_𝑛( 1

2𝜃^𝑘𝑅)^𝑛𝐶²_𝑠󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩²𝐻^1,𝑠( 1

2𝜃^𝑘𝑅)^{2(1−(𝑛/𝑠))},

(57)

and, hence,

𝐼 (𝑦, 𝜌) ≤ 𝐶₁₁𝐼 (𝑦,𝜃^𝑘𝑅

2 ) , (58)

for𝐶₁₁= 8𝜃^−𝑛𝐶²_𝑠+ 1. Combining this with (55) and (51), we have

𝐼 (𝑦, 𝜌)

≤ 𝐶₁₁𝜃^2𝑘𝜎𝐼 (𝑦,𝑅

2) ≤ 𝐶₈𝐶₁₁𝜃^−2𝜎(2𝜌

𝑅)^2𝜎𝐼 (𝑥₀, 𝑅)

≤ 𝐶₈𝐶₁₁(2

𝜃) 𝐼 (𝑥₀, 𝑅) (𝜌 𝑅)^2𝜎,

(59)

and more particularly

𝜇∈𝑅inf^𝑁∫

𝐵⁺𝜌(𝑦)󵄨󵄨󵄨󵄨𝑢 − 𝜇󵄨󵄨󵄨󵄨²𝑑𝑥 ≤ 𝐶₁₂𝐼 (𝑥₀, 𝑅) (𝜌

𝑅)^2𝜎, (60) for 𝐶₁₂ = 𝐶₈𝐶₁₁(2/𝜃)^2𝜎. Recall that this estimate is valid for all𝑦 ∈ 𝐷and 𝜌with𝐷_𝜌(𝑦) ⊂ 𝐷_𝑅/2(𝑥₀); assume only the condition (49) on𝐼(𝑥₀, 𝑅). This yields after replacing𝑅 with6𝑅the 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 H¨older continuity.

(8)

Acknowledgments

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

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