Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 676215, 8 pages
http://dx.doi.org/10.1155/2013/676215
Research Article
Regularity Result for Quasilinear Elliptic Systems with Super
Quadratic Natural Growth Condition
Shuhong Chen1 and Zhong Tan2
1
2
Department of Mathematics and Information Science, Zhangzhou Normal University, Zhangzhou, Fujian 363000, China
School of Mathematical Science, Xiamen University, Xiamen, Fujian 361005, China
Correspondence should be addressed to Shuhong Chen; shiny0320@163.com
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.
𝛼𝛽
1. Introduction
This paper considers boundary regularity for weak solutions
of quasilinear elliptic systems
𝛼𝛽
−𝐷𝛼 (𝐴 𝑖𝑗 (𝑥, 𝑢) 𝐷𝛽 𝑢𝑗 ) = 𝐵𝑖 (𝑥, 𝑢, 𝐷𝑢) ,
𝑥 ∈ Ω,
(H2) 𝐴 𝑖𝑗 (𝑥, 𝜉) is uniformly strongly elliptic; that is, for
some 𝜆 > 0 we have
𝛼𝛽
󵄨 󵄨2
𝐴 𝑖𝑗 (𝑥, 𝜉) (], ]) ≥ 𝜆(1 + 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 )
𝑁
(𝑚−2)/2
|]|2
for all (𝑥, 𝜉) ∈ Ω × 𝑅 , ] ∈ 𝑅
(1)
𝑛𝑁
(3)
.
𝛼𝛽
where Ω is a bounded domain in 𝑅𝑛 with boundary of class
𝛼𝛽
𝐶1 , 𝑛 ≥ 2 and 𝑢 takes value in 𝑅𝑁, 𝑁 > 1. Each 𝐴 𝑖𝑗 maps
Ω × 𝑅𝑁 into 𝑅, and each 𝐵𝑖 maps Ω × 𝑅𝑁 × 𝑅𝑛𝑁 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
𝛼𝛽
further impose certain structural conditions on 𝐴 𝑖𝑗 and 𝐵𝑖
with 𝑚 > 2 as follows.
(H1) There exists 𝐿 > 0 such that
𝛼𝛽
󵄨 󵄨2 (𝑚−2)/2
𝐴 𝑖𝑗 (𝑥, 𝜉) (], ̃]) ≤ 𝐿(1 + 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 )
|]| |̃]|
for all (𝑥, 𝜉) ∈ Ω × 𝑅𝑁, ], ̃] ∈ 𝑅𝑛𝑁.
(2)
(H3) Assume that 𝐴 𝑖𝑗 ∈ 𝐶0 (Ω × 𝑅𝑁, 𝑅𝑛𝑁) and further that
𝛼𝛽
𝐴 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
󵄨󵄨󵄨𝐵𝑖 (𝑥, 𝜉, ])󵄨󵄨󵄨 ≤ 𝑎 (𝑀) |]|𝑚 + 𝑏
(4)
󵄨
󵄨
for all 𝑥 ∈ Ω, 𝜉 ∈ 𝑅𝑁 with |𝜉| ≤ 𝑀 and ] ∈ 𝑅𝑛𝑁.
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
󵄨
󵄨󵄨 𝛼𝛽
󵄨󵄨𝐴 (𝑥, 𝑢) − 𝐴𝛼𝛽 (𝑦, V)󵄨󵄨󵄨 ≤ 𝜔 (󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨𝑚 + |𝑢 − V|𝑚 ) , (5)
𝑖𝑗
󵄨
󵄨
󵄨󵄨
󵄨󵄨 𝑖𝑗
for all 𝑥, 𝑦 ∈ Ω, 𝑢, V ∈ 𝑅𝑁 with |𝑢|, |V| ≤ 𝑀 [1].
2
Abstract and Applied Analysis
(H5) There exist 𝑠 with 𝑠 > 𝑛 and a function 𝑔 ∈ 𝐻1,𝑠 (Ω,
𝑅𝑁), such that
𝑢|𝜕Ω = 𝑔|𝜕Ω .
(6)
Note that we trivially have 𝑔 ∈ 𝐻1,2 (Ω, 𝑅𝑁). Further, by
the Sobolev embedding theorem we have 𝑔 ∈ 𝐶0,𝜅 (Ω, 𝑅𝑁) 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 ∈
𝐻1,𝑠 (𝐵+ , 𝑅𝑁), such that
𝑢|𝐷 = 𝑔|𝐷.
(7)
Here we write 𝐵𝜌 (𝑥0 ) = {𝑥 ∈ 𝑅𝑛 : |𝑥 − 𝑥0 | < 𝜌}, and
further 𝐵𝜌 = 𝐵𝜌 (0), 𝐵 = 𝐵1 . Similarly we denote upper half
balls as follows: for 𝑥0 ∈ 𝑅𝑛−1 × {0}, we write 𝐵𝜌+ (𝑥0 ) for {𝑥 ∈
𝑅𝑛 : 𝑥𝑛 > 0, |𝑥 − 𝑥0 | < 𝜌} and set 𝐵𝜌+ = 𝐵𝜌+ (0), 𝐵+ = 𝐵1+ . For
𝑥0 ∈ 𝑅𝑛−1 × {0} we further write 𝐷𝜌 (𝑥0 ) for {𝑥 ∈ 𝑅𝑛 : 𝑥𝑛 =
0, |𝑥 − 𝑥0 | < 𝜌} and set 𝐷𝜌 = 𝐷𝜌 (0), 𝐷 = 𝐷1 .
Definition 1. By a weak solution of (1) one means a vector
valued function 𝑢 ∈ 𝑊1,𝑚 (Ω, 𝑅𝑁) ∩ 𝐿∞ (Ω, 𝑅𝑁) such that
𝛼𝛽
∫ 𝐴 𝑖𝑗 (𝑥, 𝑢) (𝐷𝛽 𝑢𝑗 , 𝐷𝛼 𝜑𝑖 ) 𝑑𝑥 = ∫ 𝐵𝑖 (𝑥, 𝑢, 𝐷𝑢) ⋅ 𝜑𝑖 𝑑𝑥
Ω
Ω
(8)
holds for all test-functions 𝜑 ∈ 𝐶0∞ (Ω, 𝑅𝑁) and, by approximation, for all 𝜑 ∈ 𝑊01,𝑚 (Ω, 𝑅𝑁) ∩ 𝐿∞ (Ω, 𝑅𝑁).
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 ∫Ω 𝐴(𝑥, 𝑢)|𝐷𝑢|2 𝑑𝑥.
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] 𝜀regularity theorem. In this context, using the A-harmonic
approximation technique, we obtain the following regularity
results.
Theorem 2. Consider a bounded domain Ω in 𝑅𝑁, 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)–
𝛼𝛽
(H3) hold for 𝐴 𝑖𝑗 and (H4) holds for 𝐵𝑖 . 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
󵄨󵄨
󵄨2
2
2(1−(𝑛/𝑠))
󵄨󵄨𝑢 − 𝑢𝑥󸀠 ,𝑅 󵄨󵄨󵄨 𝑑𝑥 + 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩H
+ 𝑅2 ≤ 𝜀02
1,𝑠 𝑅
0 󵄨
󵄨
𝐵𝑅 (𝑥0 )∩Ω
−
∫
(9)
for some 𝑅 ∈ (0, 𝑅1 ] for a given 𝑥0 ∈ 𝜕Ω implies 𝑢 ∈
𝐶0,𝛾 (𝐵𝑅/2 (𝑥0 ) ∩ Ω, 𝑅𝑁).
Note in particular that the boundary condition (H5)
means that 𝑢𝑥󸀠 0 ,𝑅 makes sense: in fact, we have 𝑢𝑥󸀠 0 ,𝑅 = 𝑔𝑥󸀠 0 ,𝑅 .
For ] ∈ 𝐿1 (𝜕Ω), 𝑥0 ∈ 𝜕Ω, we set ]󸀠𝑥0 ,𝑅 = ∫− 𝜕Ω∩𝐵 (𝑥 ) ]𝑑𝐻𝑛−1 .
𝑅
0
In particular, for ] ∈ 𝐿1 (𝐷𝜌 (𝑥0 )), 𝑥0 ∈ 𝐷, we write ]󸀠𝑥0 ,𝜌 =
∫− 𝐷 (𝑥 ) ]𝑑𝐻𝑛−1 .
𝜌
0
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
𝛼𝛽
structure conditions (H1)–(H3) hold for 𝐴 𝑖𝑗 and (H4) holds
for 𝐵𝑖 . 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
−
∫
𝐵𝑅+ (𝑥0 )
󵄨󵄨
󵄨2
󵄨󵄨𝑢 − 𝑢𝑥󸀠 ,𝑅 󵄨󵄨󵄨 𝑑𝑥 + 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2𝐻1,𝑠 𝑅2(1−(𝑛/𝑠)) + 𝑅2 ≤ 𝜀02 ,
0 󵄨
󵄨
(10)
for some 𝑅 ∈ (0, 𝑅0 ] for a given 𝑥0 ∈ 𝐷, implies that there
+
holds: 𝑢 ∈ 𝐶0,𝜎 (𝐵𝑅/2 (𝑥0 ), 𝑅𝑁).
Note that analogous to the above, the boundary condition
(H5)󸀠 ensures that 𝑢𝑥󸀠 0 ,𝑅 exists, and we have indeed 𝑢𝑥󸀠 0 ,𝑅 =
𝑔𝑥󸀠 0 ,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].
Abstract and Applied Analysis
3
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 (𝑅𝑛𝑁) satisfying
𝐴 (], ]) ≥ 𝜆|]|2
|𝐴 (], ])| ≤ 𝐿 |]| |]|
𝑓𝑜𝑟 𝑎𝑙𝑙 ] ∈ 𝑅𝑛𝑁,
𝑓𝑜𝑟 𝑎𝑙𝑙 ], ] ∈ 𝑅𝑛𝑁
(11)
for any 𝑤 ∈ 𝐻1,2 (𝐵𝜌+ (𝑥0 ), 𝑅𝑁) (for some 𝜌 > 0, 𝑥0 ∈ 𝑅𝑛 )
satisfying
𝜌
2−𝑛
𝐵𝜌+ (𝑥0 )
(12)
𝑤|𝐷𝜌 (𝑥0 ) = 0
for all 𝜑 ∈ 𝐶01 (𝐵𝜌+ (𝑥0 ), 𝑅𝑁), there exists an A-harmonic
function
{
1,2
+
𝑁
̃= 𝑤
V∈𝐻
{ ̃ ∈ 𝐻 (𝐵𝜌 (𝑥0 ) , 𝑅 )
{
󵄨󵄨
󵄨󵄨
}
󵄨
󵄨󵄨
̃󵄨󵄨󵄨󵄨
𝑤|2 𝑑𝑥 ≤ 1, 𝑤
≡ 0}
⋅󵄨󵄨󵄨𝜌2−𝑛 ∫
|𝐷̃
󵄨󵄨
󵄨󵄨
𝐵𝜌+ (𝑥0 )
󵄨𝐷𝜌 (𝑥0 )
󵄨
}
(13)
with
𝜌−𝑛 ∫
𝐵𝜌+ (𝑥0 )
|V − 𝑤|2 𝑑𝑥 ≤ 𝜀.
(14)
Lemma 6. Consider 𝑛 ∈ 𝑁, 𝑛 ≥ 2, and 𝑥0 ∈ 𝑅𝑛−1 × {0}.
Suppose that there are positive constants 𝜅 and 𝛼, with 𝛼 ∈
(0, 1] such that, for some ] ∈ 𝐿2 (B+6𝑅 (𝑥0 )), there holds the
following:
2𝛼
󵄨2
󵄨󵄨
2 𝜌
inf {𝜌−𝑛 ∫
󵄨󵄨] − 𝜇󵄨󵄨󵄨 𝑑𝑥} ≤ 𝜅 ( ) ,
𝜇∈𝑅
𝑅
𝐵+ (𝑦)
(15)
𝜌
for all 𝑦 ∈ 𝐷2𝑅 (𝑥0 ) and 𝜌 ≤ 4𝑅; and
𝐵𝜌
2𝛼
󵄨2
󵄨󵄨
2 𝜌
󵄨󵄨] − 𝜇󵄨󵄨󵄨 𝑑𝑥} ≤ 𝜅 ( ) ,
𝑅
(𝑦)
(16)
+
+
for all 𝑦 ∈ 𝐵2𝑅
(𝑥0 ) and 𝐵𝜌 (𝑦) ⊂ 𝐵2𝑅
(𝑥0 ).
Then there exists a Hölder continuous representative of the
+
𝐿2 -class of ] on 𝐵𝑅 (𝑥0 ), and for this representative ] there holds
|] (𝑥) − ] (𝑧)| ≤ 𝐶𝜅 (
|𝑥 − 𝑧| 𝛼
) ,
𝑅
(17)
for all 𝑥, 𝑧 ∈ 𝐵𝑅 (𝑥0 ), for a constant 𝐶𝜅 depending only on 𝑛
and 𝛼.
(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 𝐶𝑠 depending only on 𝑠, 𝑛, and 𝑁 such that for
𝑥0 ∈ 𝐷, 𝜌 ≤ 1 − |𝑥0 | there holds
󵄨
󵄨
󵄩 󵄩
sup 󵄨󵄨󵄨󵄨𝑔 − 𝑔𝑥󸀠 0 ,𝜌 󵄨󵄨󵄨󵄨 ≤ 𝐶𝑠 𝜌1−(𝑛/𝑠) 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐻1,𝑠 (𝐵+ (𝑥0 ),𝑅𝑁 ) .
𝜌
+
𝐵𝜌 (𝑥0 )
(19)
Obviously, the inequality remains true if we replace
‖𝑔‖𝐻1,𝑠 (𝐵𝜌+ (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
∫
𝐵𝜌+ (𝑥0 )
Next we recall a slight modification of a characterization
of Hölder continuous functions originally due to Campanato
[21].
+
𝜌2 sup |𝐷ℎ|2 ≤ 𝐶0 𝜌2−𝑛 ∫
|𝐷ℎ|2 𝑑𝑥.
+ (𝑥 )
+
𝐵
𝜌 0
𝐵𝜌/2 (𝑥0 )
|𝐷𝑤| 𝑑𝑥 ≤ 1,
󵄨󵄨
󵄨󵄨
󵄨󵄨 2−𝑛
󵄨󵄨
󵄨󵄨𝜌 ∫
󵄨󵄨 ≤ 𝛿𝜌 sup 󵄨󵄨󵄨𝐷𝜑󵄨󵄨󵄨 ,
𝐴
(𝐷𝑤,
𝐷𝜑)
𝑑𝑥
󵄨󵄨
󵄨󵄨
󵄨 󵄨
+ (𝑥 )
𝐵
󵄨󵄨
󵄨󵄨
𝜌 0
𝐵𝜌+ (𝑥0 )
𝜇∈𝑅
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 (𝑅𝑛𝑁) satisfying (11), any A-harmonic function ℎ on
𝐵𝜌+ (𝑥0 ) with ℎ|𝐷𝜌 (𝑥0 ) ≡ 0 satisfies
2
∫
inf 𝜌−𝑛 {∫
We close this section by a standard estimate for the
solutions to homogeneous second-order elliptic systems with
constant coefficients [20].
󵄨󵄨
󵄨𝑚
󵄨󵄨 󵄨󵄨𝑚
󵄨󵄨𝑔 − 𝑔𝑥0 ,𝜌 󵄨󵄨󵄨 𝑑𝑥 ≤ 𝐶𝑝 𝜌𝑚 ∫
󵄨󵄨𝐷𝑔󵄨󵄨 𝑑𝑥,
󵄨
󵄨
𝐵+ (𝑥 )
𝜌
(20)
0
for a constant 𝐶𝑝 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 ∈ 𝑊1,𝑚 (Ω,
𝑅𝑁) ∩ 𝐿∞ (Ω, 𝑅𝑁) 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 𝐵𝜌+ (𝑥0 ) ⊂ 𝐵+ , with 𝑥0 ∈ 𝐷+ , 0 < 𝜌 < 𝑅 < 𝜌0 ,
there holds
󵄨2
󵄨󵄨
󵄨󵄨𝑢 (𝑥) − 𝑢𝑥󸀠 ,𝑅 󵄨󵄨󵄨
0 󵄨
󵄨
𝑑𝑥 + 𝐶2 𝛼𝑛 𝜌𝑛
∫
|𝐷𝑢| 𝑑𝑥 ≤ 𝐶1 ∫
+
𝜌2
𝐵𝜌/2
𝐵𝜌+ (𝑥0 )
(𝑥0 )
2
󵄩2
󵄩󵄩𝑔󵄩󵄩󵄩𝐻1,𝑠 ,
1−(2/𝑠) 󵄩
󵄩
+ 𝐶3 (𝛼𝑛 𝜌𝑛 )
(21)
where 𝐶1 depends only on 𝜆, 𝐿, and 𝑀 and 𝐶3 depends on these
quantities, and in addition to 𝐶𝑝 , 𝐶2 depends on 𝜆, 𝐿, 𝑀, 𝑎, 𝑏,
and ‖𝑔‖𝐿∞ (𝐵,𝑅𝑁 ) .
4
Abstract and Applied Analysis
+
Proof. Consider a cutoff function 𝜂 ∈ 𝐶0∞ (𝐵𝜌/2
(𝑥0 )), satisfy+
ing 0 ≤ 𝜂 ≤ 1, 𝜂 ≡ 0 on 𝐵𝜌/2 (𝑥0 ) and |∇𝜂| < 4/𝜌. Then the
+
(𝑥0 , 𝑅𝑁)) and thus can be
function (𝑢 − 𝑔)𝜂2 is in 𝑊01,𝑚 (𝐵𝜌/2
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)
∫
𝐵𝜌+ (𝑥0 )
𝛼𝛽
𝐴 𝑖𝑗
𝑗
𝑖
(1 + |𝑢|2 )
≤ 𝜀∫
(1 + |𝑢|2 )
𝐵𝜌+ (𝑥0 )
𝐵𝜌+ (𝑥0 )
𝐵𝜌+ (𝑥0 )
(1 + |𝑢|2 )
󸀠
+ 2𝐿 ∫
𝐵𝜌+ (𝑥0 )
+ 𝑎∫
𝐵𝜌+ (𝑥0 )
(1 + |𝑢|2 )
𝐵𝜌+ (𝑥0 )
𝐵𝜌+ (𝑥0 )
(1 + |𝑢|2 )
𝜀
+ 𝑏2 𝜂2 𝛼𝑛 𝜌𝑛+2
4
󵄨
󵄨
󵄨󵄨 󵄨󵄨
󵄨󵄨𝐷𝜂󵄨󵄨 |𝐷𝑢| 𝜂 󵄨󵄨󵄨𝑢 − 𝑔󵄨󵄨󵄨 𝑑𝑥
󵄨
󵄨
|𝐷𝑢|𝑚 󵄨󵄨󵄨𝑢 − 𝑔󵄨󵄨󵄨 𝜂2 𝑑𝑥 + 𝑏 ∫
(𝑚−2)/2
≤ 𝜀∫
󵄨󵄨 󵄨󵄨
2
󵄨󵄨𝐷𝑔󵄨󵄨 |𝐷𝑢| 𝜂 𝑑𝑥
(𝑚−2)/2
(𝑚−2)/2
+ (1 + 𝑀2 )
󵄨 2
󵄨󵄨
󵄨󵄨𝑢 − 𝑔󵄨󵄨󵄨 𝜂 𝑑𝑥
⋅∫
𝐵𝜌+ (𝑥0 )
󵄨
󵄨
+ 𝑎 sup 󵄨󵄨󵄨󵄨𝑔 − 𝑔𝑥󸀠 0 ,𝜌 󵄨󵄨󵄨󵄨 ∫
|𝐷𝑢|𝑚 𝜂2 𝑑𝑥
+ (𝑥 )
+
𝐵
𝜌 0
𝐵𝜌 (𝑥0 )
(22)
Using (H2), (19), and (20), we thus have
(𝜆 − 𝜀) ∫
𝐵𝜌+ (𝑥0 )
(𝑚−2)/2 󵄨
𝐿2
󵄨󵄨𝐷𝑔󵄨󵄨󵄨2 𝜂2 𝑑𝑥
(1 + |𝑢|2 )
∫
󵄨 󵄨
+
2𝜀 𝐵𝜌 (𝑥0 )
≤ (𝜆 − 𝜀) ∫
+
(𝑚−2)/2 󵄨
4𝐿2
󵄨󵄨𝐷𝜂󵄨󵄨󵄨2 󵄨󵄨󵄨󵄨𝑢 − 𝑢󸀠 󵄨󵄨󵄨󵄨2 𝑑𝑥
(1 + |𝑢|2 )
∫
𝑥0 ,𝜌 󵄨
󵄨 󵄨󵄨
𝜀 𝐵𝜌+ (𝑥0 )
≤
+
(𝑚−2)/2 󵄨
4𝐿2
󵄨󵄨𝐷𝜂󵄨󵄨󵄨2 󵄨󵄨󵄨󵄨𝑔 − 𝑔󸀠 󵄨󵄨󵄨󵄨2 𝑑𝑥
(1 + |𝑢|2 )
∫
𝑥0 ,𝜌 󵄨
󵄨 󵄨󵄨
𝜀 𝐵𝜌+ (𝑥0 )
1
󵄨󵄨
󵄨2
󵄨󵄨𝑔 − 𝑔𝑥󸀠 ,𝜌 󵄨󵄨󵄨 𝑑𝑥
∫
2
0
󵄨
󵄨
𝜀𝜌 𝐵𝜌+ (𝑥0 )
(𝑚−2)/2
(1 + |𝑢|2 )
󵄩 󵄩
+ 𝑎 (𝑀 + 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐿∞ (𝐵+ ,𝑅𝑁 ) ) ∫
𝐵𝜌+ (𝑥0 )
+
|𝐷𝑢|𝑚 𝜂2 𝑑𝑥
(𝑚−2)/2 1 󵄨󵄨
64𝐿2 + 1
󵄨2
󵄨𝑢 − 𝑢𝑥󸀠 ,𝜌 󵄨󵄨󵄨 𝑑𝑥
(1 + |𝑢|2 )
∫
2 󵄨󵄨
0 󵄨
+
𝜀
𝜌
𝐵𝜌 (𝑥0 )
𝜀
+ 𝑏2 𝜂2 𝛼𝑛 𝜌𝑛+2
4
+(
2
64𝐿 𝐶𝑝 4𝐶𝑝
𝐿
+
+
)
2𝜀
2𝜀
𝜀
2
(𝑥0 )
(1 + |𝑢|2 )
(𝑚−2)/2
|𝐷𝑢|2 𝜂2 𝑑𝑥
(𝑚−2)/2 1 󵄨󵄨
64𝐿2 + 1
󵄨2
󵄨𝑢 − 𝑢𝑥󸀠 󵄨󵄨󵄨 𝑑𝑥
(1 + |𝑢|2 )
∫
2 󵄨󵄨
0󵄨
+
𝜀
𝜌
𝐵𝜌 (𝑥0 )
󵄩 󵄩
+ 𝐶 (𝑎, 𝑀, 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐿∞ (𝐵+ ,𝑅𝑁 ) , ‖𝑢‖𝑊1,𝑚 (𝐵𝜌+ (𝑥0 )) , 𝑏) 𝛼𝑛 𝜌𝑛
+ (𝐿, 𝐶𝑝 , 𝑀) ∫
𝐵𝜌+ (𝑥0 )
≤
|𝐷𝑢|2 𝜂2 𝑑𝑥
|𝐷𝑢|2 𝜂2 𝑑𝑥
𝐵𝜌+
𝜀
1
󵄨󵄨
󵄨2
󵄨󵄨𝑢 − 𝑢𝑥󸀠 ,𝜌 󵄨󵄨󵄨 𝑑𝑥
+ 𝑏2 ∫
𝜌2 𝜂2 𝑑𝑥 + 2 ∫
0 󵄨
󵄨
+
+
2
𝜀𝜌 𝐵𝜌 (𝑥0 )
𝐵𝜌 (𝑥0 )
𝐵𝜌+ (𝑥0 )
2
𝐿2 64𝐿 𝐶𝑝 4𝐶𝑝
+
+
)
2𝜀
2𝜀
𝜀
󵄨󵄨 󵄨󵄨2 2
󵄨󵄨𝐷𝑔󵄨󵄨 𝜂 𝑑𝑥.
+
≤ 𝜀∫
(
|𝐷𝑢|2 𝜂2 𝑑𝑥
󵄨
󵄨
+ 𝑎 sup 󵄨󵄨󵄨󵄨𝑢 − 𝑢𝑥󸀠 0 ,𝜌 󵄨󵄨󵄨󵄨 ∫
|𝐷𝑢|𝑚 𝜂2 𝑑𝑥
+ (𝑥 )
+
𝐵
0
𝜌
𝐵𝜌 (𝑥0 )
+
2
(𝑚−2)/2 𝑢 − 𝑢𝑥0 ,𝜌
64𝐿2 + 1
+
(1 + |𝑢|2 )
(
) 𝑑𝑥
∫
𝜀
𝜌
𝐵𝜌+ (𝑥0 )
2
(𝑚−2)/2
󵄨󵄨𝐷𝑔󵄨󵄨󵄨2 𝜂2 𝑑𝑥
󵄨 󵄨
(𝑚−2)/2 󵄨
󵄩 󵄩
+ 𝑎 (𝑀 + 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐿∞ (𝐵+ ,𝑅𝑁 ) ) 𝐶 (‖𝑢‖𝑊1,𝑚 (𝐵𝜌+ (𝑥0 )) ) 𝛼𝑛 𝜌𝑛
(⋅, 𝑢) (𝐷𝛽 𝑢 , 𝐷𝛼 𝑢 ) 𝜂 𝑑𝑥
≤ 𝐿∫
󵄨󵄨𝐷𝑔󵄨󵄨󵄨2 𝜂2 𝑑𝑥
󵄨 󵄨
(𝑚−2)/2 󵄨
⋅∫
(23)
󵄨󵄨 󵄨󵄨2
󵄨󵄨𝐷𝑔󵄨󵄨 𝑑𝑥
(𝑚−2)/2
64𝐿2 + 1
1 󵄨󵄨
󵄨2
󵄨󵄨𝑢 − 𝑢𝑥󸀠 󵄨󵄨󵄨 𝑑𝑥
(1 + 𝑀2 )
∫
2
0
󵄨
󵄨
𝜀
𝐵𝜌+ (𝑥0 ) 𝜌
󵄩 󵄩
+ 𝐶 (𝑎, 𝑀, 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐿∞ (𝐵+ ,𝑅𝑁 ) , ‖𝑢‖𝑊1,𝑚 (𝐵𝜌+ (𝑥0 )) , 𝑏) 𝛼𝑛 𝜌𝑛
+ (𝐿, 𝐶𝑝 , 𝑀) (𝛼𝑛 𝜌𝑛 )
1−(2/𝑠)
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝑔󵄩󵄩𝐻1,𝑠 .
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.
Abstract and Applied Analysis
5
Lemma 9. Consider 𝑢 ∈ 𝑊1,𝑚 (Ω, 𝑅𝑁) ∩ 𝐿∞ (Ω, 𝑅𝑁) to be a
weak solution of (1), 𝑥0 ∈ 𝐷 and 𝑦 ∈ 𝐷𝑅 (𝑥0 ), 𝐷𝜌 (𝑦) ⊂⊂
+
𝐷𝑅 (𝑥0 ), for 𝑅 < 1 − |𝑥0 |, and 𝜑 ∈ 𝐶0∞ (𝐵𝜌/2
(𝑦), 𝑅𝑁) with
sup |𝐷𝜑| ≤ 1. We have
≤
𝛼𝑛 𝜌𝑛−1
{(𝑎−
∫
|𝐷𝑢|𝑚 𝑥 + 2−𝑛 𝑏) 𝜌2 }
2
𝐵𝜌+ (𝑦)
+ 𝛼𝑛 𝜌𝑛−1 𝜔 (𝜌𝑚 + 𝑀𝑚−2∫
−
𝐵𝜌+ (𝑦)
𝐵𝜌+ (𝑦)
𝜌
( )
2
2−𝑛
⋅ {𝐶1∫
−
𝛼𝛽
∫
+
𝐵𝜌/2
(𝑦)
+
𝐴 𝑖𝑗 (𝑦, 𝑢𝑦,𝜌
) (𝐷𝛽 𝑢𝑗 , 𝐷𝛼 𝜑𝑖 ) 𝑑𝑥
+ 𝐶2 𝜌2 }
≤
Here and hereafter, we define
󵄨󵄨
󸀠 󵄨󵄨2
󵄨󵄨𝑢 − 𝑢𝑦,𝜌
󵄨󵄨 𝑑𝑥 + 𝐶3 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2𝐻1,𝑠 𝜌2(1−(𝑛/𝑠))
󵄨
󵄨
1/2
(24)
󵄨 󵄨
≤ 𝐶4 √𝐼 (√𝐼 + 𝜔 (𝐼)) 𝜌 sup 󵄨󵄨󵄨𝐷𝜑󵄨󵄨󵄨 .
+
𝐵𝜌/2 (𝑥0 )
𝐵𝜌+ (𝑦)
󵄨󵄨
󸀠 󵄨󵄨2
󵄨󵄨𝑢 − 𝑢𝑦,𝜌
󵄨󵄨 𝑑𝑥)
󵄨
󵄨
𝛼𝑛 𝜌𝑛−1
𝛼 𝜌𝑛−1
𝐶5 𝐼 + 𝑛
𝐶6 𝜔 (𝐼) √𝐼
2
2
≤ 𝐶7 𝛼𝑛 𝜌𝑛−1 (𝐼 + 𝜔 (𝐼) √𝐼) ,
󵄨󵄨
󸀠 󵄨󵄨2
󵄨󵄨𝑢 − 𝑢𝑧,𝑟
󵄨󵄨 𝑑𝑥 + 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2𝐻1,𝑠 𝑟2(1−(𝑛/𝑠)) + 𝑟2 , (25)
𝐼 (𝑧, 𝑟) = ∫
−
󵄨
󵄨
+
𝐵 (𝑧)
𝑟
for 𝑧 ∈ 𝐷, 𝑟 ∈ (0, 1 − |𝑧|).
(27)
where 𝐶5 = 𝑎‖𝑢‖𝑊1,𝑚 + 𝑏, 𝐶6 = max{√𝐶1 , √𝐶2 , √𝐶3 }, and
𝐶7 = (1/2)(𝐶5 + 𝐶6 ), for 𝑧 ∈ 𝐷, 𝑟 ∈ (0, 1 − |𝑧|). We introduce
the notation
Proof. Using (8) we have
𝐼 (𝑧, 𝑟) = ∫
−
𝐵𝑟+ (𝑧)
󵄨󵄨
󸀠 󵄨󵄨2
󵄨󵄨𝑢 − 𝑢𝑧,𝑟
󵄨󵄨 𝑑𝑧 + 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2𝐻1,𝑠 𝑟2(1−(𝑛/𝑠)) + 𝑟2
󵄨
󵄨
(28)
𝛼𝛽
∫
+
𝐵𝜌/2
(𝑦)
󸀠
𝐴 𝑖𝑗 (𝑦, 𝑢𝑦,𝜌
) (𝐷𝛽 𝑢𝑗 , 𝐷𝛼 𝜑𝑖 ) 𝑑𝑥
≤ [𝑎 ∫
+
𝐵𝜌/2
󵄨 󵄨
|𝐷𝑢|𝑚 𝑑𝑥 + 2−𝑛−1 𝛼𝑛 𝑏𝜌𝑛 ] ⋅ 𝜌 sup 󵄨󵄨󵄨𝐷𝜑󵄨󵄨󵄨
+
(𝑦)
𝐵𝜌/2
(𝑦)
∫
󵄨
󵄨󵄨 𝛼𝛽
󵄨󵄨𝐴 (𝑦, 𝑢󸀠 ) − 𝐴𝛼𝛽 (𝑥, 𝑢)󵄨󵄨󵄨
+∫
𝑦,𝜌
𝑖𝑗
𝑖𝑗
󵄨󵄨
󵄨
+
󵄨
𝐵𝜌/2 (𝑦)
󵄨 󵄨
⋅ |𝐷𝑢| 𝑑𝑥 sup 󵄨󵄨󵄨𝐷𝜑󵄨󵄨󵄨 .
+
𝐵𝜌/2 (𝑦)
𝛼𝛽
󸀠
𝐴 𝑖𝑗 (𝑦, 𝑢𝑦,𝜌
) (𝐷𝛽 𝑢𝑗 , 𝐷𝛼 𝜑𝑖 ) 𝑑𝑥
≤ [𝑎 ∫
𝑚
+
𝐵𝜌/2
(𝑦)
−𝑛−1
|𝐷𝑢| 𝑑𝑥 + 2
𝛼𝑛 𝑏𝜌 ] ⋅ 𝜌
1/2
⋅ [∫
+
𝐵𝜌/2
(𝑦)
(29)
𝑛−1 √
𝐼 (√𝐼 + 𝜔 (𝐼)) .
Multiplying (29) through by (𝜌/2)2−𝑛 yields
󵄨󵄨
󵄨󵄨
󵄨󵄨 𝜌 2−𝑛
󵄨󵄨
𝛼𝛽
󸀠
𝑗
𝑖
󵄨󵄨( ) ∫
󵄨󵄨
𝐴
(𝑦,
𝑢
)
(𝐷
𝑢
,
𝐷
𝜑
)
𝑑𝑥
𝛽
𝛼
𝑦,𝜌
𝑖𝑗
󵄨󵄨 2
󵄨󵄨
+
𝐵𝜌/2
(𝑦)
󵄨󵄨
󵄨󵄨
󵄨 󵄨
≤ 𝐶4 √𝐼 (√𝐼 + 𝜔 (𝐼)) 𝜌 sup 󵄨󵄨󵄨𝐷𝜑󵄨󵄨󵄨 ,
+
𝐵𝜌/2 (𝑥0 )
(30)
for 𝐶4 defined by 𝐶4 = 2𝑛−3 𝛼𝑛 𝐶7 .
𝑛
󵄨1/2
󵄨󵄨 𝛼𝛽
󵄨󵄨𝐴 (𝑦, 𝑢󸀠 ) − 𝐴𝛼𝛽 (𝑥, 𝑢)󵄨󵄨󵄨 𝑑𝑥]
+ [∫
𝑦,𝜌
𝑖𝑗
𝑖𝑗
󵄨󵄨
󵄨
+
󵄨
𝐵𝜌/2 (𝑦)
2
󸀠
𝐴 𝑖𝑗 (𝑦, 𝑢𝑦,𝜌
) (𝐷𝛽 𝑢𝑗 , 𝐷𝛼 𝜑𝑖 ) 𝑑𝑥
≤ 𝐶7 𝛼𝑛 𝜌
Applying in turn Young’s inequality, (H3), the Caccioppoli inequality (Theorem 8), and Jensen’s inequality, we
calculate from (26)
+
𝐵𝜌/2
(𝑦)
𝛼𝛽
+
𝐵𝜌/2
(𝑦)
(26)
∫
and further write 𝐼 for 𝐼(𝑦, 𝜌). For arbitrary 𝜑 ∈ 𝐶0∞ (Ω, 𝑅𝑁)
we thus have, by rescalling,
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) ≤ 𝐶8−1 min {𝛿2 /4, 𝑠0 } together, imply the
growth condition
1/2
|𝐷𝑢| 𝑑𝑥]
𝐼 (𝑦, 𝜃𝜌) ≤ 𝜃2𝜎 𝐼 (𝑦, 𝜌) .
(31)
6
Abstract and Applied Analysis
Proof. We now set 𝑤 = 𝑢 − 𝑔, using in turn (H1), Young’s
inequality, and Hölder’s inequality. We have from (30)
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨 𝜌 2−𝑛
𝛼𝛽
󸀠
𝑗
𝑖
󵄨󵄨
󵄨󵄨( ) ∫
𝐴
(𝑦,
𝑢
)
(𝐷
𝑤
,
𝐷
𝜑
)
𝑑𝑥
𝛽
𝛼
𝑦,𝜌
𝑖𝑗
󵄨󵄨
󵄨󵄨 2
+
𝐵
(𝑦)
󵄨󵄨
󵄨󵄨
𝜌/2
󵄨󵄨
󵄨󵄨
󵄨󵄨 𝜌 2−𝑛
󵄨󵄨
𝛼𝛽
󸀠
𝑗
𝑖
󵄨
≤ 󵄨󵄨( ) ∫
𝐴 𝑖𝑗 (𝑦, 𝑢𝑦,𝜌 ) (𝐷𝛽 𝑢 , 𝐷𝛼 𝜑 ) 𝑑𝑥󵄨󵄨󵄨
+
󵄨󵄨 2
󵄨󵄨
𝐵𝜌/2 (𝑦)
󵄨
󵄨
󵄨󵄨󵄨 𝜌 2−𝑛
󵄨󵄨󵄨
𝛼𝛽
󵄨
󵄨
󸀠
+ 󵄨󵄨󵄨( ) ∫
𝐴 𝑖𝑗 (𝑦, 𝑢𝑦,𝜌
) (𝐷𝛽 𝑔𝑗 , 𝐷𝛼 𝜑𝑖 ) 𝑑𝑥󵄨󵄨󵄨
+
󵄨󵄨 2
󵄨󵄨
𝐵
(𝑦)
𝜌/2
󵄨
󵄨
󵄨
󵄨
≤ 𝐶9 √𝐼 [√𝐼 + 𝜔 (𝐼)] 𝜌 sup 󵄨󵄨󵄨𝐷𝜑󵄨󵄨󵄨 ,
+
𝐵𝜌/2
(𝑥0 )
(32)
for 𝐶9 = max {𝐶4 , (𝛼𝑛 /2)1−(𝑛/𝑠) }.
We now set V = 𝑤/𝛾, for 𝛾 = 𝐶9 √𝐼. From (32) we then
have
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨 𝜌 2−𝑛
𝛼𝛽
󸀠
𝑗
𝑖
󵄨󵄨
󵄨󵄨( ) ∫
𝐴
(𝑦,
𝑢
)
(𝐷
V
,
𝐷
𝜑
)
𝑑𝑥
𝛽
𝛼
𝑦,𝜌
𝑖𝑗
󵄨󵄨
󵄨󵄨 2
+
𝐵𝜌/2
(𝑦)
󵄨󵄨
󵄨󵄨
(33)
󵄨 󵄨
≤ (√𝐼 + 𝜔 (𝐼)) 𝜌 sup 󵄨󵄨󵄨𝐷𝜑󵄨󵄨󵄨 ,
+
𝐵𝜌/2
(𝑥0 )
and from (32) we observe from the definition of 𝐶9 (recalling
also the definition of 𝛾)
𝜌 2−𝑛
( ) ∫
|𝐷V|2 𝑑𝑥 < 1.
+
2
𝐵𝜌/2
(𝑦)
(34)
Using (38) and (39) we observe
(𝜃𝜌)
−𝑛
∫
+
𝐵𝜃𝜌
(𝑦)
−𝑛
≤ 2(𝜃𝜌)
|V|2 𝑑𝑥
[∫
+
𝐵𝜃𝜌
−𝑛
≤ 2(𝜃𝜌)
V|𝐷𝜌 (𝑦)
(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
𝛼𝛽
+
󸀠
(𝑦), 𝑅𝑁) which is 𝐴 𝑖𝑗 (𝑦, 𝑢𝑦,𝜌
)-harmonic, with
ℎ ∈ 𝐻1,2 (𝐵𝜌/2
ℎ|𝐷𝜌/2 (𝑦) ≡ 0 such that
𝜌 2−𝑛
( ) ∫
|𝐷ℎ|2 𝑑𝑥 ≤ 1,
+
2
𝐵𝜌/2 (𝑦)
(37)
𝜌 −𝑛
( ) ∫
|V − ℎ|2 𝑑𝑥 ≤ 𝜀.
+
2
𝐵𝜌/2
(𝑦)
(38)
sup |ℎ|2 ≤ 𝜃2 𝜌2 sup |𝐷ℎ|2 ≤ 4𝐶0 𝜃2 .
+
𝐵𝜌/4
(𝑦)
(𝑦)
|ℎ|2 𝑑𝑥]
(40)
≤ 21−𝑛 𝜃−𝑛 𝜀 + 4𝛼𝑛 𝐶0 𝜃2 ,
and, hence, on multiplying this through by 𝛾2 , we obtain the
estimate
(𝜃𝜌)
−𝑛
∫
+
𝐵𝜃𝜌
(𝑦)
|𝑤|2 𝑑𝑥 ≤ 𝐶92 (21−𝑛 𝜃−𝑛 𝜀 + 4𝛼𝑛 𝐶0 𝜃2 ) 𝐼.
(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
󸀠
󸀠
= 𝑔𝑦,𝜃𝜌
, thus from (41) we get
noting also that 𝑢𝑦,𝜃𝜌
−𝑛
(𝜃𝜌)
∫
+
𝐵𝜃𝜌
󵄨󵄨
󵄨2
󵄨󵄨𝑢 − u󸀠𝑦,𝜃𝜌 󵄨󵄨󵄨 𝑑𝑥
󵄨
󵄨
(𝑦)
≤ 2(𝜃𝜌)
−𝑛
[∫
+
𝐵𝜃𝜌
(𝑦)
2
󵄨󵄨
󸀠 󵄨󵄨
󵄨󵄨𝑔 − 𝑔𝑦,𝜃𝜌
󵄨󵄨 𝑑𝑥]
󵄨
󵄨
𝑦
𝜃𝜌 ( )
󵄨2
󵄨󵄨
󵄨󵄨𝑢 − 𝑔󵄨󵄨󵄨 𝑑𝑥 + ∫
𝐵+
≤ 2𝐶92 (21−𝑛 𝜃−𝑛 𝜀 + 4𝛼𝑛 𝐶0 𝜃2 ) 𝐼
+ 2𝐶𝑝 (𝜃𝜌)
2−𝑛
1−(2/𝑠)
1
𝑛
󵄩󵄩 󵄩󵄩2
[ 𝛼𝑛 (𝜃𝜌) ]
󵄩󵄩𝑔󵄩󵄩𝐻1,𝑠
2
≤ 𝐶10 (𝜃−𝑛 𝜀 + 𝜃2 ) 𝐼 + 𝐶10 𝜃2(1−(𝑛/𝑠)) 𝐼,
(42)
for 𝐶10 = max {8𝛼𝑛 𝐶0 𝐶92 , 22/𝑠 𝐶𝑝 𝛼𝑛1−(2/𝑠) }, and provided 𝜀 =
𝜃𝑛+2 , we have
(𝜃𝜌)
−𝑛
∫
+
𝐵𝜃𝜌
(𝑦)
󵄨󵄨
󵄨󵄨2
󸀠
󵄨󵄨𝑢 − 𝑢𝑦,𝜃𝜌
󵄨󵄨 𝑑𝑥 ≤ 3𝐶10 𝜃2(1−(𝑛/𝑠)) 𝐼.
󵄨
󵄨
(43)
Note that fix 𝜀 = 𝜃𝑛+2 , which is also fixed 𝛿. Since 𝜌 ≤ 1,
we see from the definition of 𝐼
2(1−(𝑛/𝑠))
󵄩󵄩 󵄩󵄩2
≤ 𝜃2(1−(𝑛/𝑠)) 𝐼,
󵄩󵄩𝑔󵄩󵄩𝐻1,𝑠 (𝜃𝜌)
(44)
and further
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,
+
𝐵𝜃𝜌
(𝑦)
+
𝐵𝜃𝜌
𝑛
[( 𝜌 ) 𝜀 + 1 𝛼𝑛 (𝜃𝜌)𝑛 sup |ℎ|2 ]
2
2
B+𝜃𝜌 (𝑦)
[
]
Further we note
1
1
= 𝑤|𝐷𝜌 (𝑦) = (𝑢 − 𝑔) |𝐷𝜌 (𝑦) ≡ 0.
𝛾
𝛾
(𝑦)
|V − ℎ|2 𝑑𝑥 + ∫
(39)
2
(𝜃𝜌) ≤ 𝜃2 𝐼.
(45)
Combining these estimates with (43), we can get
𝐼 (𝑦, 𝜃𝜌) ≤ 3 (𝐶10 + 1) 𝜃2(1−(𝑛/𝑠)) 𝐼.
(46)
Choose 𝜃 ∈ (0, 1/4] sufficiently small that there holds:
3(𝐶10 + 1)𝜃2(1−(𝑛/𝑠)) ≤ 𝜃2𝜎 .
Abstract and Applied Analysis
7
+
(𝑦) as well, yielding
meaning that we can apply (46) on 𝐵𝜃𝜌/2
We can see from (46)
𝐼 (𝑦, 𝜃𝜌) ≤ 𝜃2𝜎 𝐼.
𝐼 (𝑦,
𝑅
𝜃2 𝑅
) ≤ 𝜃4𝜎 𝐼 (𝑦, ) ,
2
2
(54)
𝐼 (𝑦,
𝜃𝑘 𝑅
𝑅
) ≤ 𝜃2𝑘𝜎 𝐼 (𝑦, ) .
2
2
(55)
(47)
We now choose 𝑠0 > 0 such that 0 < 𝜔(𝑠0 ) < (𝛿/2) and
define 𝐶8 by
𝐶8 = max {2𝑛−1 , 2𝐶92 + 1, 2𝐶𝑠2 + 1} .
and inductively
(48)
Suppose that we have
𝐼 (𝑥0 , 𝑅) ≤
𝐶8−1
𝛿2
min { , 𝑠0 } ,
4
(49)
for some 𝑅 ∈ (0, 𝑅0 ], where 𝑅0 = min{√2𝑠0 , 1 − |𝑥0 |}.
For any 𝑦 ∈ 𝐷𝑅/2 (𝑥0 ) we use the Sobolev inequality to
calculate
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 𝜃𝑘+1 𝑅/2 < 𝜌 ≤ 𝜃𝑘 𝑅/2. Firstly we use the
Sobolev inequality, to see
∫
𝐵𝜌+ (𝑦)
𝛼𝑛 𝑅𝑛 󵄨󵄨 󸀠
󵄨󵄨2
󸀠
󵄨󵄨𝑢
󵄨󵄨
− 𝑢𝑦,𝑅/2
󵄨
2𝑛+1 󵄨 𝑥0 ,𝑅
=∫
+
𝐵𝑅/2
≤ 2∫
󵄨󵄨 󸀠
󵄨󵄨2
󸀠
󵄨󵄨𝑢𝑥 ,𝑅 − 𝑢𝑦,𝑅/2
󵄨󵄨 𝑑𝑥 = ∫
󵄨 0
󵄨
𝐵+
𝑅/2
+
𝐵𝑅/2
󵄨󵄨 󸀠
󵄨󵄨2
󸀠
󵄨󵄨𝑔𝑥 ,𝑅 − 𝑔𝑦,𝑅/2
󵄨󵄨 𝑑𝑥
󵄨 0
󵄨
󵄨󵄨
󵄨2
󵄨
󵄨󵄨2
󸀠
󵄨󵄨𝑔 − 𝑔𝑥󸀠 ,𝑅 󵄨󵄨󵄨 𝑑𝑥 + 2 ∫ 󵄨󵄨󵄨𝑔 − 𝑔𝑦,𝑅/2
󵄨󵄨 𝑑𝑥
0 󵄨
󵄨
󵄨
󵄨
𝐵+
1 𝑛 󵄩 󵄩2
1 2(1−(𝑛/𝑠))
≤ 2𝛼𝑛 ( 𝑘 ) 𝐶𝑠2 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐻1,𝑠 ( 𝑘 )
,
2𝜃 𝑅
2𝜃 𝑅
which allows us to deduce
󵄨󵄨
󸀠 󵄨󵄨2
󵄨󵄨𝑢 − 𝑢𝑦,𝜌
󵄨󵄨 𝑑𝑥
∫
󵄨
󵄨
+
𝐵 (𝑦)
≤ 2∫
𝐵𝜌+ (𝑦)
(50)
+ 4𝛼𝑛 (
Then we can calculate
1
𝐼 (𝑦, 𝑅)
2
−
≤ 2𝑛−1∫
󵄨󵄨
󵄨2
󵄨󵄨𝑢 − 𝑢𝑦,𝜃𝑘 𝑅/2󸀠 󵄨󵄨󵄨 𝑑𝑥
󵄨
󵄨
(57)
1 𝑛 2 󵄩󵄩 󵄩󵄩2
1 2(1−(𝑛/𝑠))
𝑔
𝐶
(
,
)
)
󵄩
󵄩
1,𝑠
𝑠 󵄩 󵄩𝐻
2𝜃𝑘 𝑅
2𝜃𝑘 𝑅
and, hence,
+
𝐵𝑅/2
󵄨󵄨
󵄨2
󵄨󵄨𝑢 − 𝑢𝑥󸀠 ,𝑅 󵄨󵄨󵄨 𝑑𝑥
0 󵄨
󵄨
(𝑦)
𝐼 (𝑦, 𝜌) ≤ 𝐶11 𝐼 (𝑦,
(51)
1
󵄩 󵄩2
+ (2𝐶𝑠2 + 1) 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐻1,𝑠 𝑅2(1−(𝑛/𝑠)) + 𝑅2
4
(52)
and more particularly
inf𝑁 ∫
1
≤ 𝛿 + 𝜔 (𝑠0 ) ≤ 𝛿,
2
𝜇∈𝑅
which means that the condition (49) is sufficient to guarantee
the smallness condition (37) for 𝜌 = 𝑅/2, for all 𝑦 ∈ 𝐷𝑅/2 (𝑥0 ).
We can thus conclude that (46) holds in this situation. From
(46) we thus have
1
1
≤ √ 𝐼 (y, 𝑅) + √ 𝜔 (𝐼 (𝑦, 𝑅)) ≤ 𝛿,
2
2
2𝜌 2𝜎
𝑅
) ≤ 𝐶8 𝐶11 𝜃−2𝜎 ( ) 𝐼 (𝑥0 , 𝑅) (59)
2
𝑅
𝜌 2𝜎
2
≤ 𝐶8 𝐶11 ( ) 𝐼 (𝑥0 , 𝑅) ( ) ,
𝜃
𝑅
1
1
√ 𝐼 (𝑦, 𝑅) + 𝜔 (𝐼 (𝑦, 𝑅))
2
2
𝜃𝜌
𝜃𝜌
) + √ 𝜔 (𝐼 (𝑦, ))
2
2
(58)
for 𝐶11 = 8𝜃−𝑛 𝐶𝑠2 + 1. Combining this with (55) and (51), we
have
≤ 𝐶11 𝜃2𝑘𝜎 𝐼 (𝑦,
Then we have
≤ √𝐶8 𝐼 (𝑥0 , 𝑅) + √𝜔 (𝐶8 𝐼 (𝑥0 , 𝑅))
𝜃𝑘 𝑅
),
2
𝐼 (𝑦, 𝜌)
≤ 𝐶8 𝐼 (𝑥0 , 𝑅) .
√ 𝐼 (𝑦,
(56)
𝜌
𝑅/2
󵄩 󵄩2
≤ 2𝛼𝑛 𝐶𝑠2 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐻1,𝑠 𝑅𝑛+2(1−(𝑛/𝑠)) .
󵄨󵄨 󸀠
󵄨󵄨2
󸀠
󵄨󵄨𝑢𝑦,𝜌 − 𝑢𝑦,𝜃
𝑘 𝑅/2 󵄨
󵄨󵄨 𝑑𝑥
󵄨
(53)
𝐵𝜌+ (𝑦)
𝜌 2𝜎
󵄨2
󵄨󵄨
󵄨󵄨𝑢 − 𝜇󵄨󵄨󵄨 𝑑𝑥 ≤ 𝐶12 𝐼 (𝑥0 , 𝑅) ( ) ,
𝑅
(60)
for 𝐶12 = 𝐶8 𝐶11 (2/𝜃)2𝜎 . Recall that this estimate is valid
for all 𝑦 ∈ 𝐷 and 𝜌 with 𝐷𝜌 (𝑦) ⊂ 𝐷𝑅/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.
8
Abstract and Applied Analysis
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).
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