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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

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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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M1">View MathML</a>

(1.1)

where Ω is a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M4">View MathML</a>, and u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M5">View MathML</a> take values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M6">View MathML</a>. Here each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M7">View MathML</a> maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M8">View MathML</a> into R, and each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M5">View MathML</a> maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M10">View MathML</a> into R. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M11">View MathML</a>, we have the following.

(H1) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M12">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M13">View MathML</a>

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M14">View MathML</a> is uniformly strongly elliptic, that is, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M15">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M16">View MathML</a>

(H3) There exists a monotone nondecreasing concave function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M17">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M18">View MathML</a>, continuous at 0, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M19">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M21">View MathML</a>.

(H4) The <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M5">View MathML</a> fulfill the following controllable growth condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M23">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M24">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M25">View MathML</a>, or any exponent if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M26">View MathML</a>; for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M28">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M29">View MathML</a>.

(H5) There exist s with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M30">View MathML</a> and a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M31">View MathML</a>, such that we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M32">View MathML</a>

Note that we trivially have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M33">View MathML</a>. Further, by Sobolev’s embedding theorem we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M34">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M35">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M36">View MathML</a>, we will take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M37">View MathML</a> on Ω.

If the domain we consider is an upper half unit ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M38">View MathML</a>, the boundary condition is the following.

(H5)′ There exist s with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M30">View MathML</a> and a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M40">View MathML</a>, such that we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M41">View MathML</a>

Here we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M42">View MathML</a>, and further <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M44">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M45">View MathML</a> we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M46">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M47">View MathML</a>, and we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M49">View MathML</a>. We further write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M50">View MathML</a>, and we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M52">View MathML</a>. For bounded <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M53">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M54">View MathML</a> we denote the average of a given function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M55">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M56">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M57">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M59">View MathML</a> we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M60">View MathML</a>. In particular, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M62">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M63">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M64">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M65">View MathML</a>

(1.2)

holds for all test-functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M66">View MathML</a> and, by approximation, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M67">View MathML</a>, where we have introduced the notation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M68">View MathML</a>

(1.3)

In the current situation, Sobolev’s embedding theorem yields the existence of a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M69">View MathML</a> depending only on s, n, and N such that we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M70">View MathML</a>

(1.4)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M72">View MathML</a>. Obviously, the inequality remains true if we replace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M73">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M74">View MathML</a>, which we will henceforth abbreviate simply as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M75">View MathML</a>.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M76">View MathML</a>

(1.5)

for a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M77">View MathML</a> depending only on n.

Finally, we fix an exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M78">View MathML</a> as follows: if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M79">View MathML</a>, σ can be chosen arbitrary (but henceforth fixed); otherwise we take σ fixed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M80">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M81">View MathML</a>.

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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M6">View MathML</a>, with boundary of class<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M83">View MathML</a>. Letube a weak solution of (1.1) satisfying the structural conditions (H1)-(H5). Consider a fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M84">View MathML</a>. Then there exist positive<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M85">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M86">View MathML</a> (depending only onn, N, λ, L, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M87">View MathML</a>andγ) with the property that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M88">View MathML</a>

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M90">View MathML</a>, which implies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M91">View MathML</a>.

Note in particular that the boundary condition (H5) means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M92">View MathML</a> makes sense: in fact, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M93">View MathML</a>.

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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M94">View MathML</a>-dimensional Hausdorff measure zero in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M95">View MathML</a>.

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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M38">View MathML</a>which satisfies the structural conditions (H1)-(H4) and (H5)′. Then there exist positive<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M97">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M86">View MathML</a> (depending only onn, N, λ, L, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M87">View MathML</a>andγ) with the property that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M100">View MathML</a>

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M90">View MathML</a>, which implies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M103">View MathML</a>.

Analogously to above, the boundary condition (H5)′ ensures that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M92">View MathML</a> exists, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M93">View MathML</a>.

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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M106">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M3">View MathML</a>. Then for any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M108">View MathML</a>there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M109">View MathML</a>with the following property: for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M110">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M111">View MathML</a>

(2.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M112">View MathML</a>

(2.2)

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M113">View MathML</a> (for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M115">View MathML</a>) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M116">View MathML</a>

(2.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M117">View MathML</a>

(2.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M118">View MathML</a>

(2.5)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M119">View MathML</a>, there exists anA-harmonic function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M120">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M121">View MathML</a>

(2.6)

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

Lemma 2.2Consider<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M3">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M45">View MathML</a>. Suppose that there are positive constantsκandα, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M125">View MathML</a>such that, for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M126">View MathML</a>, we have the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M127">View MathML</a>

(2.7)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M128">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M129">View MathML</a>; and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M130">View MathML</a>

(2.8)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M131">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M132">View MathML</a>.

Then there exists a Hölder continuous representative of the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M133">View MathML</a>-class ofνon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M134">View MathML</a>, and for this representative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M135">View MathML</a>we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M136">View MathML</a>

(2.9)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M137">View MathML</a>, with the constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M138">View MathML</a>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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M139">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M3">View MathML</a>. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M141">View MathML</a>depending only onn, N, λ, andL (without loss of generality we take<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M142">View MathML</a>) such that for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M110">View MathML</a>satisfying (2.1) and (2.2), anyA-harmonic functionhon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M144">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M145">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M146">View MathML</a>

3 Caccioppoli inequality

In this section we prove Caccioppoli’s inequality.

Theorem 3.1 (Caccioppoli inequality)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M64">View MathML</a>be a weak solution of systems (1.1) under the conditions (H1)-(H5). Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M148">View MathML</a>depending only onL, s, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M149">View MathML</a>, such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M150">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M152">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M153">View MathML</a>

(3.1)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M156">View MathML</a>depend only onλ, L, m, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M157">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M155">View MathML</a>additionally on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M69">View MathML</a>, and also ons.

Proof Now we consider a cut off function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M161">View MathML</a>, satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M163">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M46">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M165">View MathML</a>. Then the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M166">View MathML</a> is in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M167">View MathML</a>, 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M168">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M93">View MathML</a>, yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M170">View MathML</a>

(3.3)

Using Young’s inequality again,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M171">View MathML</a>

here we have used the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M172">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M173">View MathML</a>.

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M174">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M175">View MathML</a>

Combining these estimates in (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M176">View MathML</a>

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M177">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M178">View MathML</a>

(3.4)

Using (H2), we thus have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M179">View MathML</a>

Recalling that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M180">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M166">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M182">View MathML</a>

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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M183">View MathML</a>to be a weak solution of (1.1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M187">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M188">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M189">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M190">View MathML</a>

(4.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M191">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M193">View MathML</a>.

Proof From the definition of weak solution we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M194">View MathML</a>

(4.2)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M195">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M196">View MathML</a>

(4.3)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M198">View MathML</a>, we introduce the notation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M199">View MathML</a>

and further write I for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M200">View MathML</a>. Defining the constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M201">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M202">View MathML</a>, from (4.3) and Theorem 3.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M203">View MathML</a>

For arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M204">View MathML</a> we thus have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M205">View MathML</a>

Multiplying through by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M206">View MathML</a>, this yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M207">View MathML</a>

(4.4)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M208">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M209">View MathML</a>. □

Lemma 4.2Considerusatisfying the conditions of Theorem 1.1 andσfixed, then we can findδand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M210">View MathML</a>together with positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M211">View MathML</a>such that the smallness conditions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M212">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M213">View MathML</a>together imply the growth condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M214">View MathML</a>

Proof Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M215">View MathML</a>, using in turn (H1), Young’s inequality and Hölder’s inequality, from (4.4) we can see that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M216">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M217">View MathML</a>

(4.5)

We now set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M218">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M219">View MathML</a>. From (4.5) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M220">View MathML</a>

(4.6)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M221">View MathML</a>

(4.7)

Further we note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M222">View MathML</a>

(4.8)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M108">View MathML</a> we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M224">View MathML</a> to be the corresponding δ from the A-harmonic approximation lemma. Suppose that we could ensure that the smallness condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M225">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M226">View MathML</a>, which is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M227">View MathML</a>-harmonic, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M228">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M229">View MathML</a>

(4.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M230">View MathML</a>

(4.11)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M231">View MathML</a> arbitrary (to be fixed later), from Lemma 2.3 and (4.10), recalling also that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M232">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M233">View MathML</a>

(4.12)

Using (4.11) and (4.12) we observe

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M234">View MathML</a>

and hence, on multiplying this through by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M235">View MathML</a>, we obtain the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M236">View MathML</a>

(4.13)

For the time being, we restrict ourselves to the case that g does not vanish identically. Recalling that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M215">View MathML</a>, (4.13) yields, using in turn Poincaré’s, Sobolev’s and then Hölder’s inequalities, noting also that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M238">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M239">View MathML</a>

(4.14)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M240">View MathML</a> and provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M241">View MathML</a> together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M242">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M243">View MathML</a>

(4.15)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M244">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M245">View MathML</a>) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M246">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M247">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M248">View MathML</a>. Therefore we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M249">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M250">View MathML</a>.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M251">View MathML</a>

(4.16)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M252">View MathML</a>.

We then fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M241">View MathML</a>, note that this also fixes δ. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M254">View MathML</a>, we see from the definition of γ: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M255">View MathML</a>, and further <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M256">View MathML</a>.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M257">View MathML</a>

(4.17)

We choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M231">View MathML</a> small enough, such that we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M259">View MathML</a>.

Thus from (4.17) we can see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M260">View MathML</a>

(4.18)

We now choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M261">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M262">View MathML</a>, and we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M263">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M264">View MathML</a>. Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M265">View MathML</a>

(4.19)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M101">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M267">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M268">View MathML</a> we use Sobolev’s inequality to calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M269">View MathML</a>

(4.20)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M270">View MathML</a>

(4.21)

Using these estimations, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M271">View MathML</a>

(4.22)

Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M272">View MathML</a>

which means that the condition (4.18) is sufficient to guarantee the smallness condition (4.9) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M273">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M274">View MathML</a>. Thus, we can conclude that (4.17) holds in this situation. From (4.18) we thus have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M275">View MathML</a>

which means that we can apply (4.18) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M276">View MathML</a> as well, and this yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M277">View MathML</a>

and inductively we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M278">View MathML</a>

(4.23)

The next step is to go from a discrete to a continuous version of the decay estimate. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M279">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M280">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M281">View MathML</a>. Then we calculate in a similar manner to above. Firstly, we use Sobolev’s inequality (1.4) to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M282">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M283">View MathML</a>

which allows us to deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M284">View MathML</a>

and hence, finally

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M285">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M286">View MathML</a>. We combine these estimates with (4.22) and (4.23):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M287">View MathML</a>

and more particularly

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M288">View MathML</a>

(4.24)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M289">View MathML</a> given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M290">View MathML</a>. We recall that this estimate is valid for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M291">View MathML</a> and ρ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M292">View MathML</a>, and we assume only the smallness condition (4.19) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/88/mathml/M293">View MathML</a>. 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).

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