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Optimal partial regularity of second-order parabolic systems under natural 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 2013, 2013:152  doi:10.1186/1687-2770-2013-152

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/152


Received:17 August 2012
Accepted:10 May 2013
Published:26 June 2013

© 2013 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 cited.

Abstract

We consider the regularity for weak solutions of second-order nonlinear parabolic systems under a natural growth condition when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M1">View MathML</a>, and obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particular, we get the optimal regularity by the method of A-caloric approximation introduced by Duzaar and Mingione.

Keywords:
nonlinear parabolic systems; natural growth condition; A-caloric approximation; optimal partial regularity

1 Introduction

Electrorheological fluids are special viscous liquids, that are characterized by their ability to undergo significant changes in their mechanical properties when an electric field is applied. This property can be exploited in technological applications, e.g., actuators, clutches, shock absorbers, and rehabilitation equipment to name a few [1].

A model was developed for these liquids within the framework of rational mechanics [2,3]; it takes into account the complex interactions between the electro-magnetic fields and the moving liquid. If the fluid is assumed to be incompressible, it turns out that the relevant equations of the model are the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M2">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M3">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M4">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M5">View MathML</a>

(1.4)

where E is the electric field, P is the polarization, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M6">View MathML</a> is the density, v is the velocity, S is the extra stress, ϕ is the pressure, and f is the mechanical force. In fact, in a model capable of explaining many of the observed phenomena, the extra stress has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M7">View MathML</a>

(1.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M8">View MathML</a> are material constants, and where the material function p depends on the strength of the electric field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M9">View MathML</a> and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M10">View MathML</a>

(1.6)

Since the material function p, which essentially determines S, depends on the magnitude of the electric field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M9">View MathML</a>, we have to deal with an elliptic or parabolic system of partial differential equations with the so-called non-standard growth conditions, i.e., the elliptic operator S satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M12">View MathML</a>

(1.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M13">View MathML</a>

(1.8)

Equality (1.5) of electrorheological fluids with the conditions (1.7) and (1.8) encouraged us to considered the partial regularity of a more simple and standard model as the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M14">View MathML</a>

(1.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M15">View MathML</a> is a bounded domain and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M17">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M19">View MathML</a>, denote a point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M20">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M21">View MathML</a> be a vector-valued function defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M22">View MathML</a>. Denote by Du the gradient of u, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M23">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M1">View MathML</a> is a real number.

In order to define the weak solution of (1.9), one needs to impose some regularity conditions and constructer conditions to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M26">View MathML</a>. For a vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M27">View MathML</a>, we shall denote the coefficients by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M28">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M31">View MathML</a>. We assume that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M32">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M33">View MathML</a> are continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M34">View MathML</a> and that the following growth and ellipticity conditions are satisfied:

(H1) There exists a constant L such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M35">View MathML</a>

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M36">View MathML</a> are differentiable functions in p and there exists a constant L such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M37">View MathML</a>

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M25">View MathML</a> is uniformly strongly elliptic, that is, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M39">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M40">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M41">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M42">View MathML</a>. Now we shall specify the regularity assumptions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M36">View MathML</a> with respect to the ‘coefficient’ <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M44">View MathML</a> and assume that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M45">View MathML</a> is Hölder continuous with respect to the parabolic metric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M46">View MathML</a> with Hölder exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M47">View MathML</a> but not necessarily uniformly Hölder continuous; namely we shall assume that:

(H4) There exists a constant L such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M48">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M50">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M22">View MathML</a>. u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M52">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M53">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M31">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M56">View MathML</a> is a given non-decreasing function. Note that θ is concave in the argument. This is the standard way to prescribe (non-uniform) Hölder continuity of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M36">View MathML</a>. We find it a bit difficult to handle, therefore, in many points of the paper, we shall use:

(H4′) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M59">View MathML</a> monotone nondecreasing such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M60">View MathML</a>

valid for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M50">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M22">View MathML</a>, u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M52">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M31">View MathML</a>.

(H5) There exist constants a and b such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M67">View MathML</a>

Finally, we remark a trial consequence of the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M68">View MathML</a>; this implies the existence of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M69">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M70">View MathML</a> for all t such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M71">View MathML</a> is nondecreasing for fixed s, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M72">View MathML</a> is concave and nondecreasing for fixed t, and such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M73">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M50">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M22">View MathML</a>, any u, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M52">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M79">View MathML</a> whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M80">View MathML</a>.

From (H2) and (H3) we immediately deduce the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M81">View MathML</a>

(1.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M82">View MathML</a>

(1.11)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M85">View MathML</a>.

Definition 1.1 By a weak solution of (1.9) under the assumptions (H1)-(H5), we mean a vector-valued function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M86">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M87">View MathML</a>

(1.12)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M88">View MathML</a>.

In [4] Duzaar and Mingione considered the partial regularity of homogeneous systems of (1.9) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M89">View MathML</a> under the natural growth condition. In this paper, we extend their results to the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M1">View MathML</a>. We have to overcome the difficulty of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M1">View MathML</a>. Motivated by the works of Duzaar [4,5], Chen and Tan [6-9] and Tan [10], we use the technique of ‘A-caloric approximation’ to establish the optimal partial regularity of nonlinear parabolic systems (1.9). In fact, the use of the ‘A-caloric approximation lemma’ allows optimal regularity, without the use of Reverse-Hölder inequalities and (parabolic) Gehring’s lemma. The method is based on an approximation result that we called the ‘A-caloric approximation lemma’. This is the parabolic analogue of the classical harmonic approximation lemma of De Giorgi [11,12] and allows to approximate functions with solutions to parabolic systems with constant coefficients in a similar way as the classical harmonic approximation lemma does with harmonic functions. And we can obtain the following theorem.

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M92">View MathML</a>be a weak solution to system (1.9) under the assumptions (H1)-(H4) and the natural growth condition (H5) and denote by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M93">View MathML</a>the set of regularity points ofuin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M22">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M95">View MathML</a>

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M93">View MathML</a>is an open subset with full measure, and therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M97">View MathML</a>

At the end of the section, we summarize some notions which we will be used in this paper. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M99">View MathML</a>, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M101">View MathML</a>. If v is an integrable function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M103">View MathML</a>, we will denote its average by , where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M105">View MathML</a> denotes the volume of the unit ball in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M53">View MathML</a>. We remark that in the following, when not crucial, the ‘center’ of the cylinder will be often unspecified, e.g., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M107">View MathML</a>; the same convention will be adopted for balls in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M53">View MathML</a> therefore denoting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M109">View MathML</a>. Finally, in the rest of the paper, the symbol C will denote a positive, finite constant that may vary from line to line; the relevant dependencies will be specified.

2 The A-caloric approximation technique and preliminaries

In this section we introduce the A-caloric approximation lemma [4] and some preliminaries. Recall a strongly elliptic bilinear form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M25">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M111">View MathML</a> with an ellipticity constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M41">View MathML</a>, and upper bound <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M113">View MathML</a> means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M115">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M116">View MathML</a>, we define A-caloric approximation function.

Definition 2.1 We shall say that a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M117">View MathML</a> is A-caloric on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M118">View MathML</a> if it satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M119">View MathML</a>

Remark 2.2 Obviously, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M120">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M121">View MathML</a>, then an A-caloric function is just a caloric function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M122">View MathML</a>.

Lemma 2.3 (A-caloric approximation lemma)

There exists a positive function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M123">View MathML</a>with the following property: WheneverAis a bilinear form on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M111">View MathML</a>, which is strongly ellipticity constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M41">View MathML</a>and upper bound Λ, εis a positive number, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M126">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M127">View MathML</a>

(2.1)

is approximativelyA-caloric in the sense that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M128">View MathML</a>

(2.2)

then there exists anA-caloric functionhsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M129">View MathML</a>

(2.3)

Actually, we could have directly applied Theorem 5 of [13] with the choice <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M134">View MathML</a> to conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M135">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M136">View MathML</a>.

Lemma 2.4There exists a positive function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M137">View MathML</a>with the following property: WheneverAis a bilinear form on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M111">View MathML</a>which is strongly ellipticity constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M41">View MathML</a>and upper bound Λ, εis a positive number, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M140">View MathML</a>with

(2.4)

is approximativelyA-caloric in the sense that

(2.5)

then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M143">View MathML</a>A-caloric on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M144">View MathML</a>such that

(2.6)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M146">View MathML</a> we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M147">View MathML</a> the unique affine function (in space) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M148">View MathML</a> minimizing , amongst all affine functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M150">View MathML</a> which are independent of t. To get an explicit formula for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M147">View MathML</a>, we note that such a unique minimum point exists and takes the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M152">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M153">View MathML</a>. A straightforward computation yields that , for any affine function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M155">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M156">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M157">View MathML</a>. This implies in particular that and .

For convenience we recall from [14] the following.

Lemma 2.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M161">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M147">View MathML</a>respectively<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M163">View MathML</a>the unique affine functions minimizingrespectively. Then there holds

Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M167">View MathML</a>, we have

3 Caccioppoli second inequality

In this section we prove Caccioppoli’s second inequality.

Theorem 3.1 (Caccioppoli second inequality)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M169">View MathML</a>be a weak solution to (1.9) under the assumptions (H1)-(H4) and the natural growth condition (H5). Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M170">View MathML</a>, any affine function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M148">View MathML</a>independent oftand satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M172">View MathML</a>, and any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M173">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M174">View MathML</a>, we have

Proof We take the test function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M176">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M177">View MathML</a> is a cut-off function in space such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M179">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M181">View MathML</a>. While <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M182">View MathML</a> is a cut-off function in time such that, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M183">View MathML</a> being arbitrary,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M184">View MathML</a>

Thus, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M185">View MathML</a>

We further have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M186">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M187">View MathML</a>

Adding these equations and using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M188">View MathML</a>, we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M189">View MathML</a>

(3.1)

By (1.10) and Young’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M190">View MathML</a>

(3.2)

By the condition (H4′) and Young’s inequality, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M191">View MathML</a>

(3.3)

Similarly, we can estimate III as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M192">View MathML</a>

(3.4)

Using the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M193">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M194">View MathML</a>, taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M195">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M196">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M197">View MathML</a>, we infer

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M198">View MathML</a>

(3.5)

and for μ positive to be fixed later, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M199">View MathML</a>

(3.6)

By (1.11) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M200">View MathML</a>

(3.7)

Combining (3.2)-(3.7) in (3.1) and noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M201">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M202">View MathML</a>), that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M203">View MathML</a>, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M204">View MathML</a> (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M205">View MathML</a>), choosing ε sufficiently small and taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M206">View MathML</a>, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M207">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M208">View MathML</a>, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M179">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M180">View MathML</a>, we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M211">View MathML</a>

Then the desired result follows by taking the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M212">View MathML</a>. □

4 The proof of the main theorem

The next lemma is a prerequisite for applying the A-caloric approximation technique.

Lemma 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M92">View MathML</a>be a weak solution to (1.9) under the assumptions (H1)-(H6). Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M170">View MathML</a>, we have

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M173">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M217">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M218">View MathML</a>and any affine function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M148">View MathML</a>independent of time, satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M172">View MathML</a>. Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M221">View MathML</a>and we write

Proof Without loss of generality, we can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M223">View MathML</a>. From (1.12) and the fact that and , we deduce

In turn, we split the first integral as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M227">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M229">View MathML</a>.

We proceed estimating the two resulting pieces. As for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M230">View MathML</a>, using (H6), the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M72">View MathML</a> is concave and Jensen’s inequality (note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M232">View MathML</a>), we get

To estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M234">View MathML</a>, we preliminarily observe that, using Hölder inequality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M235">View MathML</a>

and therefore

Similarly, we also have

Using (H1), (H2) and the previous inequality, we then conclude the estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M234">View MathML</a> as follows:

Combining the estimates found for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M230">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M234">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M242">View MathML</a>

For the remaining pieces, using (H4′), we deduce

Here we have used that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M205">View MathML</a> and the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M245">View MathML</a>. Using again (H4′) and Young’s inequality, we estimate

and

Noting the definition of H and combining the estimates just found for I, II, III and IV, we obtain

A simple scaling argument yields the result for general φ. □

The next lemma is a standard estimate for weak solutions to linear parabolic systems with constant coefficients [15], Lemma 5.1.

Lemma 4.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M249">View MathML</a>be a weak solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M250">View MathML</a>of the following linear parabolic system with constant coefficients:

where the coefficients<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M25">View MathML</a>satisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M253">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M254">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M255">View MathML</a>. Thenhis smooth in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M144">View MathML</a>and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M257">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M258">View MathML</a>

Here we write

In the following we consider a weak solution u of the nonlinear parabolic system (1.9) on a fixed sub-cylinder <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M260">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M218">View MathML</a>.

Lemma 4.3Given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M170">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M263">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M264">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M265">View MathML</a>depending only onn, N, λ, L, β, αandmsuch that if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M266">View MathML</a>

on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M260">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M268">View MathML</a>and such if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M269">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M270">View MathML</a>

for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M271">View MathML</a>

Proof Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M272">View MathML</a>. And we shall always consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M218">View MathML</a>. We first want to apply Lemma 4.1 on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M274">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M275">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M148">View MathML</a> is an affine function independent of t satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M172">View MathML</a>. We observe that Ψ has the following property:

(4.1)

From Caccioppoli’s second inequality, we infer

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M279">View MathML</a>

(4.2)

From Lemma 4.1 we therefore get, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M280">View MathML</a>, that

(4.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M282">View MathML</a>.

For given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M283">View MathML</a> to be specified later, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M284">View MathML</a> to be constant from Lemma 2.3. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M285">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M286">View MathML</a>.

Then from (4.3) we deduce that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M280">View MathML</a>, the following holds:

(4.4)

Moreover, we estimate, using Caccioppoli’s second inequality, (4.1) and (4.2),

(4.5)

provided we have chosen <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M290">View MathML</a> large enough.

Assuming the smallness condition,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M291">View MathML</a>

(4.6)

satisfied. Then (4.4) and (4.5) allow us to apply Lemma 2.4, i.e., they yield the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M143">View MathML</a> solving the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M68">View MathML</a>-heat equation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M274">View MathML</a> and satisfying

(4.7)

and

(4.8)

From Lemma 4.2 we recall that h satisfies, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M161">View MathML</a>, the a priori estimate (note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M298">View MathML</a>)

Here we have used that , and and (4.7). Combining the previous estimate with (4.8), we deduce

(4.9)

Recalling back <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M303">View MathML</a> via <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M304">View MathML</a>, we arrive at

(4.10)

Next we use the minimizing property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M306">View MathML</a>

(4.11)

At the same time, from (4.11), we can see that: For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M308">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M309">View MathML</a>), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M310">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M311">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M312">View MathML</a>. Therefore we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M313">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M314">View MathML</a>.

Using Sobolev’s, Caccioppoli’s and Young’s inequalities together with (4.11), we have

(4.12)

Using Lemma 2.5, Caccioppoli’s inequality, (4.4), (4.6), (4.12) and Young’s inequality, we obtain

(4.13)

From (4.12) and (4.13), we conclude

(4.14)

provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M318">View MathML</a> and we fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M319">View MathML</a>. That it is to say,

(4.15)

Combining (4.11) and (4.15) yields the desired estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M321">View MathML</a>

(4.16)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M322">View MathML</a>. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M323">View MathML</a>, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M161">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M325">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M326">View MathML</a>. This also fixes the constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M327">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M328">View MathML</a>. Thus we have shown Lemma 4.3. □

In the following, we want to iterate Lemma 4.3. That is,

Lemma 4.4For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M272">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M173">View MathML</a>, suppose that the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M331">View MathML</a>

are satisfied. Then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M332">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M333">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M334">View MathML</a>

Moreover, the limit

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M335">View MathML</a>

exists, and the estimate

is valid for a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M337">View MathML</a>.

Proof For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M338">View MathML</a> we shall denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M339">View MathML</a>. For given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M272">View MathML</a> (and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M323">View MathML</a>), we determine <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M342">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M343">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M344">View MathML</a> according to Lemma 4.3. Then we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M345">View MathML</a> sufficiently small such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M346">View MathML</a>

(4.17)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M347">View MathML</a>

(4.18)

Given this, we can also find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M348">View MathML</a> so small that, writing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M349">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M350">View MathML</a>

(4.19)

Now, suppose that the conditions (i), (ii) and (iii) are satisfied on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M260">View MathML</a>. Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M352">View MathML</a> , we shall show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M353">View MathML</a>

Note first that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M354">View MathML</a> combined with (ii), (iii) and (4.19) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M355">View MathML</a>

Moreover, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M356">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M357">View MathML</a>. There we can apply Lemma 4.3 to conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M358">View MathML</a> holds. Furthermore, using Lemma 2.5, (iii) and (4.18), we deduce

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M360">View MathML</a> holds. We now assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M361">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M362">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M363">View MathML</a> hold. We can apply Lemma 4.3 to calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M364">View MathML</a>

showing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M354">View MathML</a>. To show <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M366">View MathML</a> we estimate

Here we have used in turn Lemma 2.5, the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M368">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M361">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M370">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M371">View MathML</a>. We are in a position to apply Theorem 3.1. We obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M372">View MathML</a>

(4.20)

We now consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M373">View MathML</a>. We fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M374">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M375">View MathML</a>. Then the previous estimate implies

Next, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M377">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M111">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M379">View MathML</a> we deduce

This proves the claim. Therefore the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M381">View MathML</a> exists and from the previous estimate, we infer (taking the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M382">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M383">View MathML</a>

Combining this with (4.20), we arrive at

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M373">View MathML</a>, we find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M374">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M387">View MathML</a>. Then the previous estimate implies

This proves the assertion of the lemma. □

An immediate consequence of the previous lemma and of isomorphism theorem of Campanato-Da Prato [16] is the following result.

Theorem 4.1 (Description of regularity points)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M169">View MathML</a>be a weak solution to the system (1.9) under the assumptions (H1)-(H3) and (H4′), (H5), and denote by Σ the singular set ofu. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M390">View MathML</a>, where

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M392">View MathML</a>

At last, we have the following.

Theorem 4.2 (Almost everywhere regularity)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M169">View MathML</a>be a weak solution to the system (1.9) under the assumptions (H1)-(H3) and (H4), (H5), and denote by Σ the singular set ofu. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M394">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M395">View MathML</a>is as in Theorem 4.1 and

Proof We start taking a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M397">View MathML</a> such that

(4.21)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M399">View MathML</a>

(4.22)

The proof is complete if we show that such points are regularity points.

Step 1: a comparison estimate. Consider the unique weak solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M400">View MathML</a> of the initial boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M401">View MathML</a>

Then the difference <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M402">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M403">View MathML</a>

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M404">View MathML</a>. We now choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M405">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M406">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M407">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M408">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M409">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M410">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M411">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M412">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M413">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M414">View MathML</a>, we easily obtain that for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M415">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M416">View MathML</a>

The second term of the left-hand side of the previous equation can be estimated by the use of monotonicity, i.e., (H3). We therefore obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M417">View MathML</a>

(4.23)

To estimate the right-hand side, we use (H4) which easily yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M418">View MathML</a>

Using the previous estimate, Young’s inequality and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M419">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M420">View MathML</a>

Having combined the previous estimate with (4.23), we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M421">View MathML</a>

(4.24)

We shall provide on estimate for III. We denote , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M423">View MathML</a>.

If we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M424">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M425">View MathML</a>

(4.25)

We now split III

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M426">View MathML</a>

and estimate IV and V. We have, using that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M419">View MathML</a>, (4.25) and (4.22)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M428">View MathML</a>

From the definition of θ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M429">View MathML</a>

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M430">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M431">View MathML</a>

We now choose the parameter t carefully, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M432">View MathML</a> and let ε suitably small. Then connecting the previous estimates for II, III, IV and V to (4.24), we easily have the estimate we were interested in, that is,

(4.26)

In particular, we see that

(4.27)

We observe that, as a consequence of (4.21) and (4.22), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M435">View MathML</a>

(4.28)

Step 2: A Poincare-type inequality. Let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M436">View MathML</a>

Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M437">View MathML</a> solves

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M438">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M439">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M31">View MathML</a>. From [17], Theorem 3.1, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M441">View MathML</a> and that

In view of the previous estimate, using the Poincare inequality for v and (4.26), we find

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M444">View MathML</a>.

Finally, by comparison, we get the Poincare inequality for u via (4.26) and the previous estimate

(4.29)

for a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M444">View MathML</a>.

Step 3: Conclusion. From the previous estimate and (4.28), the assertion readily follows. Indeed if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M447">View MathML</a> satisfies (4.21) and (4.22), then we have

therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/152/mathml/M338">View MathML</a> is a regular point in view of Theorem 4.1. □

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 conceived of the study and the amendment of the paper. All authors read and approved the final manuscript.

Acknowledgements

Supported by the National Natural Science Foundation of China (Nos: 11201415, 11271305), the Natural Science Foundation of Fujian Province (2012J01027) and the Training Programme Foundation for Excellent Youth Researching Talents of Fujian’s Universities (JA12205).

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