Open Access

Global Optimal Regularity for the Parabolic Polyharmonic Equations

Boundary Value Problems20102010:879821

DOI: 10.1155/2010/879821

Received: 21 February 2010

Accepted: 3 June 2010

Published: 14 June 2010

Abstract

We show the global regularity estimates for the following parabolic polyharmonic equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq1_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq2_HTML.gif under proper conditions. Moreover, it will be verified that these conditions are necessary for the simplest heat equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq3_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq4_HTML.gif .

1. Introduction

Regularity theory in PDE plays an important role in the development of second-order elliptic and parabolic equations. Classical regularity estimates for elliptic and parabolic equations consist of Schauder estimates, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq5_HTML.gif estimates, De Giorgi-Nash estimates, Krylov-Safonov estimates, and so on. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq6_HTML.gif and Schauder estimates, which play important roles in the theory of partial differential equations, are two fundamental estimates for elliptic and parabolic equations and the basis for the existence, uniqueness, and regularity of solutions.

The objective of this paper is to investigate the generalization of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq7_HTML.gif estimates, that is, regularity estimates in Orlicz spaces, for the following parabolic polyharmonic problems:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq9_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq10_HTML.gif is a positive integer. Since the 1960s, the need to use wider spaces of functions than Sobolev spaces arose out of various practical problems. Orlicz spaces have been studied as the generalization of Sobolev spaces since they were introduced by Orlicz [1] (see [26]). The theory of Orlicz spaces plays a crucial role in many fields of mathematics (see [7]).

We denote the distance in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq11_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ3_HTML.gif
(1.3)
and the cylinders in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq12_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ4_HTML.gif
(1.4)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq13_HTML.gif is an open ball in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq14_HTML.gif . Moreover, we denote
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ5_HTML.gif
(1.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq15_HTML.gif is a multiple index, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq16_HTML.gif . For convenience, we often omit the subscript https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq17_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq18_HTML.gif and write https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq19_HTML.gif .

Indeed if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq20_HTML.gif , then (1.1) is simplified to be the simplest heat equation. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq21_HTML.gif estimates and Schauder estimates for linear second-order equations are well known (see [8, 9]). When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq22_HTML.gif , the corresponding regularity results for the higher-order parabolic equations are less. Solonnikov [10] studied https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq23_HTML.gif and Schauder estimates for the general linear higher-order parabolic equations with the help of fundamental solutions and Green functions. Moreover, in [11] we proved global Schauder estimates for the initial-value parabolic polyharmonic problem using the uniform approach as the second-order case. Recently we [6] generalized the local https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq24_HTML.gif estimates to the Orlicz space
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ6_HTML.gif
(1.6)
for
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ7_HTML.gif
(1.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq25_HTML.gif (see Definition 1.2) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq26_HTML.gif is an open bounded domain in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq27_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq28_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq29_HTML.gif , (1.6) is reduced to the local https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq30_HTML.gif estimates. In fact, we can replace https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq31_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq32_HTML.gif in (1.6) by the power of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq33_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq34_HTML.gif .

Our purpose in this paper is to extend local regularity estimate ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq35_HTML.gif ) in [6] to global regularity estimates, assuming that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq36_HTML.gif . Moreover, we will also show that the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq37_HTML.gif condition is necessary for the simplest heat equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq38_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq39_HTML.gif . In particular, we are interested in the estimate like
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ8_HTML.gif
(1.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq40_HTML.gif is a constant independent from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq42_HTML.gif . Indeed, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq43_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq44_HTML.gif , (1.8) is reduced to classical https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq45_HTML.gif estimates. We remark that although we use similar functional framework and iteration-covering procedure as in [6, 12], more complicated analysis should be carefully carried out with a proper dilation of the unbounded domain.

Here for the reader's convenience, we will give some definitions on the general Orlicz spaces.

Definition 1.1.

A convex function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq46_HTML.gif is said to be a Young function if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ9_HTML.gif
(1.9)

Definition 1.2.

A Young function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq47_HTML.gif is said to satisfy the global https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq48_HTML.gif condition, denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq49_HTML.gif , if there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq50_HTML.gif such that for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq51_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ10_HTML.gif
(1.10)
Moreover, a Young function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq52_HTML.gif is said to satisfy the global https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq53_HTML.gif condition, denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq54_HTML.gif , if there exists a number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq55_HTML.gif such that for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq56_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ11_HTML.gif
(1.11)
Example 1.3.
  1. (i)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq57_HTML.gif , but https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq58_HTML.gif .

     
  2. (ii)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq59_HTML.gif , but https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq60_HTML.gif .

     
  3. (iii)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq61_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq62_HTML.gif .

     

Remark 1.4.

If a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq63_HTML.gif satisfies (1.10) and (1.11), then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ12_HTML.gif
(1.12)

for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq64_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq65_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq66_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq67_HTML.gif .

Remark 1.5.

Under condition (1.12), it is easy to check that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq68_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ13_HTML.gif
(1.13)

Definition 1.6.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq69_HTML.gif is a Young function. Then the Orlicz class https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq70_HTML.gif is the set of all measurable functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq71_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ14_HTML.gif
(1.14)

The Orlicz space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq72_HTML.gif is the linear hull of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq73_HTML.gif .

Lemma 1.7 (see [2]).

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq75_HTML.gif . Then

(1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq76_HTML.gif ,

(2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq77_HTML.gif is dense in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq78_HTML.gif ,
  1. (3)
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ15_HTML.gif
    (1.15)
     

Now let us state the main results of this work.

Theorem 1.8.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq79_HTML.gif is a Young function and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq80_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ16_HTML.gif
(1.16)
Then if the following inequality holds
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ17_HTML.gif
(1.17)
One has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ18_HTML.gif
(1.18)

Theorem 1.9.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq81_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq82_HTML.gif is the solution of (1.1)-(1.2) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq83_HTML.gif , then (1.8) holds.

Remark 1.10.

We would like to point out that the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq84_HTML.gif condition is necessary. In fact, if the local https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq85_HTML.gif estimate (1.6) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq86_HTML.gif is true, then by choosing
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ19_HTML.gif
(1.19)
we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ20_HTML.gif
(1.20)
which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ21_HTML.gif
(1.21)

2. Proof of Theorem 1.8

In this section we show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq87_HTML.gif satisfies the global https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq88_HTML.gif condition if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq89_HTML.gif satisfies (1.16) and estimate (1.17) is true.

Proof.

Now we consider the special case in (1.16) when
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ22_HTML.gif
(2.1)
for any constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq90_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq91_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq92_HTML.gif is a cutoff function satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ23_HTML.gif
(2.2)
Therefore the problem (1.16) has the solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ24_HTML.gif
(2.3)
It follows from (1.17), (2.1), and (2.2) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ25_HTML.gif
(2.4)
We know from (2.3) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ26_HTML.gif
(2.5)
Define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ27_HTML.gif
(2.6)
Then when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq93_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq94_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ28_HTML.gif
(2.7)
since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ29_HTML.gif
(2.8)
Therefore, since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq95_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq97_HTML.gif , we conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ30_HTML.gif
(2.9)
Recalling estimate (2.4) we find that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ31_HTML.gif
(2.10)
which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ32_HTML.gif
(2.11)
By changing variable we conclude that, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq98_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ33_HTML.gif
(2.12)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq99_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq100_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq101_HTML.gif . Then we conclude from (2.12) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ34_HTML.gif
(2.13)
Now we use (2.12) and (2.13) to obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ35_HTML.gif
(2.14)
where we choose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq102_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq103_HTML.gif in (2.13). Set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq104_HTML.gif . Then we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ36_HTML.gif
(2.15)

when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq105_HTML.gif is chosen large enough. This implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq106_HTML.gif satisfies the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq107_HTML.gif condition. Thus this completes our proof.

3. Proof of the Main Result

In this section, we will finish the proof of the main result, Theorem 1.9. Just as [6], we will use the following two lemmas. The first lemma is the following integral inequality.

Lemma 3.1 (see [6]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq108_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq109_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq110_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq111_HTML.gif is defined in (1.12). Then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq112_HTML.gif one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ37_HTML.gif
(3.1)

Moreover, we recall the following result.

Lemma 3.2 (see [10, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq113_HTML.gif ]).

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq114_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq115_HTML.gif . There exists a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq116_HTML.gif of (1.1)-(1.2) with the estimate
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ38_HTML.gif
(3.2)
Moreover, we give one important lemma, which is motivated by the iteration-covering procedure in [12]. To start with, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq117_HTML.gif be a solution of (1.1)-(1.2). Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ39_HTML.gif
(3.3)
In fact, in the subsequent proof we can choose any constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq118_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq119_HTML.gif . Now we write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ40_HTML.gif
(3.4)
while https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq120_HTML.gif is a small enough constant which will be determined later. Set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ41_HTML.gif
(3.5)
for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq121_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq122_HTML.gif is still the solution of (1.1)-(1.2) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq123_HTML.gif replacing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq124_HTML.gif . Moreover, we write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ42_HTML.gif
(3.6)
for any domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq125_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq126_HTML.gif and the level set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ43_HTML.gif
(3.7)

Next, we will decompose the level set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq127_HTML.gif .

Lemma 3.3.

For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq128_HTML.gif , there exists a family of disjoint cylinders https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq129_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq130_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq131_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ44_HTML.gif
(3.8)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ45_HTML.gif
(3.9)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq132_HTML.gif . Moreover, one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ46_HTML.gif
(3.10)

Proof.

( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq133_HTML.gif ) Fix any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq134_HTML.gif . We first claim that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ47_HTML.gif
(3.11)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq135_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq136_HTML.gif . To prove this, fix any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq137_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq138_HTML.gif . Then it follows from (3.4) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ48_HTML.gif
(3.12)
( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq139_HTML.gif ) For a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq140_HTML.gif , from Lebesgue's differentiation theorem we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ49_HTML.gif
(3.13)
which implies that there exists some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq141_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ50_HTML.gif
(3.14)
Therefore from (3.11) we can select a radius https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq142_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ51_HTML.gif
(3.15)
Therefore, applying Vitali's covering lemma, we can find a family of disjoint cylinders https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq143_HTML.gif such that (3.8) and (3.9) hold.
  1. (3)
    Equation (3.8) implies that
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ52_HTML.gif
    (3.16)
     
Therefore, by splitting the two integrals above as follows we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ53_HTML.gif
(3.17)

Thus we can obtain the desired result (3.10).

Now we are ready to prove the main result, Theorem 1.9.

Proof.

In the following by the elementary approximation argument as [3, 12] it is sufficient to consider the proof of (1.8) under the additional assumption that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq144_HTML.gif . In view of Lemma 3.3, given any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq145_HTML.gif , we can construct a family of cylinders https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq146_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq147_HTML.gif . Fix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq148_HTML.gif . It follows from (3.6) and (3.8) in Lemma 3.3 that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ54_HTML.gif
(3.18)
We first extend https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq149_HTML.gif from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq150_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq151_HTML.gif by the zero extension and denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq152_HTML.gif . From Lemma 3.2, there exists a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq153_HTML.gif of
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ55_HTML.gif
(3.19)
with the estimate
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ56_HTML.gif
(3.20)
Therefore we see that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ57_HTML.gif
(3.21)
Set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq154_HTML.gif . Then we know that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ58_HTML.gif
(3.22)
Moreover, by (3.18) and (3.21) we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ59_HTML.gif
(3.23)
Thus from the elementary interior https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq155_HTML.gif regularity, we know that there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq156_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ60_HTML.gif
(3.24)
Set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq157_HTML.gif . Therefore, we deduce from (3.5) and (3.24) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ61_HTML.gif
(3.25)
Then according to (3.18) and (3.21), we discover
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ62_HTML.gif
(3.26)
Therefore, from (3.10) in Lemma 3.3 we find that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ63_HTML.gif
(3.27)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq158_HTML.gif . Recalling the fact that the cylinders https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq159_HTML.gif are disjoint,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ64_HTML.gif
(3.28)
and then summing up on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq160_HTML.gif in the inequality above, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ65_HTML.gif
(3.29)
Therefore, from Lemma 1.7(3) and the inequality above we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ66_HTML.gif
(3.30)
Consequently, from Lemma 3.1 we conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ67_HTML.gif
(3.31)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq161_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq162_HTML.gif . Finally selecting a suitable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq163_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq164_HTML.gif , we finish the proof.

Declarations

Acknowledgments

The author wishes to thank the anonymous referee for offering valuable suggestions to improve the expressions. This work is supported in part by Tianyuan Foundation (10926084) and Research Fund for the Doctoral Program of Higher Education of China (20093108120003). Moreover, the author wishes to thank the department of mathematics at Shanghai university which was supported by the Shanghai Leading Academic Discipline Project (J50101) and Key Disciplines of Shanghai Municipality (S30104).

Authors’ Affiliations

(1)
Department of Mathematics, Shanghai University

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Copyright

© Fengping Yao. 2010

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