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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq1_HTML.gif in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq3_HTML.gif in http://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, http://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. http://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 http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ1_HTML.gif
(1.1)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ2_HTML.gif
(1.2)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq9_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq11_HTML.gif as
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ3_HTML.gif
(1.3)
and the cylinders in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq12_HTML.gif as
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ4_HTML.gif
(1.4)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq13_HTML.gif is an open ball in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq14_HTML.gif . Moreover, we denote
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ5_HTML.gif
(1.5)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq15_HTML.gif is a multiple index, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq17_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq18_HTML.gif and write http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq19_HTML.gif .

Indeed if http://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. http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq24_HTML.gif estimates to the Orlicz space
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ6_HTML.gif
(1.6)
for
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ7_HTML.gif
(1.7)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq25_HTML.gif (see Definition 1.2) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq26_HTML.gif is an open bounded domain in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq27_HTML.gif . When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq28_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq30_HTML.gif estimates. In fact, we can replace http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq31_HTML.gif of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq33_HTML.gif for any http://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 ( http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq38_HTML.gif in http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ8_HTML.gif
(1.8)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq40_HTML.gif is a constant independent from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq42_HTML.gif . Indeed, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq43_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq44_HTML.gif , (1.8) is reduced to classical http://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 http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq47_HTML.gif is said to satisfy the global http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq48_HTML.gif condition, denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq49_HTML.gif , if there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq50_HTML.gif such that for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq51_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ10_HTML.gif
(1.10)
Moreover, a Young function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq52_HTML.gif is said to satisfy the global http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq53_HTML.gif condition, denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq54_HTML.gif , if there exists a number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq55_HTML.gif such that for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq56_HTML.gif ,
http://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)

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

     
  2. (ii)

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

     
  3. (iii)

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

     

Remark 1.4.

If a function http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ12_HTML.gif
(1.12)

for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq65_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq66_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq68_HTML.gif satisfies
http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq71_HTML.gif satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ14_HTML.gif
(1.14)

The Orlicz space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq72_HTML.gif is the linear hull of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq75_HTML.gif . Then

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

(2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq77_HTML.gif is dense in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq78_HTML.gif ,
  1. (3)
    http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq79_HTML.gif is a Young function and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq80_HTML.gif satisfies
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ17_HTML.gif
(1.17)
One has
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq81_HTML.gif . If http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq85_HTML.gif estimate (1.6) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq86_HTML.gif is true, then by choosing
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ19_HTML.gif
(1.19)
we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ20_HTML.gif
(1.20)
which implies that
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq87_HTML.gif satisfies the global http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq88_HTML.gif condition if http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ22_HTML.gif
(2.1)
for any constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq90_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq91_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq92_HTML.gif is a cutoff function satisfying
http://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
http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ26_HTML.gif
(2.5)
Define
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ27_HTML.gif
(2.6)
Then when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq93_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq94_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ28_HTML.gif
(2.7)
since
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ29_HTML.gif
(2.8)
Therefore, since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq95_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq96_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq97_HTML.gif , we conclude that
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ31_HTML.gif
(2.10)
which implies that
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq98_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ33_HTML.gif
(2.12)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq99_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq100_HTML.gif and http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ35_HTML.gif
(2.14)
where we choose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq102_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq103_HTML.gif in (2.13). Set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq104_HTML.gif . Then we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ36_HTML.gif
(2.15)

when http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq106_HTML.gif satisfies the http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq108_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq109_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq110_HTML.gif , where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq112_HTML.gif one has
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq113_HTML.gif ]).

Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq114_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq115_HTML.gif . There exists a unique solution http://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
http://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 http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq118_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq119_HTML.gif . Now we write
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ40_HTML.gif
(3.4)
while http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ41_HTML.gif
(3.5)
for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq121_HTML.gif . Then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq123_HTML.gif replacing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq124_HTML.gif . Moreover, we write
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ42_HTML.gif
(3.6)
for any domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq125_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq126_HTML.gif and the level set
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq127_HTML.gif .

Lemma 3.3.

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

Proof.

( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq133_HTML.gif ) Fix any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq134_HTML.gif . We first claim that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ47_HTML.gif
(3.11)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq135_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq136_HTML.gif . To prove this, fix any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq137_HTML.gif and http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ48_HTML.gif
(3.12)
( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq139_HTML.gif ) For a.e. http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq141_HTML.gif satisfying
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq142_HTML.gif such that
http://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 http://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
    http://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
http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq146_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq147_HTML.gif . Fix http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ54_HTML.gif
(3.18)
We first extend http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq149_HTML.gif from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq150_HTML.gif to http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq153_HTML.gif of
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ55_HTML.gif
(3.19)
with the estimate
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ56_HTML.gif
(3.20)
Therefore we see that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ57_HTML.gif
(3.21)
Set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq154_HTML.gif . Then we know that
http://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
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq156_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ60_HTML.gif
(3.24)
Set http://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
http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ63_HTML.gif
(3.27)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq158_HTML.gif . Recalling the fact that the cylinders http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq159_HTML.gif are disjoint,
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq160_HTML.gif in the inequality above, we have
http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_Equ67_HTML.gif
(3.31)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq161_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq162_HTML.gif . Finally selecting a suitable http://static-content.springer.com/image/art%3A10.1155%2F2010%2F879821/MediaObjects/13661_2010_Article_965_IEq163_HTML.gif such that http://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

References

  1. Orlicz W: Über eine gewisse Klasse von Räumen vom Typus B. Bulletin International de l'Académie Polonaise Série A 1932, 8: 207-220.
  2. Adams RA, Fournier JJF: Sobolev Spaces. Volume 140. 2nd edition. Academic Press, Amsterdam, The Netherlands; 2003:xiv+305.
  3. Byun S, Yao F, Zhou S: Gradient estimates in Orlicz space for nonlinear elliptic equations. Journal of Functional Analysis 2008,255(8):1851-1873.MathSciNetView Article
  4. Jia H, Li D, Wang L: Regularity of Orlicz spaces for the Poisson equation. Manuscripta Mathematica 2007,122(3):265-275. 10.1007/s00229-006-0066-yMathSciNetView Article
  5. Kokilashvili V, Krbec M: Weighted inequalities in Lorentz and Orlicz spaces. World Scientific, River Edge, NJ, USA; 1991:xii+233.View Article
  6. Yao F: Regularity theory in Orlicz spaces for the parabolic polyharmonic equations. Archiv der Mathematik 2008,90(5):429-439. 10.1007/s00013-008-2576-1MathSciNetView Article
  7. Rao MM, Ren ZD: Applications of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics. Volume 250. Marcel Dekker, New York, NY, USA; 2002:xii+464.
  8. Gilbarg D, Trudinger N: Elliptic Partial Differential Equations of Second Order. 3rd edition. Springer, Berlin, Germany; 1998.
  9. Lieberman GM: Second Order Parabolic Differential Equations. World Scientific, River Edge, NJ, USA; 1996:xii+439.View Article
  10. Solonnikov VA: On boundary value problems for linear parabolic systems of differential equations of general form. Trudy Matematicheskogo Instituta Imeni V. A. Steklova 1965, 83: 3-163.MathSciNet
  11. Yao F-P, Zhou S-L: Schauder estimates for parabolic equation of bi-harmonic type. Applied Mathematics and Mechanics 2007,28(11):1503-1516. 10.1007/s10483-007-1110-zMathSciNetView Article
  12. Acerbi E, Mingione G: Gradient estimates for a class of parabolic systems. Duke Mathematical Journal 2007,136(2):285-320. 10.1215/S0012-7094-07-13623-8MathSciNetView Article

Copyright

© Fengping Yao. 2010

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