# Global Optimal Regularity for the Parabolic Polyharmonic Equations

- Fengping Yao
^{1}Email author### Affiliated with

**2010**:879821

**DOI: **10.1155/2010/879821

© Fengping Yao. 2010

**Received: **21 February 2010

**Accepted: **3 June 2010

**Published: **14 June 2010

Research Article

- Fengping Yao
^{1}Email author### Affiliated with

Boundary Value Problems2010**2010**:879821

**DOI: **10.1155/2010/879821

© Fengping Yao. 2010

**Received: **21 February 2010

**Accepted: **3 June 2010

**Published: **14 June 2010

We show the global regularity estimates for the following parabolic polyharmonic equation in under proper conditions. Moreover, it will be verified that these conditions are necessary for the simplest heat equation in .

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, estimates, De Giorgi-Nash estimates, Krylov-Safonov estimates, and so on. 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
estimates, that is, regularity estimates in Orlicz spaces, for the following parabolic polyharmonic problems:

(1.1)

(1.2)

where , and 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 [2–6]). The theory of Orlicz spaces plays a crucial role in many fields of mathematics (see [7]).

We denote the distance in
as

(1.3)

and the cylinders in
as

(1.4)

where
is an open ball in
. Moreover, we denote

(1.5)

where is a multiple index, . For convenience, we often omit the subscript in and write .

Indeed if
, then (1.1) is simplified to be the simplest heat equation.
estimates and Schauder estimates for linear second-order equations are well known (see [8, 9]). When
, the corresponding regularity results for the higher-order parabolic equations are less. Solonnikov [10] studied
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
estimates to the Orlicz space

(1.6)

for

(1.7)

where (see Definition 1.2) and is an open bounded domain in . When with , (1.6) is reduced to the local estimates. In fact, we can replace of in (1.6) by the power of for any .

Our purpose in this paper is to extend local regularity estimate (
) in [6] to global regularity estimates, assuming that
. Moreover, we will also show that the
condition is necessary for the simplest heat equation
in
. In particular, we are interested in the estimate like

(1.8)

where is a constant independent from and . Indeed, if with , (1.8) is reduced to classical 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
is said to be a Young function if

(1.9)

Definition 1.2.

A Young function
is said to satisfy the global
condition, denoted by
, if there exists a positive constant
such that for every
*,*

(1.10)

Moreover, a Young function
is said to satisfy the global
condition, denoted by
, if there exists a number
such that for every
,

(1.11)

Example 1.3.

- (i)
, but .

- (ii)
, but .

- (iii)
, .

Remark 1.4.

If a function
satisfies (1.10) and (1.11), then

(1.12)

for every and , where and .

Remark 1.5.

Under condition (1.12), it is easy to check that
satisfies

(1.13)

Definition 1.6.

Assume that
is a Young function. Then the Orlicz class
is the set of all measurable functions
satisfying

(1.14)

The Orlicz space is the linear hull of .

Lemma 1.7 (see [2]).

Assume that and . Then

(1) ,

(2)
is dense in
,

- (3)(1.15)

Now let us state the main results of this work.

Theorem 1.8.

Assume that
is a Young function and
satisfies

(1.16)

Then if the following inequality holds

(1.17)

One has

(1.18)

Theorem 1.9.

Assume that . If is the solution of (1.1)-(1.2) with , then (1.8) holds.

Remark 1.10.

We would like to point out that the
condition is necessary. In fact, if the local
estimate (1.6)
is true, then by choosing

(1.19)

we have

(1.20)

which implies that

(1.21)

In this section we show that satisfies the global condition if satisfies (1.16) and estimate (1.17) is true.

Proof.

Now we consider the special case in (1.16) when

(2.1)

for any constant
, where
and
is a cutoff function satisfying

(2.2)

Therefore the problem (1.16) has the solution

(2.3)

It follows from (1.17), (2.1), and (2.2) that

(2.4)

We know from (2.3) that

(2.5)

Define

(2.6)

Then when
and
, we have

(2.7)

since

(2.8)

Therefore, since
for
and
, we conclude that

(2.9)

Recalling estimate (2.4) we find that

(2.10)

which implies that

(2.11)

By changing variable we conclude that, for any
,

(2.12)

where
. Let
and
. Then we conclude from (2.12) that

(2.13)

Now we use (2.12) and (2.13) to obtain that

(2.14)

where we choose that
,
in (2.13). Set
. Then we have

(2.15)

when is chosen large enough. This implies that satisfies the condition. Thus this completes our proof.

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
,
and
, where
is defined in (1.12). Then for any
one has

(3.1)

Moreover, we recall the following result.

Lemma 3.2 (see [10, Theorem ]).

Assume that
for
. There exists a unique solution
of (1.1)-(1.2) with the estimate

(3.2)

Moreover, we give one important lemma, which is motivated by the iteration-covering procedure in [12]. To start with, let
be a solution of (1.1)-(1.2). Let

(3.3)

In fact, in the subsequent proof we can choose any constant
with
. Now we write

(3.4)

while
is a small enough constant which will be determined later. Set

(3.5)

for any
. Then
is still the solution of (1.1)-(1.2) with
replacing
. Moreover, we write

(3.6)

for any domain
in
and the level set

(3.7)

Next, we will decompose the level set .

Lemma 3.3.

For any
, there exists a family of disjoint cylinders
with
and
such that

(3.8)

(3.9)

where
. Moreover, one has

(3.10)

Proof.

(
) Fix any
. We first claim that

(3.11)

where
satisfies
. To prove this, fix any
and
. Then it follows from (3.4) that

(3.12)

(
) For a.e.
, from Lebesgue's differentiation theorem we have

(3.13)

which implies that there exists some
satisfying

(3.14)

Therefore from (3.11) we can select a radius
such that

(3.15)

Therefore, applying Vitali's covering lemma, we can find a family of disjoint cylinders
such that (3.8) and (3.9) hold.

- (3)Equation (3.8) implies that(3.16)

Therefore, by splitting the two integrals above as follows we have

(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
. In view of Lemma 3.3, given any
, we can construct a family of cylinders
, where
. Fix
. It follows from (3.6) and (3.8) in Lemma 3.3 that

(3.18)

We first extend
from
to
by the zero extension and denote by
. From Lemma 3.2, there exists a unique solution
of

(3.19)

with the estimate

(3.20)

Therefore we see that

(3.21)

Set
. Then we know that

(3.22)

Moreover, by (3.18) and (3.21) we have

(3.23)

Thus from the elementary interior
regularity, we know that there exists a constant
such that

(3.24)

Set
. Therefore, we deduce from (3.5) and (3.24) that

(3.25)

Then according to (3.18) and (3.21), we discover

(3.26)

Therefore, from (3.10) in Lemma 3.3 we find that

(3.27)

where
. Recalling the fact that the cylinders
are disjoint,

(3.28)

and then summing up on
in the inequality above, we have

(3.29)

Therefore, from Lemma 1.7(3) and the inequality above we have

(3.30)

Consequently, from Lemma 3.1 we conclude that

(3.31)

where and . Finally selecting a suitable such that , we finish the proof.

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).

(1)

Department of Mathematics, Shanghai University

- Orlicz W:
**Über eine gewisse Klasse von Räumen vom Typus B.***Bulletin International de l'Académie Polonaise Série A*1932,**8:**207-220. - Adams RA, Fournier JJF:
*Sobolev Spaces*.*Volume 140*. 2nd edition. Academic Press, Amsterdam, The Netherlands; 2003:xiv+305. - 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 - 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 - Kokilashvili V, Krbec M:
*Weighted inequalities in Lorentz and Orlicz spaces*. World Scientific, River Edge, NJ, USA; 1991:xii+233.View Article - 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 - 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. - Gilbarg D, Trudinger N:
*Elliptic Partial Differential Equations of Second Order*. 3rd edition. Springer, Berlin, Germany; 1998. - Lieberman GM:
*Second Order Parabolic Differential Equations*. World Scientific, River Edge, NJ, USA; 1996:xii+439.View Article - 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 - 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 - 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

© Fengping Yao. 2010

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