Global Optimal Regularity for the Parabolic Polyharmonic Equations

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:

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

and the cylinders in as

where is an open ball in . Moreover, we denote

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

for

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

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

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 ,

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

Example 1.3.

(i) , but .

(ii) , but .

(iii) , .

Remark 1.4.

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

for every and , where and .

Remark 1.5.

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

Definition 1.6.

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

The Orlicz space is the linear hull of .

Lemma 1.7 (see [2]).

Assume that and . Then

(1),

(2) is dense in ,

(3)

Now let us state the main results of this work.

Theorem 1.8.

Assume that is a Young function and satisfies

Then if the following inequality holds

One has

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

we have

which implies that

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

for any constant , where and is a cutoff function satisfying

Therefore the problem (1.16) has the solution

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

We know from (2.3) that

Define

Then when and , we have

since

Therefore, since for and , we conclude that

Recalling estimate (2.4) we find that

which implies that

By changing variable we conclude that, for any ,

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

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

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

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

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

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

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

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

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

for any domain in and the level set

Next, we will decompose the level set .

Lemma 3.3.

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

where . Moreover, one has

Proof.

() Fix any . We first claim that

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

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

which implies that there exists some satisfying

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

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

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

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

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

with the estimate

Therefore we see that

Set . Then we know that

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

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

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

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

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

where . Recalling the fact that the cylinders are disjoint,

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

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

Consequently, from Lemma 3.1 we conclude that

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).

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