Global Optimal Regularity for the Parabolic Polyharmonic Equations
© 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.
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  (see [2–6]). The theory of Orlicz spaces plays a crucial role in many fields of mathematics (see ).
where is a multiple index, . For convenience, we often omit the subscript in and write .
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 .
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.
, but .
, but .
for every and , where and .
The Orlicz space is the linear hull of .
Lemma 1.7 (see ).
Assume that and . Then
Now let us state the main results of this work.
Assume that . If is the solution of (1.1)-(1.2) with , then (1.8) holds.
2. Proof of Theorem 1.8
In this section we show that satisfies the global condition if satisfies (1.16) and estimate (1.17) is true.
when is chosen large enough. This implies that satisfies the 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 , we will use the following two lemmas. The first lemma is the following integral inequality.
Lemma 3.1 (see ).
Moreover, we recall the following result.
Lemma 3.2 (see [10, Theorem ]).
Next, we will decompose the level set .
- (3)Equation (3.8) implies that(3.16)
Thus we can obtain the desired result (3.10).
Now we are ready to prove the main result, Theorem 1.9.
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).
- Orlicz W: Über eine gewisse Klasse von Räumen vom Typus B. Bulletin International de l'Académie Polonaise Série A 1932, 8: 207-220.MATH
- Adams RA, Fournier JJF: Sobolev Spaces. Volume 140. 2nd edition. Academic Press, Amsterdam, The Netherlands; 2003:xiv+305.MATH
- 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 ArticleMATH
- 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 ArticleMATH
- Kokilashvili V, Krbec M: Weighted inequalities in Lorentz and Orlicz spaces. World Scientific, River Edge, NJ, USA; 1991:xii+233.View ArticleMATH
- 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 ArticleMATH
- 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.MATH
- Lieberman GM: Second Order Parabolic Differential Equations. World Scientific, River Edge, NJ, USA; 1996:xii+439.View ArticleMATH
- 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.MathSciNetMATH
- 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 ArticleMATH
- 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 ArticleMATH
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.