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Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 879821, 12 pages doi:10.1155/2010/879821 Research Article Global Optimal Regularity for the Parabolic Polyharmonic Equations Fengping Yao Department of Mathematics, Shanghai University, Shanghai 200436, China Correspondence should be addressed to Fengping Yao, yfp1123@math.pku.edu.cn Received 21 February 2010; Accepted June 2010 Academic Editor: Vicentiu D Radulescu Copyright q 2010 Fengping Yao 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 We show the global regularity estimates for the following parabolic polyharmonic equation −Δ m u f in Rn × 0, ∞ , m ∈ Z under proper conditions Moreover, it will be verified ut that these conditions are necessary for the simplest heat equation ut − Δu f in Rn × 0, ∞ 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, Lp estimates, De Giorgi-Nash estimates, KrylovSafonov estimates, and so on Lp 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 Lp estimates, that is, regularity estimates in Orlicz spaces, for the following parabolic polyharmonic problems: ut x, t −Δ m u x, t u x, f x, t in Rn , in Rn × 0, ∞ , 1.1 1.2 n 2 where x x1 , , xn , Δ i ∂ /∂xi and m 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 Boundary Value Problems We denote the distance in Rn and the cylinders in Rn where BR as max |x1 − x2 |, |t1 − t2 |1/2m δ z1 , z2 QR 1 for z1 x1 , t1 , z2 x2 , t2 1.3 as BR × −R2m , R2m , QR z QR z, z x, t ∈ Rn , 1.4 {x ∈ Rn : |x| < R} is an open ball in Rn Moreover, we denote ν Dx u ∂|ν| u , ∂ν1 · · · ∂νn x1 xn 1.5 n where ν ν1 , ν2 , , νn is a multiple index, νi ≥ i 1, 2, , n , and |ν| i |νi | For ν k ν convenience, we often omit the subscript x in Dx u and write D u {D u : |ν| k} Indeed if m 1, then 1.1 is simplified to be the simplest heat equation Lp estimates and Schauder estimates for linear second-order equations are well known see 8, When m / 1, the corresponding regularity results for the higher-order parabolic equations are less Solonnikov 10 studied Lp 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 generalized the local Lp estimates to the Orlicz space φ D2m u Q1/6 φ |ut |2 dz ≤ C dz Q1/6 φ f φ |u|2 dz dz Q1/2 1.6 Q1/2 for ut z −Δ m u z f z in Ω × 0, T , 1.7 where φ ∈ Δ2 ∩ ∇2 see Definition 1.2 and Ω is an open bounded domain in Rn When φ x |x|p/2 with p > 2, 1.6 is reduced to the local Lp estimates In fact, we can replace of φ | · | in 1.6 by the power of p1 for any p1 > Our purpose in this paper is to extend local regularity estimate 1.3 in to global regularity estimates, assuming that φ ∈ Δ2 ∩ ∇2 Moreover, we will also show that the Δ2 ∩ ∇2 condition is necessary for the simplest heat equation ut − Δu f in Rn × 0, ∞ In particular, we are interested in the estimate like Rn × 0,∞ φ D2m u dz Rn × 0,∞ φ |ut | dz ≤ C Rn × 0,∞ φ f dz, 1.8 where C is a constant independent from u and f Indeed, if φ x |x|p with p > 1, 1.8 p is reduced to classical L estimates We remark that although we use similar functional Boundary Value Problems 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 φ : R → R is said to be a Young function if φ −s φ s , φ 0, ∞ lim φ s s→∞ 1.9 Definition 1.2 A Young function φ is said to satisfy the global Δ2 condition, denoted by φ ∈ Δ2 , if there exists a positive constant K such that for every s > 0, φ 2s ≤ Kφ s 1.10 Moreover, a Young function φ is said to satisfy the global ∇2 condition, denoted by φ ∈ ∇2 , if there exists a number a > such that for every s > 0, φ s ≤ φ as 2a 1.11 Example 1.3 i φ1 s |s| log |s| − |s| ∈ Δ2 , but φ1 s / ∇2 ∈ e|s| − |s| − ∈ ∇2 , but φ2 s / Δ2 ∈ ii φ2 s |s|α | log |s|| ∈ Δ2 ∩ ∇2 , α > iii φ3 s Remark 1.4 If a function φ satisfies 1.10 and 1.11 , then α φ θ1 s ≤ Kθ1 φ s , α φ θ2 s ≤ 2aθ2 φ s , for every s > and < θ2 ≤ ≤ θ1 < ∞, where α1 log2 K and α2 1.12 loga Remark 1.5 Under condition 1.12 , it is easy to check that φ satisfies φ 0, ∞, lim φ s s→∞ lim s→0 φ s s lim s→ ∞φ s s 1.13 Definition 1.6 Assume that φ is a Young function Then the Orlicz class K φ Rn is the set of all measurable functions g : Rn → R satisfying Rn φ g dx < ∞ The Orlicz space Lφ Rn is the linear hull of K φ Rn 1.14 Boundary Value Problems Lemma 1.7 see Assume that φ ∈ Δ2 ∩ ∇2 and g ∈ Lφ Ω Then Kφ Ω Lφ Ω , ∞ C0 Ω is dense in Lφ Ω , ∞ Ω x∈Ω: g >μ d φ μ φ g dx 1.15 Now let us state the main results of this work Theorem 1.8 Assume that φ is a Young function and u satisfies ut x, t − Δu x, t in Rn × 0, ∞ , f x, t u x, 1.16 in Rn Then if the following inequality holds Rn × 0,∞ φ D2 u dz Rn × 0,∞ φ |ut | dz ≤ C Rn × 0,∞ φ f dz, 1.17 One has φ ∈ Δ2 ∩ ∇2 1.18 Theorem 1.9 Assume that φ ∈ Δ2 ∩ ∇2 If u is the solution of 1.1 - 1.2 with f ∈ Lφ Rn × 0, ∞ , then 1.8 holds Remark 1.10 We would like to point out that the Δ2 condition is necessary In fact, if the local Lφ estimate 1.6 m is true, then by choosing u √ 1/2 2s x1 x2 , f ∀s > 1.19 we have ⎛ φ 2s dz Q1/6 ⎞ ∂2 u ⎠dz φ⎝ ∂x1 ∂x2 Q1/6 ≤C φ f Q1/2 ≤C φ s dz, Q1/2 φ |u|2 dz dz Q1/2 1.20 Boundary Value Problems which implies that φ 2s ≤ Cφ s , for any s > 1.21 Proof of Theorem 1.8 In this section we show that φ satisfies the global ∇2 condition if u satisfies 1.16 and estimate 1.17 is true Proof Now we consider the special case in 1.16 when f z 2.1 ∞ x, t and η ∈ C0 Rn for any constant ρ > 0, where z ≤ η ≤ 1, ρη z η≡1 in B1 × −1, , is a cutoff function satisfying η ≡ in Rn {B2 × −2, } 2.2 Therefore the problem 1.16 has the solution t u x, t 4π t − s n/2 Rn e−|x−y| /4 t−s f y, s dy ds 2.3 φ f dz ≤ C1 φ ρ 2.4 It follows from 1.17 , 2.1 , and 2.2 that Rn × 0,∞ φ |ut | dz ≤ C Rn × 0,∞ We know from 2.3 that ut x, t t 4π n/2 t−s n /2 B2 x−y n − t−s e−|x−y| /4 t−s f y, s dy ds 2.5 Define D : z √ x, t ∈ Rn × 0, ∞ : |x| > 4, |x| ≥ nt 2.6 Then when z ∈ D, t > s and y ∈ B2 , we have x−y |x|2 ≥ ≥ n, t−s 16t 2.7 Boundary Value Problems since x − y ≥ |x| − y ≥ |x| − Therefore, since |x − y| ≤ |x| |ut x, t | ≥ ≥ ≥ |x| |x| 2.8 |y| ≤ 2|x| for z ∈ D and y ∈ B2 , we conclude that t nρ · 4π n/2 t nρ · 4π n/2 t nρ · 4π n/2 t ≥ Cρ|x|−n t−s t−s e−|x−y| /4 t−s dy ds n /2 B1 y∈B1 t − s n−2 /2 2|x| n t−s x−y t−s e−|ξ| /4 dξ ds ξ n−2 /2 n −|ξ|2 /4 |ξ| e 2.9 dξ ds y∈B1 ds ≥ C2 ρ|x|−n tn/2 Recalling estimate 2.4 we find that φ C2 ρ|x|−n tn/2 dx dt ≤ C1 φ ρ , 2.10 D which implies that 1/n ∞ √ n φ C2 ρr −n r n−1 dr dt ≤ C1 φ ρ 2.11 By changing variable we conclude that, for any ρ > 0, αρ where α C3 φ ρ φ σ , dσ ≤ ρ σ 2.12 C2 4−n n−n/2 Let ρ2 ≥ ρ1 and < ε ≤ α/2 Then we conclude from 2.12 that φ ρ2 ≥ ρ2 C3 αρ2 φ ερ1 ≥ C3 φ σ dσ ≥ C3 σ2 1 − ερ1 αρ1 αρ1 ερ1 φ σ dσ σ2 2.13 φ ερ1 ≥ 2C3 ερ1 Now we use 2.12 and 2.13 to obtain that φ ρ ≥ ρ C3 αρ ερ φ ε2 ρ φ σ α dσ ≥ ln , 2 σ σ ε 2C3 ε ρ 2.14 Boundary Value Problems ερ, ρ2 where we choose that ρ1 1/ε2 Then we have σ in 2.13 Set a φ ρ ≥ √ ln α a ρ ρ ≥ 2aφ , a a aφ 2C3 2.15 when a is chosen large enough This implies that φ satisfies the ∇2 condition Thus this completes our proof 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 Let φ ∈ Δ2 ∩ ∇2 , g ∈ Lφ Rn Then for any b1 , b2 > one has ∞ μp , and p ∈ 1, α2 , where α2 is defined in 1.12 p {z∈Rn :|g|>b1 μ} ≤ C b1 , b2 , φ g dz d φ b2 μ Rn φ g dz 3.1 Moreover, we recall the following result Lemma 3.2 see 10, Theorem 5.5 Assume that g ∈ Lp Rn × 0, ∞ 2m,1 Rn × 0, ∞ of 1.1 - 1.2 with the estimate unique solution v ∈ Wp D2m v vt Lp Rn × 0,∞ Lp Rn × 0,∞ ≤C g for p > There exists a Lp Rn × 0,∞ 3.2 Moreover, we give one important lemma, which is motivated by the iteration-covering procedure in 12 To start with, let u be a solution of 1.1 - 1.2 Let p α2 3.3 > In fact, in the subsequent proof we can choose any constant p with < p < α2 Now we write p p λ0 while Rn × D2m u dz p Rn × 0,∞ f dz, 3.4 0,∞ ∈ 0, is a small enough constant which will be determined later Set uλ u , λ0 λ fλ f λ0 λ 3.5 Boundary Value Problems for any λ > Then uλ is still the solution of 1.1 - 1.2 with fλ replacing f Moreover, we write p − Jλ Q D2m uλ dz for any domain Q in Rn p − Q fλ dz 3.6 Q and the level set z ∈ Rn × 0, ∞ : D2m uλ > Eλ 3.7 Next, we will decompose the level set Eλ Lemma 3.3 For any λ > 0, there exists a family of disjoint cylinders {Qρi zi }i∈N with zi Eλ and ρi ρ zi , λ > such that Jλ Qρi zi 1, Jλ Qρ zi Eλ ⊂ < for any ρ > ρi , 3.8 Q5ρi zi ∪ negligible set, 3.9 i∈N where Q5ρi zi Qρi zi : B5ρi xi × ti − 5ρi ≤2 2m , ti 5ρi 2m p {z∈Qρi zi :|D2m uλ |p >1/4} D2m uλ dz xi , ti ∈ Moreover, one has p {z∈Qρi zi :|fλ |p > /4} fλ dz 3.10 Proof Fix any λ > We first claim that sup sup Jλ Qρ w w∈Rn × 0,∞ ρ≥ρ0 where ρ0 ρ0 λ > satisfies λp |Qρ0 | Then it follows from 3.4 that Jλ Qρ w ≤ p Qρ w ≤ p λ Qρ0 3.11 To prove this, fix any w ∈ Rn × 0, ∞ and ρ ≥ ρ0 p λ0 λp ≤ 1, Rn × 0,∞ D2m u dz p Rn × 0,∞ f dz 3.12 For a.e w ∈ Eλ , from Lebesgue’s differentiation theorem we have lim Jλ Qρ w ρ→0 > 1, 3.13 Boundary Value Problems which implies that there exists some ρ > satisfying Jλ Qρ w > 3.14 Therefore from 3.11 we can select a radius ρw ∈ 0, ρ0 such that Jλ Qρw w 1, Jλ Qρ w for any ρ > ρw 1/4} p {z∈Qρi zi :|fλ |p > fλ dz /4} 3.17 Qρi zi 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 D2m u ∈ Lφ Rn × 0, ∞ In view of Lemma 3.3, given any λ > 0, we can construct a family of cylinders {Qρi zi }i∈N , xi , ti ∈ Eλ Fix i ≥ It follows from 3.6 and 3.8 in Lemma 3.3 that where zi p D2m uλ dz ≤ 1, − p fλ dz ≤ − Q10ρi zi Q10ρi zi 3.18 We first extend fλ from Q10ρi zi to Rn by the zero extension and denote by f λ From 2m,1 Rn × 0, ∞ of Lemma 3.2, there exists a unique solution v ∈ Wp vt −Δ v 2m v fλ in Rn × 0, ∞ , in Rn × {t 3.19 0} with the estimate D2m v Lp Rn × 0,∞ ≤ C fλ Lp Rn × 0,∞ 3.20 10 Boundary Value Problems Therefore we see that D2m v Lp Q10ρi zi ≤ D2m v ≤ C fλ C fλ Set w Lp Rn × 0,∞ 3.21 Lp Rn × 0,∞ Lp Q10ρi zi uλ − v Then we know that −Δ wt 2m w 3.22 in Q10ρi zi Moreover, by 3.18 and 3.21 we have p p D2m w dz ≤ 2p − − Q10ρi zi D2m v dz Q10ρi zi ≤2 p p − D2m v dz Q10ρi zi 3.23 p fλ dz ≤ C C− Q10ρi zi 2m,1 Thus from the elementary interior W∞ regularity, we know that there exists a constant N1 > such that sup D2m w ≤ N1 3.24 Q5ρi zi Set μ λλ0 Therefore, we deduce from 3.5 and 3.24 that z ∈ Q5ρi zi : D2m u > 2N1 μ z ∈ Q5ρi zi : D2m uλ > 2N1 ≤ z ∈ Q5ρi zi : D2m w > N1 z ∈ Q5ρi zi : D2m v > N1 z ∈ Q5ρi zi : D2m v > N1 ≤ p N1 3.25 p D2m v dz Q5ρi zi Then according to 3.18 and 3.21 , we discover z ∈ Q5ρi zi : D2m u > 2N1 μ p ≤C fλ dz ≤ C Q10ρi zi Q10ρi zi 3.26 C Qρi zi Boundary Value Problems 11 Therefore, from 3.10 in Lemma 3.3 we find that z ∈ Q5ρi zi : D2m u > 2N1 μ ≤ C μp p {z∈Qρi zi :|D2m u|p >μp /4} 3.27 p D2m u dz f dz , {z∈Qρi zi :|f|> μp /4} C n, m Recalling the fact that the cylinders {Qρi zi }i∈N are disjoint, where C z ∈ Rn × 0, ∞ : D2m uλ z Q5ρi zi ∪ negligible set ⊃ Eλ >1 , i∈N 3.28 and then summing up on i ∈ N in the inequality above, we have z ∈ Rn × 0, ∞ : D2m u > 2N1 μ z ∈ Q5ρi zi : D2m u > 2N1 μ ≤ 3.29 i∈N ≤ C μp {z∈Rn × D2m u p dz {z∈Rn × :|D2m u|p >μp /4} 0,∞ f 0,∞ :|f|> p dz μp /4} Therefore, from Lemma 1.7 and the inequality above we have φ D2m u dz Rn × 0,∞ ∞ z ∈ Rn × 0, ∞ : D2m u > 2N1 μ d φ 2N1 μ ∞ ≤C μp ∞ C {z∈Rn × 0,∞ :|D2m u|p >μp /4} μp D 2m 3.30 p u dz d φ 2N1 μ p {z∈Rn × 0,∞ :|f|> μp /4} f dz d φ 2N1 μ Consequently, from Lemma 3.1 we conclude that Rn × 0,∞ φ D2m u dz ≤ C1 where C1 C1 n, m, φ and C2 C1 ≤ 1/2, we finish the proof Rn × 0,∞ φ D2m u dz C2 Rn × 0,∞ φ f dz, C2 n, m, , φ Finally selecting a suitable 3.31 ∈ 0, such that 12 Boundary Value Problems 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 References ă W Orlicz, Uber eine gewisse Klasse von Ră umen vom Typus B, Bulletin International de lAcad mie a e Polonaise S´ rie A, vol 8, pp 207–220, 1932 e R A Adams and J J F Fournier, Sobolev Spaces, vol 140, Academic Press, Amsterdam, The Netherlands, 2nd edition, 2003 S Byun, F Yao, and S Zhou, “Gradient estimates in Orlicz space for nonlinear elliptic equations,” Journal of Functional Analysis, vol 255, no 8, pp 1851–1873, 2008 H Jia, D Li, and L Wang, “Regularity of Orlicz spaces for the Poisson equation,” Manuscripta Mathematica, vol 122, no 3, pp 265–275, 2007 V Kokilashvili and M Krbec, Weighted inequalities in Lorentz and Orlicz spaces, World Scientific, River Edge, NJ, USA, 1991 F Yao, “Regularity theory in Orlicz spaces for the parabolic polyharmonic equations,” Archiv der Mathematik, vol 90, no 5, pp 429–439, 2008 M M Rao and Z D Ren, Applications of Orlicz Spaces, vol 250 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2002 D Gilbarg and N Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, Germany, 3rd edition, 1998 G M Lieberman, Second Order Parabolic Differential Equations, World Scientific, River Edge, NJ, USA, 1996 10 V A Solonnikov, “On boundary value problems for linear parabolic systems of differential equations of general form,” Trudy Matematicheskogo Instituta Imeni V A Steklova, vol 83, pp 3–163, 1965 11 F.-P Yao and S.-L Zhou, “Schauder estimates for parabolic equation of bi-harmonic type,” Applied Mathematics and Mechanics, vol 28, no 11, pp 1503–1516, 2007 12 E Acerbi and G Mingione, “Gradient estimates for a class of parabolic systems,” Duke Mathematical Journal, vol 136, no 2, pp 285–320, 2007  ... 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. .. 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 Let φ ∈ Δ2 ∩ ∇2 , g ∈ Lφ Rn Then for. .. World Scientific, River Edge, NJ, USA, 1991 F Yao, ? ?Regularity theory in Orlicz spaces for the parabolic polyharmonic equations,” Archiv der Mathematik, vol 90, no 5, pp 429–439, 2008 M M Rao and