SUBSPACE INTERPOLATION WITH ... - Semantic Scholar
SUBSPACE INTERPOLATION WITH APPLICATIONS TO ELLIPTIC REGULARITY CONSTANTIN BACUTA
Abstract. In this paper, we prove new embedding results by means of subspace interpolation theory and apply them to establishing regularity estimates for the biharmonic Dirichlet problem, and for the Stokes and the Navier-Stokes systems on polygonal domains. The main result of the paper gives a stability estimate for the biharmonic problem at the threshold index of smoothness. The classical regularity estimates for the biharmonic problem are deduced as a simple corollary of the main result. The subspace interpolation tools and techniques presented in this paper can be applied to establishing sharp regularity estimates for other elliptic boundary value problems on polygonal domains.
1. Introduction Regularity estimates or shift estimates for the solutions of elliptic Boundary Value Problems (BVPs) in terms of Sobolev-Besov norms, are of a significant interest in discretizing BVPs by finite differences methods, spectral methods and finite element methods. Shift estimates for the Laplace operator with Dirichlet boundary conditions on non-smooth domains are widely studied in the literature (see, e.g., [3, 8, 19, 22, 26, 5, 6, 7]). Shift estimates for the biharmonic problem on non-smooth domains, are presented in [12, 14, 4]. In this paper we extend the results of [4] and [14] by studying the regularity at the critical case. The shift results are proved by using the real method of interpolation of Lions and Peetre [9, 23, 24] and the subspace interpolation theory introduced by Kellogg in [19]. The approach based on subspace interpolation theory proves to be essential for the threshold regularity case. We describe a typical subspace interpolation problem as follows. Let X and Y be Sobolev spaces of integer order with X embedded in Y (X ⊂ Y ), and let XK be a subspace of X of finite codimension. Then, we need to characterize the interpolation spaces between Y and XK . In particular, we are looking for embedding relations between the intermediate subspaces [Y, XK ]s,q0 and the standard interpolation spaces [Y, X]s,q1 . Here, s ∈ (0, 1) can be viewed as the smoothness index, and q0 , q1 can be viewed as tuning parameters. In proving shift estimates for the biharmonic problem, we follow Kellogg’s approach in dealing with subspace interpolation. First, we study subspace interpolation embeddings on Sobolev spaces defined on all of R2 . Then, we apply the results to the polygonal domain case by finding ”extension” and “restriction” operators connecting the Sobolev spaces defined on bounded domains and the Sobolev spaces Date: December 14, 2007. 1991 Mathematics Subject Classification. 35B30, 35B65, 35Q30, 46B70. Key words and phrases. subspace interpolation, elliptic regularity, biharmonic operator, shift theorems, threshold index of smoothness, Stokes systems, Navier-Stokes systems. This work was supported by NSF, DMS-0713125. 1
2
C. BACUTA
defined on R2 . In contrast with Kellogg interpolation results presented in [19], we consider subspaces of codimension higher than one, and allow the tuning parameter q to be not necessarily 2. The shift estimates for Stokes and Navier-Stokes systems are based on the main regularity estimate for the biharmonic problem. Next, we describe the main regularity result of the paper. Let Ω be a polygonal domain in R2 with boundary ∂Ω. Let ∂Ω be the polygonal curve P1 P2 · · · Pm P1 . Assume that ∂Ω does not self intersect. For each point Pj , we denote by ωj the measure of the angle with the vertex at Pj and sides aligned with ∂Ω (measured from inside Ω). Let ω := max{ωj : j = 1, 2, . . . , m}. We consider the biharmonic problem: Given f ∈ L2 (Ω), find u such that 2 ∆ u = f in Ω, u = 0 on ∂Ω, (1.1) ∂u ∂n = 0 on ∂Ω. It is known that the solution of (1.1) satisfies kukH 2+2s (Ω) ≤ ckf kH −2+2s (Ω) ,
(1.2)
for all f ∈ H −2+2s (Ω), 0 ≤ s < s0 ,
where c is a positive constant, and s0 = s0 (ω) ∈ (0, 1) depends on the maximum angle of ∂Ω only, (see e.g., [4, 14, 25]). We will prove in this paper that for some positive constant c, we have 2+2s0 kukB∞ ≤ ckf kB −2+2s0 (Ω) , (Ω)
(1.3)
1
for all f ∈ B1−2+2s0 (Ω),
where 2+2s0 B∞ (Ω) := H 2 (Ω), H 4 (Ω) s
0 ,∞
and B1−2+2s0 (Ω) := H −2 (Ω), L2 (Ω) s
0 ,1
.
Note that H 2+2s (Ω) = H 2 (Ω), H 4 (Ω) s,2 and H −2+2s (Ω) = H −2 (Ω), L2 (Ω) s,2 . In other words, we prove that the estimate (1.2) holds for the threshold regularity index s = s0 , provided the second index of interpolation q is taken q = 1 for measuring the data f , and q is taken q = ∞ for measuring the solution u. The precise definitions of the interpolation spaces are given in the next section. We note that (1.2) can be easily proved as a consequence of (1.3) and the reiteration theorem. In addition, the proof of (1.3) is considerably simpler than the proof of (1.2) as given in [4], (see Section 4). Section 2 and Section 3 contain interesting theoretical embedding results which can be used in deriving regularity estimates for other elliptic BVPs that are not contained in this paper. For example, one can recover the regularity estimates at the critical case for the Laplace operator with Dirichlet boundary conditions which are presented in [8] by means of multilevel representation of norms. In addition, one can use the approach of this paper and prove estimates at the threshold regularity index for the Laplace operator with mixed Dirichlet-Neuman boundary conditions. The remaining part of the paper is organized as follows. In Section 2 we present subspace interpolation results in an abstract setting. In Section 3, we prove embedding results for interpolation spaces defined on the whole R2 . The main shift estimate (1.3) is proved in Section 4. Regularity estimates for the Stokes and Navier-Stokes systems are presented in Section 5. In the Appendix, we present the technical proofs that were omitted in the prior sections.
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
3
Acknowledgments. Many thanks go to Jim Bramble, Bruce Kellogg, Joe Pasciak, and Jinchao Xu for guidance and valuable discussions on the subject. 2. Interpolation results In this section, we start with some basic definitions and results concerning interpolation between subspaces of Hilbert spaces using the real method of interpolation of Lions and Peetre (see [23]). In the second subsection, we present subspace interpolation theory. 2.1. Interpolation between Banach and Hilbert spaces. Let X, Y be Banach spaces satisfying the following conditions: X is a dense subset of Y and (2.1) kukY ≤ ckukX for all u ∈ X, for some positive constant c. For s ∈ (0, 1), 1 ≤ q ≤ ∞, the intermediate space [Y, X]s,q is defined using the K function. For u ∈ Y and t > 0, we define K(t, u, Y, X) = K(t, u) := inf (ku − u0 k2Y + t2 ku0 k2X )1/2 . u0 ∈X
For s ∈ (0, 1) and 1 ≤ q ≤ ∞, the space [Y, X]s,q is defined by [Y, X]s,q := {u ∈ Y : kuk[Y,X]s,q < ∞}, where Z kuk[Y,X]s,q =
∞
(t
−s
q dt
K(t, u))
0
1/q
t
for q < ∞,
and kuk[Y,X]s,∞ = sup t−s K(t, u). t>0
By definition, we take [Y, X]0,q := Y and [Y, X]1,q := X. The main properties of the interpolation between Banach spaces can be found in [9, 10, 13]. Next, we will focus on interpolation between Hilbert spaces. Let us assume that, in addition, X, Y are separable Hilbert spaces with inner products (·, ·)X and (·, ·)Y , and the norms on the two spaces are the norms induced by the corresponding inner products (kuk2X = (u, u)X and kuk2Y = (u, u)Y ). Let D(S) denote the subset of X consisting of all elements u such that the anti-linear form (2.2)
v → (u, v)X , v ∈ X,
is continuous in the topology induced by Y . For any u in D(S) the anti-linear form (2.2) can be extended to a continuous anti-linear form on Y . Then, by the Riesz representation theorem (see e.g., [1, 29]), there exists an element Su in Y such that (2.3)
(u, v)X = (Su, v)Y
for all v ∈ X.
In this way, S is a well defined operator in Y with domain D(S). The next result illustrates the properties of S, see, e.g. [23, 28].
4
C. BACUTA
Proposition 2.1. The domain D(S) of the operator S is dense in X and consequently D(S) is dense in Y . The operator S : D(S) ⊂ Y → Y is a bijective, selfadjoint and positive definite operator. The inverse operator S −1 : Y → D(S) ⊂ Y is a bounded symmetric positive definite operator and (2.4)
(S −1 z, u)X = (z, u)Y
for all z ∈ Y, u ∈ X.
If X,Y are Hilbert spaces, we have a better representations of the intermediate space [Y, X]s,q . The next lemma provides the relation between K(t, u) and the connecting operator S, and consequently, gives the representation of [Y, X]s,q in terms of S. A similar result can be found in [23]. Lemma 2.2. For all u ∈ Y and t > 0 , K(t, u)2 = (I + (t2 S)−1 )−1 u, u
(2.5)
∞
Z (2.6)
kuk[Y,X]s,q =
t
sq
2
(I + t S
0
−1 −1
)
,
Y
q/2 dt u, u Y t
1/q , 1 ≤ q < ∞,
and (2.7)
kuk[Y,X]s,∞ = sup ts (I + t2 S −1 )−1 u, u t>0
1/2 Y
.
Proof. Using the density of D(S) in X, we have 2
K(t, u) =
inf
u0 ∈D(S)
(ku0 k2X + t2 ku − u0 k2Y )
Let v = Su0 . Then (2.8)
2
K(t, u) = inf (ku − S −1 vk2Y + t2 (S −1 v, v)Y ). v∈Y
By solving the minimization problem (2.8), we obtain that the element v, which gives the optimum, satisfies the following equations: (t2 I + S −1 )v = u, and ku − S −1 vk2Y + t2 (S −1 v, v)Y = (I + (t2 S)−1 )−1 u, u Y . To obtain (2.6) and (2.7) we replace K in the definition of [Y, X]s,q with the expression given by (2.5) and apply the change of variable 1/t = τ . Lemma 2.3. Let X0 be a closed subspace of X, and let Y0 be a closed subspace of Y . Let X0 and Y0 be equipped with the topologies and the geometries induced by X and Y , respectively, and assume that the pair (X0 , Y0 ) satisfies (2.1). Then, for s ∈ [0, 1], 1 ≤ q ≤ ∞, [Y0 , X0 ]s,q ⊂ [Y, X]s,q ∩ Y0 . Proof. For any u ∈ Y0 , we have K(t, u, Y, X) ≤ K(t, u, Y0 , X0 ). Thus, (2.9)
kuk[Y,X]s,q ≤ kuk[Y0 ,X0 ]s,q
which proves the lemma.
for all u ∈ [Y0 , X0 ]s,q ,
The validity of the other inclusion will be addressed in the next subsection for cases related to the regularity for the biharmonic problem.
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
5
2.2. Subspace interpolation on Hilbert spaces. Let K = span{ϕ1 , . . . , ϕn } be a n-dimensional subspace of X, and let XK be the orthogonal complement of K in X in the (·, ·)X inner product. We are interested in determining the interpolation spaces of Y and XK , where on XK we consider again the (·, ·)X inner product. For certain spaces X, K, Y , n = 1, and q = 2, this problem was studied in [19]. To apply the interpolation results from the previous section we need to check that the density part of the condition (2.1) is satisfied for the pair (XK , Y ). For ϕ ∈ K, we define the linear functional Λϕ : X → C by Λϕ u := (u, ϕ)X , u ∈ X. The following lemma can be found in [4]. Lemma 2.4. The space XK is dense in Y if and only if the following condition is satisfied: Λϕ is not bounded in the topology of Y (2.10) for all ϕ ∈ K, ϕ 6= 0. For the remaining part of this section, we assume that the condition (2.10) holds. By the above lemma, the condition (2.1) is satisfied. It follows from the previous section that the operator SK : D(SK ) ⊂ Y → Y defined by (2.11)
(u, v)X = (SK u, v)Y
for all v ∈ XK ,
has the same properties as S has. Consequently, the norm on the intermediate space [Y, XK ]s,q is given by Z ∞ q/2 dt 1/q sq 2 −1 −1 , 1 ≤ q < ∞, (2.12) kuk[Y,XK ]s,q = t (I + t SK ) u, u Y t 0 and (2.13)
−1 −1 kuk[Y,XK ]s,∞ = sup ts (I + t2 SK ) u, u t>0
1/2 Y
.
In this section, our aim is to study the embedding relation between [Y, XK ]s,q1 and [Y, X]s,q2 . Lemma 2.5. Assume that for the fixed parameters s0 ∈ (0, 1), q0 , q1 ∈ [1, ∞], we have the following continuous embedding [Y, X]s0 ,q1 ⊂ [Y, XK ]s0 ,q0 .
(2.14) Then,
[Y, XK ]s,q = [Y, X]s,q , 0 < s < s0 , q ∈ [1, ∞]. Proof. From Lemma 2.3, we have that [Y, XK ]s,q ⊂ [Y, X]s,q , 0 ≤ s < s0 . On the other hand, from (2.14), the inclusion operator is continuous from [Y, X]s0 ,q1 to [Y, XK ]s0 ,q0 , and clearly, it is continuous from [Y, X]0,2 = Y to [Y, XK ]0,2 = Y . By interpolation, and by the reiteration theorem (see Appendix 6.2), we get [Y, X]s,q ⊂ [Y, XK ]s,q , 0 < s < s0 , which completes the proof.
6
C. BACUTA
We note here that, for a fixed s0 ∈ (0, 1), the weakest assumption of type (2.14) is for q0 = ∞ and q1 = 1. Next, we study the validity of (2.14) for q0 = ∞ and q1 = 1. First, we note that the operators SK and S are related by the following identity: −1 SK = (I − QK )S −1 ,
(2.15)
where QK : X → K is the orthogonal projection onto K. The proof of (2.15) follows easily from the definitions of the operators S, QK and SK . The identity (2.15) leads to a formula relating the norms on [Y, XK ]s,q and [Y, X]s,q . To get the formula, we introduce first the following notation. Let (2.16) (u, v)X,t := (I + t2 S −1 )−1 u, v X for all u, v ∈ X, and (u, v)Y,t := (I + t2 S −1 )−1 u, v
(2.17)
Y
for all u, v ∈ Y.
Denote by Mt the Gram matrix associated with the set of vectors {ϕ1 , . . . , ϕn } in the (·, ·)X,t inner product, i.e., (Mt )ij := (ϕj , ϕi )X,t , i, j ∈ {1, . . . , n}. Theorem 2.6. For any u ∈ Y , we have kuk2[Y,XK ]s,2
(2.18)
=
kuk2[Y,X]s,2
∞
Z + 0
dt
t2+2s Mt−1 yt , yt , t
kuk2[Y,XK ]s,∞ = sup t2s (u, u)Y,t + t2+2s Mt−1 yt , yt ,
(2.19)
t>0
where h·, ·i is the inner product on Cn , and yt is the n-dimensional vector in Cn whose components are (yt )i := (u, ϕi )Y,t , i = 1, . . . , n. The proof is given in Appendix 6.1. For n = 1, let K = span{ϕ} and denote XK by Xϕ . Then, the formulas (2.18) and (2.19) become Z ∞ 2 |(u, ϕ)Y,t | dt (2.20) kuk2[Y,Xϕ ]s,2 = kuk2[Y,X]s,2 + t2+2s , (ϕ, ϕ)X,t t 0 and kuk2[Y,Xϕ ]s,∞
(2.21)
2s
= sup t (u, u)Y,t + t
2 2+2s |(u, ϕ)Y,t |
t>0
(ϕ, ϕ)X,t
! ,
respectively. The next theorem gives sufficient conditions for (2.14) to be satisfied. Before we state the result, we fix si ∈ (0, 1), i = 0, 1, . . . n, and define s0 = min{s1 , s2 , . . . sn }. We introduce the following two conditions: (A.1) [Y, X]s0 ,1 ⊂ [Y, Xϕi ]s0 ,∞ , for all i = 1, . . . , n. (A.2) There exist δ > 0 and γ > 0 such that n X i=1
2
|αi | (ϕi , ϕi )X,t ≤ γ hMt α, αi
for all α = (α1 , . . . , αn )t ∈ Cn , t ∈ (δ, ∞).
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
7
Theorem 2.7. Assume that the conditions (A.1) and (A.2) hold. Then [Y, X]s0 ,1 ⊂ [Y, XK ]s0 ,∞ ,
(2.22) and
[Y, XK ]s,q = [Y, X]s,q , 0 < s < s0 , q ∈ [1, ∞].
(2.23)
Proof. Due to (A.1), for a fixed u in [Y, X]s0 ,1 , we have that kuk[Y,Xϕi ]si ,∞ < ∞. Using the notation of the previous theorem, we get kuk2[Y,XK ]s
= sup(t2s0 (wK , u)Y ).
0 ,∞
t>0
On the other hand, −1 −1 (wK , u)Y = (I + t2 SK ) u, u
Y
−1 −1 −1 (I + t2 SK ) u, u = (u, u)Y − t2 SK
Y
2
≤ (u, u)Y ≤ ckuk[Y,X]s
0 ,1
Thus, from (6.2) and (2.7), we obtain kuk2[Y,XK ]s
0 ,∞
≤ sup (t2s0 (wK , u)Y ) + sup(t2s0 (wK , u)Y ) 0δ
≤
2 c(δ)kuk[Y,X]s ,1 0
≤
2 ckuk[Y,X]s ,1 0
+ sup(t2s0 (u, u)Y,t ) + sup(t2+2s0 Mt−1 yt , yt ) t>δ
+ sup(t
t>δ
2+2s0
t>δ
Mt−1 yt , yt
).
Next, (A.2) is equivalent to
n X 2 Mt−1 α, α ≤ γ |αi | (ϕi , ϕi )−1 X,t
for all α = (α1 , . . . , αn )t ∈ Cn , t ∈ (δ, ∞).
i=1
In particular, for αi = (yt )i = (u, ϕi )Y,t , i = 1, . . . , n, we obtain
Mt−1 yt , yt
≤γ
n 2 X |(u, ϕi )Y,t | i=1
(ϕi , ϕi )X,t
for all t ∈ (δ, ∞), u ∈ [Y, X]s0 ,1 .
Consequently, using (2.21) and (A.1), we have kuk2[Y,XK ]s ,∞ 0
≤
2 ckuk[Y,X]s ,1 0 2
≤ ckuk[Y,X]s
0 ,1
+ γ sup t t>δ
+γ
n X
2+2s0
n 2 X |(u, ϕi )Y,t | i=1
kuk2[Y,Xϕ
!
(ϕi , ϕi )X,t 2
] i s0 ,∞
≤ c(n, δ, γ)kuk[Y,X]s
0 ,1
.
i=1
This concludes (2.22). The identity (2.23) follows from (2.22) and Lemma 2.5. 3. Interpolation between subspaces of H β (RN ) and H α (RN ). In this section, we apply our abstract interpolation result to the particular case Y = H α (RN ), X = H β (RN ). The codimension one case with q = 2 was analyzed for the first time by Kellogg in [19]. A finite codimension case with q = 2 is considered in [4]. Using Theorem 2.7, we prove a new subspace interpolation embedding and recover the main interpolation results of [19] and [4]. The proofs are based on techniques used in [19] and [4], but are considerably simplified. For the proof of the regularity for the biharmonic problem we will need only the last theorem of this
8
C. BACUTA
section. The reader not interested in the proof of the embedding tools might skip this section. Let α ∈ R and let H α (RN ) be defined by means of the Fourier transform. For a smooth function u with compact support in RN , the Fourier transform u ˆ is defined by Z u ˆ(ξ) = (2π)−N/2
u(x)e−ix·ξ dx,
where the integral is taken over the whole RN . For u and v smooth functions the α-inner product is defined by Z hu, viα = (1 + |ξ|2 )α u ˆ(ξ)ˆ v (ξ) dξ. The space H α (RN ) is the closure of smooth functions with compact support in the norm induced by the α-inner product. For α, β real numbers, α < β, and s ∈ [0, 1] it is easy to check, using (2.6), that α N H (R ), H β (RN ) s,2 = H (1−s)α+sβ (RN ). β N For ), we in determining αϕ ∈NH (R are interested the embedding relation between β H (R ), Hϕ (RN ) s,q and H α (RN ), H β (RN ) s,q . We recall that Hϕβ is the or1
0
thogonal complement of span{ϕ} in H β . For simplicity we shall start with the case α = 0. We note that H 0 (RN) = L2 (RN ). The operator S associated with the pair (X, Y ) = H β (RN ), H 0 (RN ) is given by c = µ2β u Su ˆ, u ∈ D(S) = H 2β (RN ), 1
where µ(ξ) = (1 + |ξ|2 ) 2 , ξ ∈ RN . For the remaining part of this chapter, H β ˆ β is the space {ˆ ˆ β , we denotes the space H β (RN ) and H u |u ∈ H β }. For u ˆ, vˆ ∈ H define the inner product and the norm by Z 1/2 (ˆ u, vˆ)β = µ2β u ˆvˆ dζ, ||ˆ u||β = (ˆ u, u ˆ)β . To simplify the notation, we denote the the inner products (·, ·)0 and h·, ·i0 by (·, ·) ˆ 0 is simply || · ||. Thus, for and h·, ·i, respectively. The norm || · ||0 on H 0 or H ˆ β and Y = H ˆ 0 , we have X=H 2β 4β µ u ˆ µ u ˆ (ˆ u, φ)Y,t = , φ and (ˆ u , φ) = , φ . X,t µ2β + t2 µ2β + t2 ˆ β be such that for some constants > 0 and c > 0, Let φ ∈ H N N |φ(ξ) − b(ω)ρ− 2 −2β+α0 | < cρ− 2 −2β+α0 − for all ρ > 1, (3.1) 0 < α0 < β, where ρ ≥ 0 and ω ∈ S N −1 (the unit sphere of RN ) are the spherical coordinates of ξ ∈ RN , and where b(ω) is a bounded measurable function on S N −1 , which is non zero on a set of positive measure. Remark 3.1. By using Lemma 2.4, under the assumption (3.1) about φ, we have that ˆ β is dense in H ˆ α1 if and only if α1 ≤ α0 . H (3.2) φ ˆ β is dense in H ˆ 0 and the pair (H ˆ β, H ˆ 0 ) satisfies Thus, in particular, we have that H φ φ (2.1).
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
9
Theorem 3.2. Let ϕ ∈ H β be such that its Fourier transform φ satisfies (3.1), and let θ0 = α0 /β. Then 0 β (3.3) H , H θ ,1 ⊂ H 0 , Hϕβ θ ,∞ , 0
0
and (3.4)
0 β H , Hϕ θ,q = H 0 , H β θ,q , 0 < θ < θ0 , q ∈ [1, ∞].
Proof. By the definition of the inner product on H β , we have that the embedding (3.3) is equivalent to i h h i ˆ 0, H ˆβ ˆ 0, H ˆβ (3.5) H ⊂ H . φ θ0 ,1
θ0 ,∞
The identity (3.4) follows from (3.3) and Lemma 2.5. Thus, we have to prove only (3.5). From (2.6), ∞
Z (3.6)
kˆ uk[Hˆ 0 ,Hˆ β ]θ
0 ,1
tθ0
=
Z
0
µ(ξ)2β |ˆ u(ξ)|2 dξ µ(ξ)2β + t2
1/2
dt , t
and due to (2.21) we get 2
(3.7)
kˆ uk2Hˆ 0 ,Hˆ β ] [
φ θ ,∞ 0
= sup t
2θ0
(ˆ u, u ˆ)Y,t + t
2+2θ0
t>0
|(ˆ u, φ)Y,t | (φ, φ)X,t
! ,
ˆ β and Y = H 0 . Using (3.1), it is easy to see that, for a large enough where X = H δ ≥ 1, one can find positive constants c1 , c2 , such that
(3.8)
c1 tθ0 −1 ≤ ((φ, φ)X,t )1/2 =
µ4β φ ,φ 2β µ + t2
1/2
N
|φ(ξ)| < c2 |ρ|− 2 −2β+α0
≤ c2 tθ0 −1 , t ≥ δ,
for |ξ| > 1.
Following the proof of Theorem 2.7, we see that, in order to prove (3.5), it is enough to verify the following inequality (3.9) M := sup t1+θ0 t>δ
|(ˆ u, φ)Y,t | ≤ ckˆ uk[Hˆ 0 ,Hˆ β ]θ ,1 0 ((φ, φ)X,t )1/2
ˆ 0, H ˆ β ]θ ,1 , for all u ˆ ∈ [H 0
for some positive constants c = c(δ). From (3.8), we get Z µ(ξ)2β M ≤ c sup t2 |(ˆ u, φ)Y,t | ≤ c sup t2 |ˆ u(ξ)||φ(ξ)| dξ µ(ξ)2β + t2 t>δ t>δ Z Z Z (3.10) 2β 2β ≤ µ(ξ) |ˆ u(ξ)||φ(ξ)| dξ = µ(ξ) |ˆ u||φ| dξ + µ(ξ)2β |ˆ u||φ| dξ. |ξ|1
The last two integrals can be estimated as follows: Z (3.11) µ(ξ)2β |ˆ u||φ| dξ ≤ c kˆ ukkφk ≤ c(φ)kˆ uk[Hˆ 0 ,Hˆ β ]θ |ξ|1
|ξ|>1
2 = π
Z
2 π
Z
=
∞
0
µ3β |ˆ u||φ| dξ dt µ2β + t2
Z |ξ|>1
∞
µβ |ˆ u| µ2β |φ| dξ dt 2 1/2 2β + t ) (µ + t2 )1/2
Z
(µ2β
0
|ξ|>1
1/2
≤
2 π
∞
Z
2β
0
2
µ |ˆ u| dξ µ2β + t2
Z
∞
Z 0
0
2
µ |φ| dξ µ2β + t2
dt
|ξ|>1
µ2β |ˆ u|2 dξ µ2β + t2
Z
1/2
((φ, φ)X,t )
dt
|ξ|>1
1/2
≤c
4β
1/2
∞
Z
|ξ|>1
2 ≤ π
1/2 Z
tθ0
Z
2β
2
µ |ˆ u| dξ µ2β + t2
dt ≤ ckˆ uk[Hˆ 0 ,Hˆ β ]θ ,1 . 0 t
|ξ|>1
Combining (3.10)-(3.12), we conclude the validity of (3.9) and the proof of the theorem. Next, we prepare for the generalization of the previous result. Let φ1 , φ2 , . . . , φn ∈ ˆ β (RN ) be such that for some constants > 0 and c > 0, we have H N |φi (ξ) − φ˜i (ξ)| < cρ− 2 −2β+αi − f or |ξ| > 1 (3.13) 0 < αi < β, i = 1, . . . , n, where N φ˜i (ξ) = bi (ω)ρ− 2 −2β+αi , ξ = (ρ, ω),
and bi (·) is a bounded measurable function on S N −1 , which is non zero on a set of positive measure. Using Lemma 2.4 and the Riesz representation theorem, it is β easy to check that, under the assumption (3.1), the space HK is dense in H 0 . Theorem 3.3. Let ϕ1 , ϕ2 , . . . , ϕn be functions in H β such that the corresponding Fourier transforms φ1 , φ2 , . . . , φn are well defined and satisfy (3.13). Assume that the functions φ˜1 , φ˜2 , . . . , φ˜n defined in (3.13) are linearly independent. Let K = span{ϕ1 , ϕ2 , . . . , ϕn }, α0 := min{α1 , α2 , . . . , αn }, and let θ0 = α0 /β. Then, h i 0 β β (3.14) H , H θ ,1 ⊂ H 0 , HK , 0
θ0 ,∞
and (3.15)
h
β H 0 , HK
i θ,q
= H 0 , H β θ,q , 0 < θ < θ0 , q ∈ [1, ∞].
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
11
Proof. We apply the Theorem 2.7 for X = H β , Y = H 0 , K = span{ϕ1 , . . . , ϕn } and s0 = θ0 . By using the hypothesis (3.13) and Theorem 3.2, we get 0 β H , H θ ,1 ⊂ H 0 , Hϕβi θ ,∞ , f or i = 1, 2, . . . , n. 0
0
Thus, (A1) is satisfied. The proof of (A2) follows from (3.13) by using elementary linear algebra and calculus tools, and it is presented in [4]. The identity (3.15), is a direct consequence of Lemma 2.5. The proof is completed. The interpolation problem between H α and a subspace of H β of finite codimension with α < β arbitrary real numbers, can be approached in the light of the previous theorem. Let ϕ1 , ϕ2 , . . . , ϕn ∈ H β and let α < β be such that the corresponding Fourier transform, φ1 , φ2 , . . . , φn are well defined and satisfy for some positive constants c and , N |φi (ξ) − φ˜i (ξ)| < cρ− 2 −2β+γi − f or |ξ| > 1 (3.16) α < γi < β, i = 1, . . . , n, where N φ˜i (ξ) = bi (ω)ρ− 2 −2β+γi , ξ = (ρ, ω),
and bi (·) is a bounded measurable function on S N −1 , which is non zero on a set of positive measure. Theorem 3.4. Let ϕ1 , ϕ2 , . . . , ϕn ∈ H β be such that the corresponding Fourier transforms φ1 , φ2 , . . . , φn are well defined and satisfy (3.16). Assume that the functions φ˜1 , φ˜2 , . . . , φ˜n are linearly independent. Let L = span{ϕ1 , ϕ2 , . . . , ϕn }, γ0 := min{γ1 , γ2 , . . . , γn }, and let θ0 = (γ0 − α)/(β − α). Then, h i α β (3.17) H , H θ ,1 ⊂ H α , HLβ , 0
θ0 ,∞
and (3.18)
h
H α , HLβ
i θ,q
= H α , H β θ,q , 0 < θ < θ0 , q ∈ [1, ∞].
Proof. The proof is a direct consequence of Theorem 3.3 and the fact that T : H α → H 0 defined by Tˆu = µα u ˆ, u ∈ H α , is an isometry from H α to H 0 and from β β−α H to H . 4. Shift estimates for the Biharmonic operator on polygonal domains Let Ω be a polygonal domain in R2 with boundary ∂Ω. Let ∂Ω be the polygonal curve P1 P2 · · · Pm P1 . At each point Pj , we denote by ωj the measure of the angle at Pj measured from inside Ω. Let ω := max{ωj : j = 1, 2, . . . , m}. We consider the biharmonic problem: Given f ∈ L2 (Ω), find u such that (1.1). Let V = H02 (Ω) and X Z ∂2v ∂2u dx, u, v, ∈ V. a(u, v) := Ω ∂xi ∂xj ∂xi ∂xj 1≤i,j≤2
The bilinear form a(·, ·) defines a scalar product on V and the induced norm is equivalent to the standard norm on H02 (Ω). The variational form of (1.1) is : Find u ∈ V such that Z (4.1) a(u, v) = f v dx for all v ∈ V. Ω
12
C. BACUTA
Let Sω denote a sector domain defined by Sω = {(r, θ), 0 < r < rω , −ω/2 < θ < ω/2}, where, without loss of generality, we assume that Sω ⊂ Ω for a sufficiently small rω , and that the vertex corresponding to the largest angle of Ω is the vertex of the sector domain Sω . We associate to (1.1), with Ω = Sω , the following characteristic equation sin2 (zω) = z 2 sin2 ω.
(4.2)
In order to simplify the exposition of the proof, we assume that r r ω2 sin ω 2 − 1 6= + 1 − (4.3) sin 2 − sin ω ω2 and Rez 6= 2 for any solution z of (4.2).
(4.4)
The restriction (4.3) assures that the equation (4.2) has only simple roots. Let z1 , z2 , . . . , zn be all the roots of (4.2) such that 0 < Re(zj ) < 2. It is known that, under the assumptions (4.3) and (4.4), the solution u of (4.1) can be written as n X (4.5) u = uR + kj Sj , j=1 4
where uR ∈ H (Sω ) is the regular part of u, the Sj ’s are called the singular functions, and the kj ’s are the coefficients of the singular functions. More precisely, for j = 1, 2, . . . , n, we have Sj (r, θ) = r1+zj uj (θ), with uj smooth func0 0 tion on [−ω/2, R ω/2] satisfying uj (−ω/2) = uj (ω/2) = uj (−ω/2) = uj (ω/2) = 0, and kj = cj Sω f ϕj dx, with cj being nonzero and depending only on ω. The function ϕj is called the dual singular function of the singular function Sj and ϕj (r, θ) = η(r) r1−zj uj (θ) − wj , where wj ∈ V is defined for a smooth truncation function η to be the solution of (4.1) with f = ∆2 (η(r) r1−zj uj (θ)). In addition, we have that the regular part uR satisfies (4.6)
kuR kH 4 (Sω ) ≤ ckf k,
for all f ∈ L2 (Sω ).
For the proof of the expansion (4.5), see, e.g., [14, 16, 17, 21, 25]. Next, we define K = span{ϕ1 , ϕ2 , . . . , ϕn } and 1 (4.7) s0 := min{Re(zj ) | j = 1, 2, . . . , n}. 2 The main theorem of the paper is based on the following embedding result: Lemma 4.1. For Ω = Sω and K defined above, we have (4.8)
[H −2 (Ω), L2 (Ω)]s0 ,1 ⊂ [H −2 (Ω), L2 (Ω)K ]s0 ,∞ .
To prove it, we first reduce the problem to the case Ω = R2 and then apply the interpolation results of the previous section. A detailed proof is given in Appendix 6.1. Now, we are ready to state the main result of the paper. Theorem 4.2. Let Ω be a polygonal domain in R2 with boundary ∂Ω. Assume that all the angles ωj of ∂Ω satisfy (4.3) and (4.4). Let s0 = s0 (ω) be the threshold defined by (4.7), where ω is the largest inner angle of the polygon ∂Ω. If u is the variational solution of (1.1), then the regularity estimate (1.3) holds. Consequently, the classical estimate (1.2) holds also.
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
13
Proof. We consider a covering of Ω with m + 1 subdomains. For each angle of ∂Ω, we associate a sector domain Sωj ⊂ Ω and complete the covering of Ω with a domain Ω0 ⊂ Ω such that Ω0 has smooth boundary and ∂Ω0 does not contain any vertex of ∂Ω. As done in [3], by using a smooth partition of unity subordinated to the described covering of Ω, the problem of deriving a shift estimate on Ω can be reduced to deriving similar estimates on each of the subdomains of the covering. Since the amount of regularity of the solution of (1.1) for a sector domain decreases as the angle of the a sector domain increases, (see Figure 1), to prove (1.3), it will be enough to prove the estimate for Ω = Sω and for Ω = Ω0 . First, let us consider the case Ω = Ω0 . Then, it is known that the solution u of (4.1) satisfies kukH 4 (Ω) ≤ ckf k,
for all f ∈ L2 (Ω),
and for all f ∈ H −2 (Ω).
kukH 2 (Ω) ≤ ckf kH −2 (Ω) ,
Interpolating these two inequalities with 0 < s < 1 and 1 ≤ q ≤ ∞, we obtain (4.9) kuk[H 2 (Ω),H 4 (Ω)]s,q ≤ ckf k[H −2 (Ω),L2 (Ω)]s,q , f ∈ H −2 (Ω), L2 (Ω) s,q . In particular, taking s = s0 and q = ∞ we get (4.10)
kuk[H 2 (Ω),H 4 (Ω)]s
0 ,∞
≤ ckf k[H −2 (Ω),L2 (Ω)]s
0 ,∞
, f ∈ H −2 (Ω), L2 (Ω) s
0 ,∞
.
Using the standard embedding result [H −2 (Ω), L2 (Ω)]s0 ,1 ⊂ [H −2 (Ω), L2 (Ω)]s0 ,∞ , and (4.10), we get , f ∈ H −2 (Ω), L2 (Ω) s ,1 . 0 2 2+s0 4 Thus, by introducing the Besov spaces B∞ (Ω) := H (Ω), H (Ω) s ,∞ and 0 B1−2+s0 (Ω) := H −2 (Ω), L2 (Ω) s ,1 , the estimate (4.11) becomes (1.3). 0 Let us consider now the case Ω = Sω. From the expansion (4.5) and the estimate (4.6), we have (4.11)
(4.12)
kuk[H 2 (Ω),H 4 (Ω)]s
0 ,∞
≤ ckf k[H −2 (Ω),L2 (Ω)]s
kukH 4 (Ω) ≤ ckf k,
0 ,1
for all f ∈ L2 (Ω)K .
Combining (4.12) with the standard estimate (4.13)
kukH 2 (Ω) ≤ ckf kH −2 (Ω) ,
for all f ∈ H −2 (Ω),
by interpolation, we obtain (4.14)
kuk[H 2 (Ω),H 4 (Ω)]s0 ,∞ ≤ ckf k[H −2 (Ω),L2 (Ω)K ]s0 ,∞ .
From (4.14) and the embedding result (4.8) of Lemma 4.1, we conclude that (1.3) holds for Ω = Sω . Here, s0 corresponds to the largest inner angle ω of the polygon. The function ω → 2 + 2s0 (ω) represents the regularity threshold for the biharmonic problem on Sω . From the plot of the function, given in Figure 1, we see that function decreases on the interval (0, 2π). Thus, by involving interpolation and the reiteration theorem, (see part iii) and part iv) of Proposition 6.2), the estimate (1.3) holds for all sector domains Sωj , j = 1, 2, . . . , m.
14
C. BACUTA
6
5
4
3
2
1
0
omega
.7Pi
1.23Pi
1.43Pi
2Pi
Figure1: Regularity for the biharmonic problem Therefore, we have proved (1.3) for any polygonal domain with not self intersecting boundary. Next, we prove the standard estimate (1.2). Let T : H −2 (Ω) → H02 (Ω) be defined by T f = u, where u is the solution of (4.1). Then, (1.3) becomes (4.15)
kT f k[H 2 ,H 4 ]s0 ,∞ ≤ ckf k[H −2 ,H 0 ]s0 ,1 ,
and the standard estimate (4.13) can be written us (4.16)
kT f k[H 2 ,H 4 ]0,2 ≤ ckf k[H −2 ,H 0 ]0,2 .
Let 0 < s < s0 . By Proposition 6.2 part iii) (“By interpolation” Theorem) , applied with λ = ss0 and q = 2, we obtain (4.17)
kT f k[[H 2 ,H 4 ]0,2 ,[H 2 ,H 4 ]s
]
0 ,∞ λ,2
≤ ckf k[[H −2 ,H 0 ]0,2 ,[H −2 ,H 0 ]s
]
0 ,1 λ,2
.
By applying the reiteration theorem, we recover the classical estimate (1.2). The proof is completed. Remark 4.3. Figure 1 gives the graph of the function ω → 2 + 2s0 (ω) which represents also the regularity threshold for the biharmonic problem in terms of the largest inner angle of the polygon ω ∈ (0, π) ∪ (π, 2π). From (4.2) and (4.7), it follows that s0 (ω) → 1/2 for ω → π and s0 (ω) → 1/4 for ω → 2π. On the same graph we represent the number of singular (dual singular) functions as a function of ω ∈ (0, π) ∪ (π, 2π). Note that, if ω0 ∈ (0, 2π) is the solution of tan ω = ω (ω0 ≈ 1.43π) and ω > ω0 , then, the the dimension of the space K is six. Remark 4.4. If the coefficient of the most singular function in the expansion (4.5) is not zero, then the solution u of (1.1) does not belong to Bq2+2s0 (Ω) for any q < ∞. Thus, any possible improvement of the shift estimate (1.3) would be in the form: (4.18)
2+2s0 kukB∞ ≤ ckf kBq−2+2s0 (Ω) , f ∈ Bq−2+2s0 (Ω), q > 1. (Ω)
The validity of (4.18) remains an open problem.
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
15
5. Regularity for the Stokes and Navier-Stokes systems on convex polygonal domains As a consequence of the regularity for the biharmonic problem we prove shift estimates for the Stokes and the Navier-Stokes systems. Let Ω be a convex polygonal domain in R2 with boundary ∂Ω and let ω be the measure of the largest angle of ∂Ω. In this case, we have that s0 = s0 (ω) ∈ (1/2, 1), (see Remark 4.3). Let γ0 := 2s0 − 1. Note that γ0 ∈ (0, 1). Then, according to the estimate (1.2), the solution u of the biharmonic problem (1.1) satisfies (5.1)
kukH 3+γ ≤ ckf kH −1+γ ,
for all f ∈ H −1+γ (Ω), −1 ≤ γ < γ0 .
The parameter γ0 can be viewed as the exact amount of extra regularity due to the convexity of the domain Ω. Next, we consider the steady-state Stokes problem in the velocity-pressure formulation. More precisely we consider the following problem: Given F ∈ (H γ )2 with 0 ≤ γ < γ0 , find the vector-valued function u and the scalar-valued function p satisfying −∆u + ∇p = F in Ω, ∇·u=0 in Ω, (5.2) u = 0 on ∂Ω, R p = 0. Ω The first two equations are considered in the standard weak sense. According to a well known Kellogg-Osborn result, [20], we have that (5.2) has a unique solution (u, p) ∈ ((H 2 (Ω))2 , H 1 (Ω)). Since ∇ · u = 0 in Ω, and u = 0 on ∂Ω, one can find w ∈ H02 , (see for example I.3.1 in [15] or [2]), such that ∂w ∂w ,− . u = (u1 , u2 ) = curl w := ∂x2 ∂x1 If we substitute u = curl w in (5.2), we get ( ∂p ∂w −∆( ∂x ) + ∂x = f1 2 1 (5.3) ∂p ∂w ∆( ∂x1 ) + ∂x2 = f2
in Ω, in Ω,
∂ ∂ and ∂x to the where (f1 , f2 ) = F. Next, we apply the differential operators − ∂x 2 1 first and second equations of (5.3), respectively, and sum up the two new equations. Thus, we have that w ∈ H02 and
∆2 w =
∂f1 ∂f2 − ∂x1 ∂x2
in Ω.
Consequently, for a fixed γ ∈ (0, γ0 ), from (5.1), we have that
∂f2 ∂f1
− , kwkH 3+γ ≤ c ∂x1 ∂x2 −1+γ H
where c is a constant independent of F. It follows that kuk(H 2+γ )2 ≤ ckFk(H γ )2 ,
for all F ∈ (H γ (Ω))2 .
From the first part of (5.2), we have ∇p = ∆u + F. Hence, k∇pk(H γ )2 ≤ k∆uk(H γ )2 + kFk(H γ )2 ≤ kuk(H 2+γ )2 + kFk(H γ )2 ≤ ckFk(H γ )2 . In conclusion we obtain:
16
C. BACUTA
Theorem 5.1. Let Ω be a convex polygonal domain in R2 with ω the measure of the largest angle. Let γ0 = 2s0 (ω) − 1 and let (u, p) be the solution of (5.2). Then for any γ ∈ (0, γ0 ), there exist a constant c such that (5.4)
kuk(H 2+γ )2 + kpkH 1+γ ≤ ckFk(H γ )2
for all F ∈ (H γ )2 .
We conclude this section by a similar regularity result for solutions of the NavierStokes equations
(5.5)
−∆u + (u · ∇)u + ∇p = G in Ω, ∇·u=0 in Ω, u = 0 on ∂Ω, R p = 0. Ω
Theorem 5.2. Let Ω be a convex polygonal domain in R2 with ω the measure of the largest angle. Let γ0 = 2s0 (ω) − 1, G ∈ (H γ )2 with 0 ≤ γ < γ0 and let (u, p) ∈ (H01 )2 × L2 be a solution of (5.5). Then, (5.6)
u ∈ (H 2+γ )2
and p ∈ H 1+γ .
Proof. The main idea of the proof is due to Temam. The proof given here was inspired by the proof presented by Kellogg and Osborn in [20]. Let (u, p) ∈ (H01 )2 × L2 be a solution of (5.5) and denote F = G − (u · ∇)u. According to [20], we have that u ∈ (H 5/3 )2 . In particular we get that ∇u ∈ (H 2/3 )2 and u is bounded. Thus F ∈ (H min{γ,2/3} )2 , and by using (5.4), we have that u ∈ (H 2+min{γ,2/3} )2 and ∇u ∈ (H 1+min{γ,2/3} )2 . Then, we deduce that, in fact, F ∈ (H γ )2 . Using (5.4) again , we conclude that (5.6) holds. Remark 5.3. In the light of Theorem 4.2, we can extend the last two results to hold for the threshold value γ = γ0 by considering the interpolation norms with the second index of interpolation q 6= 2. More precisely, we have to take q = 1 for the norm of F or G and q = ∞ for the norm of (u, p). In other words, the last two theorems hold if the spaces of type H k+γ are replaced by Bqk+γ0 with the above mentioned choice for q. 6. Appendix Here, we present the postponed proofs and some auxiliary results. 6.1. The Postponed Proofs. Proof of Theorem 2.6. Let u be fixed in Y and set (6.1)
−1 −1 w := (I + t2 S −1 )−1 u and wϕ := (I + t2 SK ) u.
Then, according to (2.12), (2.6) and (2.13), we have Z Z ∞ dt kuk2[Y,XK ]s,2 = t2s (wK , u)Y , kuk2[Y,X]s,2 = t 0 and kuk2[Y,XK ]s,∞ = sup t2s (wK , u)Y ,
∞
t2s (w, u)Y 0
t>0
respectively. Thus, (2.18) and (2.19) would follow provided we establish
(6.2) (wK , u)Y = (w, u)Y + t2 Mt−1 yt , yt .
dt , t
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
17
From (6.1), we get that −1 (I + t2 SK )wK = u.
(6.3)
Combining (2.15) and (6.3) we obtain (I + t2 S −1 )wK = u + t2
n X
αk ϕk ,
k=1
where αk are constants determined by QK (S −1 wϕ ) = applying (I + t2 S −1 )−1 to both sides, we have wK = w + t2
(6.4)
n X
Pn
k=1
αk ϕk . Equivalently,
αk (I + t2 S −1 )−1 ϕk .
k=1
We calculate the coefficients αk by taking the (·, ·)X inner product with ϕ on both sides of (6.4), i.e., (wK , ϕj )X = (w, ϕj )X +t2
n X
αk (I + t2 S −1 )−1 ϕk , ϕj
X
= (w, ϕj )X +t2
n X
αk (ϕk , ϕj )X,t .
k=1
k=1
−1 From (6.3), since SK wϕk ∈ XK , one sees that (wK , ϕj )X = (u, ϕj )X . With the notation adopted in (2.16) and (6.1) we obtain the following n × n system: n X
αk (ϕk , ϕj )X,t t = t−2 ((u, ϕj )X − (u, ϕj )X,t ).
k=1
Using (2.4) and a simple manipulation of the operator S we get (6.5) Thus
t2 (u, ϕj )Y,t = (u, ϕj )X − (u, ϕj )X,t . n X
αk (ϕk , ϕj )X,t = (u, ϕj )Y,t = (yt )j .
k=1
Let α ∈ Cn be the the vector with components α1 , α2 , · · · , αn , then α = Mt−1 yt . Now, going back to (6.4), we get (wK , u)Y = (w, y) + t2
n X
αk (ϕk , u)Y,t .
k=1
By substituting the vector α we obtain (6.2) and complete the proof. Next, we present a general subspace interpolation lemma used for the proof of ˜ Y˜ ) be two pairs of Hilbert spaces satisfying (2.1) and Theorem 4.2. Let (X, Y ), (X, ∗ ∗ ∗ ˜∗ ˜ let (Y , X ), (Y , X ) be the corresponding dual pairs. Then, the dual pairs satisfy (2.1) also, see e.g., [18]. We further identify Y ∗ with Y and Y˜ ∗ with Y˜ , and assume that there are linear operators E and R such that (6.6)
˜ are bounded operators, E : Y → Y˜ , E : X → X
(6.7)
˜ → X, are bounded operators, R : Y˜ → Y, R : X
(6.8)
REu = u
for all u ∈ Y.
18
C. BACUTA
e = E(K) ⊂ Y˜ = Y˜ ∗ be closed subspaces of Y and Y˜ , Let K ⊂ Y = Y ∗ , K respectively, and denote the corresponding orthogonal complements by YK and Y˜Ke . Let θ0 ∈ (0, 1) be such that ˜ ∗ , Y˜ ]θ ,1 ⊂ [X ˜ ∗ , Y˜ e ]θ ,∞ . (6.9) [X K
0
0
Lemma 6.1. Assume that (6.6)-(6.9) are satisfied. Then, [X ∗ , Y ]θ0 ,1 ⊂ [X ∗ , YK ]θ0 ,∞
(6.10)
Proof. Using the duality, from (6.6)-(6.8), we obtain linear operators E ∗ , R∗ such that ˜ ∗ → X ∗ , are bounded operators, (6.11) E ∗ : Y˜ → Y, E ∗ : X (6.12)
˜ ∗ are bounded operators, R∗ : Y → Y˜ , R∗ : X ∗ → X E ∗ R∗ u = u
(6.13)
for all u ∈ Y,
E ∗ maps Y˜Ke to YK .
(6.14)
From (6.11) and (6.14), by interpolation, we obtain (6.15)
kE ∗ vk[X ∗ ,YK ]θ0 ,∞ ≤ ckvk[X˜ ∗ ,Y˜ e ]θ K
0 ,∞
for all v ∈ Y˜ .
For any u ∈ Y , we take v := R∗ u in (6.15), and use (6.13) to get (6.16)
kuk[X ∗ ,YK ]θ0 ,∞ ≤ ckR∗ uk[X˜ ∗ ,Y˜ e ]θ K
0 ,∞
for all u ∈ Y.
Also, from the hypothesis (6.9), we deduce that (6.17)
kR∗ uk[X˜ ∗ ,Y˜ e ]θ K
0 ,∞
≤ ckR∗ uk[X˜ ∗ ,Y˜ ]θ
0 ,1
for all u ∈ Y.
From (6.12), again by interpolation, we have in particular (6.18)
kR∗ uk[X˜ ∗ ,Y˜ ]θ
0 ,1
≤ ckuk[X ∗ ,Y ]θ0 ,1
for all u ∈ Y.
Combining (6.16)-(6.18), it follows that (6.19)
kuk[X ∗ ,YK ]θ0 ,∞ ≤ ckuk[X ∗ ,Y ]θ0 ,1
for all u ∈ Y.
Since Y is dense in both [X ∗ , Y ]θ0 ,1 and [X ∗ , YK ]θ0 ,∞ we obtain (6.10).
Proof of Lemma 4.1. In [4], it was proven that that there are bounded operators E and R, E : L2 (Ω) −→ L2 (R), E : H02 (Ω) −→ H 2 (R2 ), R : L2 (R2 ) −→ L2 (Ω), R : H 2 (R2 ), −→ H02 (Ω) such that REu = u, for all u ∈ L2 (Ω). Next, let φj be the Fourier transform of Eϕj , j = 1, . . . , n. Using the asymptotic expansion of integrals estimates presented in [4], (see also [11, 19, 27]), we have that the functions {Eϕj , j = 1, . . . , n} satisfy for some positive constants c and , |φj (ξ) − φ˜j (ξ)| < cρ−1+(−2+sj )− f or |ξ| > 1 (6.20) −2 < −2 + si < 0, i = 1, . . . , n, where sj = Re(zj ) and φ˜j (ξ) = bi (ω)ρ−1+(−2+sj ) , ξ = (ρ, ω) in polar coordinates,
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
19
and bj (·) is a bounded measurable function on the unit circle, which is non zero on a set of positive measure. Let s0 = s0 (ω) be defined by (4.7). Thus, we have that the functions {Eϕj , j = 1, . . . , n} satisfy the hypothesis (3.16) of Theorem 3.4 with N = 2, β = 0, α = −2 and γj = −2 + sj , j = 1, . . . , n. Let us define now L := span{Eϕj , j = 1, . . . , n}. By Theorem 3.4 applied with θ0 = s0 , we have that [H −2 (R2 ), L2 (R2 )L ]s0 ,1 ⊂ [H −2 (R2 ), L2 (R2 )L ]s0 ,∞ .
(6.21)
Finally, using (6.21) and Lemma 6.1, we conclude that (4.8) holds. 6.2. Appendix B. Interpolation Results. Using the notation of Section 2, we review the classical interpolation results involved in this paper. The proofs can be found, for example, in [9, 10, 23]. Proposition 6.2. Let (X, Y ) be a pair of Banach spaces satisfying (2.1). The following hold. i) The relative strength of the smoothness index s: [Y, X]s,q ⊂ [Y, X]s˜,˜q , 0 < s˜ < s < 1, and 1 ≤ q, q˜ ≤ ∞. ii) The relative strength of the second index of interpolation: [Y, X]s,q ⊂ [Y, X]s,˜q , 1 ≤ q ≤ q˜ ≤ ∞. ˜ Y˜ ) be another pair of Banach spaces iii) “By Interpolation” Theorem: Let (X, ˜ satisfy satisfying (2.1). If T : Y → Y˜ and T : X → X kT f kY˜ ≤ c0 kf kY ,
and kT f kX˜ ≤ c1 kf kX ,
then, 1−λ λ kT f k[Y˜ ,X] c1 kf k[Y,X]λ,q , 0 < λ < 1, and 1 ≤ q ≤ ∞. ˜ λ,q ≤ c0
iv) The Reiteration Theorem: For any 0 ≤ s0 < s1 ≤ 1, q0 , q1 , q ∈ [1, ∞], and 0 < λ < 1, [[Y, X]s0 ,q0 , [Y, X]s1 ,q1 ]λ,q = [Y, X](1−λ)s0 +λs1 ,q . References [1] R. A. Adams. Sobolev Spaces. Academic Press, New York, 1975. [2] C. Bacuta and J. H. Bramble. Regularity estimates for solutions of the equations of linear elasticity in convex plane polygonal domains. Z. angew. Math. Phys.(ZAMP), 54(2003), 874878. [3] C.Bacuta, J. H. Bramble and J. Pasciak. Using finite element tools in proving shift theorems for elliptic boundary value problems. Numerical Linear Algebra with Application, Volume 10, Issue no 1-2, 33-64, 2003. [4] C.Bacuta, J. H. Bramble and J. Pasciak. Shift theorems for the biharmonic Dirichlet problem, Recent Progress in Computational and Applied PDEs, Kluwer Academic/Plenum Publishers, volume of proceedings for the CAPDE conference held in Zhangjiajie, China, July 2001. [5] C. Bacuta, V. Nistor and L. Zikatanov. Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps. Numerische Mathematik, Vol. 100, No 2, 2005, pp. 165 -184 [6] C. Bacuta, V. Nistor and L. Zikatanov. Improving the rate of convergence of ‘high order finite elements’ on polyhedra I: apriori estimates. Numerical Functional Analysis and Optimization, Vol. 26, No 6, 2005, pp 613-639. [7] C. Bacuta, V. Nistor and L. Zikatanov. Improving the rate of convergence of ‘high order finite elements’ on polyhedra II: Mesh Refinements and Interpolation. Numerical Functional Analysis and Optimization, Vol. 28, No 7-8, 2007, pp 775-824.
20
C. BACUTA
[8] C.Bacuta, J. H. Bramble, and J. Xu. Regularity estimates for elliptic boundary value problems in Besov spaces, Mathematics of Computation, 72 (2003), 1577-1595. [9] C. Bennett and R. Sharpley. Interpolation of Operators. Academic Press, New-York, 1988. [10] J. Bergh and J. Lofstrom. Interpolation Spaces. Springer-Verlag, New York, 1976. [11] N. Bleistein and R. Handelsman. Asymptotic expansions of integrals. Holt, Rinehart and Winston, New York, 1975. [12] H. Blum and R. Rannacher. On the Boundary Value Problem of Biharmonic Operator on Domains with Angular Corners, Math. Meth. in the Appl. Sci., 2(1980) 556-581. [13] S. Brenner and L.R. Scott. The Mathematical Theory of Finite Element Methods. SpringerVerlag, New York, 1994. [14] M. Dauge. Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics 1341. Springer-Verlag, Berlin, 1988. [15] V. Girault and P.A. Raviart. Finite Element Methods for Navier-Stokes Equations. SpringerVerlag, Berlin, 1986. [16] P. Grisvard. Elliptic Problems in Nonsmooth Domains. Pitman, Boston, 1985. [17] P. Grisvard. Singularities in Boundary Value Problems. Masson, Paris, 1992. [18] W. Hackbusch. Elliptic Differential Equations. Springer-Verlag, Berlin Heidelberg, 1992. [19] R. B. Kellogg. Interpolation between subspaces of a Hilbert space, Technical note BN-719, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, 1971. [20] R. B. Kellogg and J. Osborn. A regularity result for Stokes Problem in a convex Polygon. Journal of Functional Analysis, 21, 1976, 397-431. [21] V. Kondratiev. Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc., 16:227-313, 1967. [22] V. A. Kozlov, V. G. Mazya and J. Rossmann. Elliptic Boundary Value Problems in Domains with Point Singularities. American Mathematical Society, Mathematical Surveys and Monographs, vol. 52, 1997. [23] J. L. Lions and E. Magenes. Non-homogeneous Boundary Value Problems and Applications, I. Springer-Verlag, New York, 1972. [24] J. L. Lions and P. Peetre. Sur une classe d’espaces d’interpolation. Institut des Hautes Etudes Scientifique. Publ.Math., 19:5-68, 1964. [25] S. A. Nazarov and B. A. Plamenevsky. Elliptic Problems in Domains with Piecewise Smooth Boundaries. Expositions in Mathematics, vol. 13, de Gruyter, New York, 1994. [26] J. Ne˘ cas. Les Methodes Directes en Theorie des Equations Elliptiques. Academia, Prague, 1967. [27] F. W. Olver. Asymptotics and Special Functions. Academic Press, New York, 1974. [28] F. Riesz and B. Sz-Nagy. Functional Analysis (Second Edition), Frederick Ungar, New York, 1969. [29] K. Yosida Functional Analysis. Springer-Verlag, Berlin Heidelberg, 1978. Mathematical Sciences, University of Delaware, Newark, DE 19716, USA. E-mail address: [email protected]
Abstract. In this paper, we prove new embedding results by means of subspace interpolation theory and apply them to establishing regularity estimates for the biharmonic Dirichlet problem, and for the Stokes and the Navier-Stokes systems on polygonal domains. The main result of the paper gives a stability estimate for the biharmonic problem at the threshold index of smoothness. The classical regularity estimates for the biharmonic problem are deduced as a simple corollary of the main result. The subspace interpolation tools and techniques presented in this paper can be applied to establishing sharp regularity estimates for other elliptic boundary value problems on polygonal domains.
1. Introduction Regularity estimates or shift estimates for the solutions of elliptic Boundary Value Problems (BVPs) in terms of Sobolev-Besov norms, are of a significant interest in discretizing BVPs by finite differences methods, spectral methods and finite element methods. Shift estimates for the Laplace operator with Dirichlet boundary conditions on non-smooth domains are widely studied in the literature (see, e.g., [3, 8, 19, 22, 26, 5, 6, 7]). Shift estimates for the biharmonic problem on non-smooth domains, are presented in [12, 14, 4]. In this paper we extend the results of [4] and [14] by studying the regularity at the critical case. The shift results are proved by using the real method of interpolation of Lions and Peetre [9, 23, 24] and the subspace interpolation theory introduced by Kellogg in [19]. The approach based on subspace interpolation theory proves to be essential for the threshold regularity case. We describe a typical subspace interpolation problem as follows. Let X and Y be Sobolev spaces of integer order with X embedded in Y (X ⊂ Y ), and let XK be a subspace of X of finite codimension. Then, we need to characterize the interpolation spaces between Y and XK . In particular, we are looking for embedding relations between the intermediate subspaces [Y, XK ]s,q0 and the standard interpolation spaces [Y, X]s,q1 . Here, s ∈ (0, 1) can be viewed as the smoothness index, and q0 , q1 can be viewed as tuning parameters. In proving shift estimates for the biharmonic problem, we follow Kellogg’s approach in dealing with subspace interpolation. First, we study subspace interpolation embeddings on Sobolev spaces defined on all of R2 . Then, we apply the results to the polygonal domain case by finding ”extension” and “restriction” operators connecting the Sobolev spaces defined on bounded domains and the Sobolev spaces Date: December 14, 2007. 1991 Mathematics Subject Classification. 35B30, 35B65, 35Q30, 46B70. Key words and phrases. subspace interpolation, elliptic regularity, biharmonic operator, shift theorems, threshold index of smoothness, Stokes systems, Navier-Stokes systems. This work was supported by NSF, DMS-0713125. 1
2
C. BACUTA
defined on R2 . In contrast with Kellogg interpolation results presented in [19], we consider subspaces of codimension higher than one, and allow the tuning parameter q to be not necessarily 2. The shift estimates for Stokes and Navier-Stokes systems are based on the main regularity estimate for the biharmonic problem. Next, we describe the main regularity result of the paper. Let Ω be a polygonal domain in R2 with boundary ∂Ω. Let ∂Ω be the polygonal curve P1 P2 · · · Pm P1 . Assume that ∂Ω does not self intersect. For each point Pj , we denote by ωj the measure of the angle with the vertex at Pj and sides aligned with ∂Ω (measured from inside Ω). Let ω := max{ωj : j = 1, 2, . . . , m}. We consider the biharmonic problem: Given f ∈ L2 (Ω), find u such that 2 ∆ u = f in Ω, u = 0 on ∂Ω, (1.1) ∂u ∂n = 0 on ∂Ω. It is known that the solution of (1.1) satisfies kukH 2+2s (Ω) ≤ ckf kH −2+2s (Ω) ,
(1.2)
for all f ∈ H −2+2s (Ω), 0 ≤ s < s0 ,
where c is a positive constant, and s0 = s0 (ω) ∈ (0, 1) depends on the maximum angle of ∂Ω only, (see e.g., [4, 14, 25]). We will prove in this paper that for some positive constant c, we have 2+2s0 kukB∞ ≤ ckf kB −2+2s0 (Ω) , (Ω)
(1.3)
1
for all f ∈ B1−2+2s0 (Ω),
where 2+2s0 B∞ (Ω) := H 2 (Ω), H 4 (Ω) s
0 ,∞
and B1−2+2s0 (Ω) := H −2 (Ω), L2 (Ω) s
0 ,1
.
Note that H 2+2s (Ω) = H 2 (Ω), H 4 (Ω) s,2 and H −2+2s (Ω) = H −2 (Ω), L2 (Ω) s,2 . In other words, we prove that the estimate (1.2) holds for the threshold regularity index s = s0 , provided the second index of interpolation q is taken q = 1 for measuring the data f , and q is taken q = ∞ for measuring the solution u. The precise definitions of the interpolation spaces are given in the next section. We note that (1.2) can be easily proved as a consequence of (1.3) and the reiteration theorem. In addition, the proof of (1.3) is considerably simpler than the proof of (1.2) as given in [4], (see Section 4). Section 2 and Section 3 contain interesting theoretical embedding results which can be used in deriving regularity estimates for other elliptic BVPs that are not contained in this paper. For example, one can recover the regularity estimates at the critical case for the Laplace operator with Dirichlet boundary conditions which are presented in [8] by means of multilevel representation of norms. In addition, one can use the approach of this paper and prove estimates at the threshold regularity index for the Laplace operator with mixed Dirichlet-Neuman boundary conditions. The remaining part of the paper is organized as follows. In Section 2 we present subspace interpolation results in an abstract setting. In Section 3, we prove embedding results for interpolation spaces defined on the whole R2 . The main shift estimate (1.3) is proved in Section 4. Regularity estimates for the Stokes and Navier-Stokes systems are presented in Section 5. In the Appendix, we present the technical proofs that were omitted in the prior sections.
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
3
Acknowledgments. Many thanks go to Jim Bramble, Bruce Kellogg, Joe Pasciak, and Jinchao Xu for guidance and valuable discussions on the subject. 2. Interpolation results In this section, we start with some basic definitions and results concerning interpolation between subspaces of Hilbert spaces using the real method of interpolation of Lions and Peetre (see [23]). In the second subsection, we present subspace interpolation theory. 2.1. Interpolation between Banach and Hilbert spaces. Let X, Y be Banach spaces satisfying the following conditions: X is a dense subset of Y and (2.1) kukY ≤ ckukX for all u ∈ X, for some positive constant c. For s ∈ (0, 1), 1 ≤ q ≤ ∞, the intermediate space [Y, X]s,q is defined using the K function. For u ∈ Y and t > 0, we define K(t, u, Y, X) = K(t, u) := inf (ku − u0 k2Y + t2 ku0 k2X )1/2 . u0 ∈X
For s ∈ (0, 1) and 1 ≤ q ≤ ∞, the space [Y, X]s,q is defined by [Y, X]s,q := {u ∈ Y : kuk[Y,X]s,q < ∞}, where Z kuk[Y,X]s,q =
∞
(t
−s
q dt
K(t, u))
0
1/q
t
for q < ∞,
and kuk[Y,X]s,∞ = sup t−s K(t, u). t>0
By definition, we take [Y, X]0,q := Y and [Y, X]1,q := X. The main properties of the interpolation between Banach spaces can be found in [9, 10, 13]. Next, we will focus on interpolation between Hilbert spaces. Let us assume that, in addition, X, Y are separable Hilbert spaces with inner products (·, ·)X and (·, ·)Y , and the norms on the two spaces are the norms induced by the corresponding inner products (kuk2X = (u, u)X and kuk2Y = (u, u)Y ). Let D(S) denote the subset of X consisting of all elements u such that the anti-linear form (2.2)
v → (u, v)X , v ∈ X,
is continuous in the topology induced by Y . For any u in D(S) the anti-linear form (2.2) can be extended to a continuous anti-linear form on Y . Then, by the Riesz representation theorem (see e.g., [1, 29]), there exists an element Su in Y such that (2.3)
(u, v)X = (Su, v)Y
for all v ∈ X.
In this way, S is a well defined operator in Y with domain D(S). The next result illustrates the properties of S, see, e.g. [23, 28].
4
C. BACUTA
Proposition 2.1. The domain D(S) of the operator S is dense in X and consequently D(S) is dense in Y . The operator S : D(S) ⊂ Y → Y is a bijective, selfadjoint and positive definite operator. The inverse operator S −1 : Y → D(S) ⊂ Y is a bounded symmetric positive definite operator and (2.4)
(S −1 z, u)X = (z, u)Y
for all z ∈ Y, u ∈ X.
If X,Y are Hilbert spaces, we have a better representations of the intermediate space [Y, X]s,q . The next lemma provides the relation between K(t, u) and the connecting operator S, and consequently, gives the representation of [Y, X]s,q in terms of S. A similar result can be found in [23]. Lemma 2.2. For all u ∈ Y and t > 0 , K(t, u)2 = (I + (t2 S)−1 )−1 u, u
(2.5)
∞
Z (2.6)
kuk[Y,X]s,q =
t
sq
2
(I + t S
0
−1 −1
)
,
Y
q/2 dt u, u Y t
1/q , 1 ≤ q < ∞,
and (2.7)
kuk[Y,X]s,∞ = sup ts (I + t2 S −1 )−1 u, u t>0
1/2 Y
.
Proof. Using the density of D(S) in X, we have 2
K(t, u) =
inf
u0 ∈D(S)
(ku0 k2X + t2 ku − u0 k2Y )
Let v = Su0 . Then (2.8)
2
K(t, u) = inf (ku − S −1 vk2Y + t2 (S −1 v, v)Y ). v∈Y
By solving the minimization problem (2.8), we obtain that the element v, which gives the optimum, satisfies the following equations: (t2 I + S −1 )v = u, and ku − S −1 vk2Y + t2 (S −1 v, v)Y = (I + (t2 S)−1 )−1 u, u Y . To obtain (2.6) and (2.7) we replace K in the definition of [Y, X]s,q with the expression given by (2.5) and apply the change of variable 1/t = τ . Lemma 2.3. Let X0 be a closed subspace of X, and let Y0 be a closed subspace of Y . Let X0 and Y0 be equipped with the topologies and the geometries induced by X and Y , respectively, and assume that the pair (X0 , Y0 ) satisfies (2.1). Then, for s ∈ [0, 1], 1 ≤ q ≤ ∞, [Y0 , X0 ]s,q ⊂ [Y, X]s,q ∩ Y0 . Proof. For any u ∈ Y0 , we have K(t, u, Y, X) ≤ K(t, u, Y0 , X0 ). Thus, (2.9)
kuk[Y,X]s,q ≤ kuk[Y0 ,X0 ]s,q
which proves the lemma.
for all u ∈ [Y0 , X0 ]s,q ,
The validity of the other inclusion will be addressed in the next subsection for cases related to the regularity for the biharmonic problem.
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
5
2.2. Subspace interpolation on Hilbert spaces. Let K = span{ϕ1 , . . . , ϕn } be a n-dimensional subspace of X, and let XK be the orthogonal complement of K in X in the (·, ·)X inner product. We are interested in determining the interpolation spaces of Y and XK , where on XK we consider again the (·, ·)X inner product. For certain spaces X, K, Y , n = 1, and q = 2, this problem was studied in [19]. To apply the interpolation results from the previous section we need to check that the density part of the condition (2.1) is satisfied for the pair (XK , Y ). For ϕ ∈ K, we define the linear functional Λϕ : X → C by Λϕ u := (u, ϕ)X , u ∈ X. The following lemma can be found in [4]. Lemma 2.4. The space XK is dense in Y if and only if the following condition is satisfied: Λϕ is not bounded in the topology of Y (2.10) for all ϕ ∈ K, ϕ 6= 0. For the remaining part of this section, we assume that the condition (2.10) holds. By the above lemma, the condition (2.1) is satisfied. It follows from the previous section that the operator SK : D(SK ) ⊂ Y → Y defined by (2.11)
(u, v)X = (SK u, v)Y
for all v ∈ XK ,
has the same properties as S has. Consequently, the norm on the intermediate space [Y, XK ]s,q is given by Z ∞ q/2 dt 1/q sq 2 −1 −1 , 1 ≤ q < ∞, (2.12) kuk[Y,XK ]s,q = t (I + t SK ) u, u Y t 0 and (2.13)
−1 −1 kuk[Y,XK ]s,∞ = sup ts (I + t2 SK ) u, u t>0
1/2 Y
.
In this section, our aim is to study the embedding relation between [Y, XK ]s,q1 and [Y, X]s,q2 . Lemma 2.5. Assume that for the fixed parameters s0 ∈ (0, 1), q0 , q1 ∈ [1, ∞], we have the following continuous embedding [Y, X]s0 ,q1 ⊂ [Y, XK ]s0 ,q0 .
(2.14) Then,
[Y, XK ]s,q = [Y, X]s,q , 0 < s < s0 , q ∈ [1, ∞]. Proof. From Lemma 2.3, we have that [Y, XK ]s,q ⊂ [Y, X]s,q , 0 ≤ s < s0 . On the other hand, from (2.14), the inclusion operator is continuous from [Y, X]s0 ,q1 to [Y, XK ]s0 ,q0 , and clearly, it is continuous from [Y, X]0,2 = Y to [Y, XK ]0,2 = Y . By interpolation, and by the reiteration theorem (see Appendix 6.2), we get [Y, X]s,q ⊂ [Y, XK ]s,q , 0 < s < s0 , which completes the proof.
6
C. BACUTA
We note here that, for a fixed s0 ∈ (0, 1), the weakest assumption of type (2.14) is for q0 = ∞ and q1 = 1. Next, we study the validity of (2.14) for q0 = ∞ and q1 = 1. First, we note that the operators SK and S are related by the following identity: −1 SK = (I − QK )S −1 ,
(2.15)
where QK : X → K is the orthogonal projection onto K. The proof of (2.15) follows easily from the definitions of the operators S, QK and SK . The identity (2.15) leads to a formula relating the norms on [Y, XK ]s,q and [Y, X]s,q . To get the formula, we introduce first the following notation. Let (2.16) (u, v)X,t := (I + t2 S −1 )−1 u, v X for all u, v ∈ X, and (u, v)Y,t := (I + t2 S −1 )−1 u, v
(2.17)
Y
for all u, v ∈ Y.
Denote by Mt the Gram matrix associated with the set of vectors {ϕ1 , . . . , ϕn } in the (·, ·)X,t inner product, i.e., (Mt )ij := (ϕj , ϕi )X,t , i, j ∈ {1, . . . , n}. Theorem 2.6. For any u ∈ Y , we have kuk2[Y,XK ]s,2
(2.18)
=
kuk2[Y,X]s,2
∞
Z + 0
dt
t2+2s Mt−1 yt , yt , t
kuk2[Y,XK ]s,∞ = sup t2s (u, u)Y,t + t2+2s Mt−1 yt , yt ,
(2.19)
t>0
where h·, ·i is the inner product on Cn , and yt is the n-dimensional vector in Cn whose components are (yt )i := (u, ϕi )Y,t , i = 1, . . . , n. The proof is given in Appendix 6.1. For n = 1, let K = span{ϕ} and denote XK by Xϕ . Then, the formulas (2.18) and (2.19) become Z ∞ 2 |(u, ϕ)Y,t | dt (2.20) kuk2[Y,Xϕ ]s,2 = kuk2[Y,X]s,2 + t2+2s , (ϕ, ϕ)X,t t 0 and kuk2[Y,Xϕ ]s,∞
(2.21)
2s
= sup t (u, u)Y,t + t
2 2+2s |(u, ϕ)Y,t |
t>0
(ϕ, ϕ)X,t
! ,
respectively. The next theorem gives sufficient conditions for (2.14) to be satisfied. Before we state the result, we fix si ∈ (0, 1), i = 0, 1, . . . n, and define s0 = min{s1 , s2 , . . . sn }. We introduce the following two conditions: (A.1) [Y, X]s0 ,1 ⊂ [Y, Xϕi ]s0 ,∞ , for all i = 1, . . . , n. (A.2) There exist δ > 0 and γ > 0 such that n X i=1
2
|αi | (ϕi , ϕi )X,t ≤ γ hMt α, αi
for all α = (α1 , . . . , αn )t ∈ Cn , t ∈ (δ, ∞).
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
7
Theorem 2.7. Assume that the conditions (A.1) and (A.2) hold. Then [Y, X]s0 ,1 ⊂ [Y, XK ]s0 ,∞ ,
(2.22) and
[Y, XK ]s,q = [Y, X]s,q , 0 < s < s0 , q ∈ [1, ∞].
(2.23)
Proof. Due to (A.1), for a fixed u in [Y, X]s0 ,1 , we have that kuk[Y,Xϕi ]si ,∞ < ∞. Using the notation of the previous theorem, we get kuk2[Y,XK ]s
= sup(t2s0 (wK , u)Y ).
0 ,∞
t>0
On the other hand, −1 −1 (wK , u)Y = (I + t2 SK ) u, u
Y
−1 −1 −1 (I + t2 SK ) u, u = (u, u)Y − t2 SK
Y
2
≤ (u, u)Y ≤ ckuk[Y,X]s
0 ,1
Thus, from (6.2) and (2.7), we obtain kuk2[Y,XK ]s
0 ,∞
≤ sup (t2s0 (wK , u)Y ) + sup(t2s0 (wK , u)Y ) 0δ
≤
2 c(δ)kuk[Y,X]s ,1 0
≤
2 ckuk[Y,X]s ,1 0
+ sup(t2s0 (u, u)Y,t ) + sup(t2+2s0 Mt−1 yt , yt ) t>δ
+ sup(t
t>δ
2+2s0
t>δ
Mt−1 yt , yt
).
Next, (A.2) is equivalent to
n X 2 Mt−1 α, α ≤ γ |αi | (ϕi , ϕi )−1 X,t
for all α = (α1 , . . . , αn )t ∈ Cn , t ∈ (δ, ∞).
i=1
In particular, for αi = (yt )i = (u, ϕi )Y,t , i = 1, . . . , n, we obtain
Mt−1 yt , yt
≤γ
n 2 X |(u, ϕi )Y,t | i=1
(ϕi , ϕi )X,t
for all t ∈ (δ, ∞), u ∈ [Y, X]s0 ,1 .
Consequently, using (2.21) and (A.1), we have kuk2[Y,XK ]s ,∞ 0
≤
2 ckuk[Y,X]s ,1 0 2
≤ ckuk[Y,X]s
0 ,1
+ γ sup t t>δ
+γ
n X
2+2s0
n 2 X |(u, ϕi )Y,t | i=1
kuk2[Y,Xϕ
!
(ϕi , ϕi )X,t 2
] i s0 ,∞
≤ c(n, δ, γ)kuk[Y,X]s
0 ,1
.
i=1
This concludes (2.22). The identity (2.23) follows from (2.22) and Lemma 2.5. 3. Interpolation between subspaces of H β (RN ) and H α (RN ). In this section, we apply our abstract interpolation result to the particular case Y = H α (RN ), X = H β (RN ). The codimension one case with q = 2 was analyzed for the first time by Kellogg in [19]. A finite codimension case with q = 2 is considered in [4]. Using Theorem 2.7, we prove a new subspace interpolation embedding and recover the main interpolation results of [19] and [4]. The proofs are based on techniques used in [19] and [4], but are considerably simplified. For the proof of the regularity for the biharmonic problem we will need only the last theorem of this
8
C. BACUTA
section. The reader not interested in the proof of the embedding tools might skip this section. Let α ∈ R and let H α (RN ) be defined by means of the Fourier transform. For a smooth function u with compact support in RN , the Fourier transform u ˆ is defined by Z u ˆ(ξ) = (2π)−N/2
u(x)e−ix·ξ dx,
where the integral is taken over the whole RN . For u and v smooth functions the α-inner product is defined by Z hu, viα = (1 + |ξ|2 )α u ˆ(ξ)ˆ v (ξ) dξ. The space H α (RN ) is the closure of smooth functions with compact support in the norm induced by the α-inner product. For α, β real numbers, α < β, and s ∈ [0, 1] it is easy to check, using (2.6), that α N H (R ), H β (RN ) s,2 = H (1−s)α+sβ (RN ). β N For ), we in determining αϕ ∈NH (R are interested the embedding relation between β H (R ), Hϕ (RN ) s,q and H α (RN ), H β (RN ) s,q . We recall that Hϕβ is the or1
0
thogonal complement of span{ϕ} in H β . For simplicity we shall start with the case α = 0. We note that H 0 (RN) = L2 (RN ). The operator S associated with the pair (X, Y ) = H β (RN ), H 0 (RN ) is given by c = µ2β u Su ˆ, u ∈ D(S) = H 2β (RN ), 1
where µ(ξ) = (1 + |ξ|2 ) 2 , ξ ∈ RN . For the remaining part of this chapter, H β ˆ β is the space {ˆ ˆ β , we denotes the space H β (RN ) and H u |u ∈ H β }. For u ˆ, vˆ ∈ H define the inner product and the norm by Z 1/2 (ˆ u, vˆ)β = µ2β u ˆvˆ dζ, ||ˆ u||β = (ˆ u, u ˆ)β . To simplify the notation, we denote the the inner products (·, ·)0 and h·, ·i0 by (·, ·) ˆ 0 is simply || · ||. Thus, for and h·, ·i, respectively. The norm || · ||0 on H 0 or H ˆ β and Y = H ˆ 0 , we have X=H 2β 4β µ u ˆ µ u ˆ (ˆ u, φ)Y,t = , φ and (ˆ u , φ) = , φ . X,t µ2β + t2 µ2β + t2 ˆ β be such that for some constants > 0 and c > 0, Let φ ∈ H N N |φ(ξ) − b(ω)ρ− 2 −2β+α0 | < cρ− 2 −2β+α0 − for all ρ > 1, (3.1) 0 < α0 < β, where ρ ≥ 0 and ω ∈ S N −1 (the unit sphere of RN ) are the spherical coordinates of ξ ∈ RN , and where b(ω) is a bounded measurable function on S N −1 , which is non zero on a set of positive measure. Remark 3.1. By using Lemma 2.4, under the assumption (3.1) about φ, we have that ˆ β is dense in H ˆ α1 if and only if α1 ≤ α0 . H (3.2) φ ˆ β is dense in H ˆ 0 and the pair (H ˆ β, H ˆ 0 ) satisfies Thus, in particular, we have that H φ φ (2.1).
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
9
Theorem 3.2. Let ϕ ∈ H β be such that its Fourier transform φ satisfies (3.1), and let θ0 = α0 /β. Then 0 β (3.3) H , H θ ,1 ⊂ H 0 , Hϕβ θ ,∞ , 0
0
and (3.4)
0 β H , Hϕ θ,q = H 0 , H β θ,q , 0 < θ < θ0 , q ∈ [1, ∞].
Proof. By the definition of the inner product on H β , we have that the embedding (3.3) is equivalent to i h h i ˆ 0, H ˆβ ˆ 0, H ˆβ (3.5) H ⊂ H . φ θ0 ,1
θ0 ,∞
The identity (3.4) follows from (3.3) and Lemma 2.5. Thus, we have to prove only (3.5). From (2.6), ∞
Z (3.6)
kˆ uk[Hˆ 0 ,Hˆ β ]θ
0 ,1
tθ0
=
Z
0
µ(ξ)2β |ˆ u(ξ)|2 dξ µ(ξ)2β + t2
1/2
dt , t
and due to (2.21) we get 2
(3.7)
kˆ uk2Hˆ 0 ,Hˆ β ] [
φ θ ,∞ 0
= sup t
2θ0
(ˆ u, u ˆ)Y,t + t
2+2θ0
t>0
|(ˆ u, φ)Y,t | (φ, φ)X,t
! ,
ˆ β and Y = H 0 . Using (3.1), it is easy to see that, for a large enough where X = H δ ≥ 1, one can find positive constants c1 , c2 , such that
(3.8)
c1 tθ0 −1 ≤ ((φ, φ)X,t )1/2 =
µ4β φ ,φ 2β µ + t2
1/2
N
|φ(ξ)| < c2 |ρ|− 2 −2β+α0
≤ c2 tθ0 −1 , t ≥ δ,
for |ξ| > 1.
Following the proof of Theorem 2.7, we see that, in order to prove (3.5), it is enough to verify the following inequality (3.9) M := sup t1+θ0 t>δ
|(ˆ u, φ)Y,t | ≤ ckˆ uk[Hˆ 0 ,Hˆ β ]θ ,1 0 ((φ, φ)X,t )1/2
ˆ 0, H ˆ β ]θ ,1 , for all u ˆ ∈ [H 0
for some positive constants c = c(δ). From (3.8), we get Z µ(ξ)2β M ≤ c sup t2 |(ˆ u, φ)Y,t | ≤ c sup t2 |ˆ u(ξ)||φ(ξ)| dξ µ(ξ)2β + t2 t>δ t>δ Z Z Z (3.10) 2β 2β ≤ µ(ξ) |ˆ u(ξ)||φ(ξ)| dξ = µ(ξ) |ˆ u||φ| dξ + µ(ξ)2β |ˆ u||φ| dξ. |ξ|1
The last two integrals can be estimated as follows: Z (3.11) µ(ξ)2β |ˆ u||φ| dξ ≤ c kˆ ukkφk ≤ c(φ)kˆ uk[Hˆ 0 ,Hˆ β ]θ |ξ|1
|ξ|>1
2 = π
Z
2 π
Z
=
∞
0
µ3β |ˆ u||φ| dξ dt µ2β + t2
Z |ξ|>1
∞
µβ |ˆ u| µ2β |φ| dξ dt 2 1/2 2β + t ) (µ + t2 )1/2
Z
(µ2β
0
|ξ|>1
1/2
≤
2 π
∞
Z
2β
0
2
µ |ˆ u| dξ µ2β + t2
Z
∞
Z 0
0
2
µ |φ| dξ µ2β + t2
dt
|ξ|>1
µ2β |ˆ u|2 dξ µ2β + t2
Z
1/2
((φ, φ)X,t )
dt
|ξ|>1
1/2
≤c
4β
1/2
∞
Z
|ξ|>1
2 ≤ π
1/2 Z
tθ0
Z
2β
2
µ |ˆ u| dξ µ2β + t2
dt ≤ ckˆ uk[Hˆ 0 ,Hˆ β ]θ ,1 . 0 t
|ξ|>1
Combining (3.10)-(3.12), we conclude the validity of (3.9) and the proof of the theorem. Next, we prepare for the generalization of the previous result. Let φ1 , φ2 , . . . , φn ∈ ˆ β (RN ) be such that for some constants > 0 and c > 0, we have H N |φi (ξ) − φ˜i (ξ)| < cρ− 2 −2β+αi − f or |ξ| > 1 (3.13) 0 < αi < β, i = 1, . . . , n, where N φ˜i (ξ) = bi (ω)ρ− 2 −2β+αi , ξ = (ρ, ω),
and bi (·) is a bounded measurable function on S N −1 , which is non zero on a set of positive measure. Using Lemma 2.4 and the Riesz representation theorem, it is β easy to check that, under the assumption (3.1), the space HK is dense in H 0 . Theorem 3.3. Let ϕ1 , ϕ2 , . . . , ϕn be functions in H β such that the corresponding Fourier transforms φ1 , φ2 , . . . , φn are well defined and satisfy (3.13). Assume that the functions φ˜1 , φ˜2 , . . . , φ˜n defined in (3.13) are linearly independent. Let K = span{ϕ1 , ϕ2 , . . . , ϕn }, α0 := min{α1 , α2 , . . . , αn }, and let θ0 = α0 /β. Then, h i 0 β β (3.14) H , H θ ,1 ⊂ H 0 , HK , 0
θ0 ,∞
and (3.15)
h
β H 0 , HK
i θ,q
= H 0 , H β θ,q , 0 < θ < θ0 , q ∈ [1, ∞].
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
11
Proof. We apply the Theorem 2.7 for X = H β , Y = H 0 , K = span{ϕ1 , . . . , ϕn } and s0 = θ0 . By using the hypothesis (3.13) and Theorem 3.2, we get 0 β H , H θ ,1 ⊂ H 0 , Hϕβi θ ,∞ , f or i = 1, 2, . . . , n. 0
0
Thus, (A1) is satisfied. The proof of (A2) follows from (3.13) by using elementary linear algebra and calculus tools, and it is presented in [4]. The identity (3.15), is a direct consequence of Lemma 2.5. The proof is completed. The interpolation problem between H α and a subspace of H β of finite codimension with α < β arbitrary real numbers, can be approached in the light of the previous theorem. Let ϕ1 , ϕ2 , . . . , ϕn ∈ H β and let α < β be such that the corresponding Fourier transform, φ1 , φ2 , . . . , φn are well defined and satisfy for some positive constants c and , N |φi (ξ) − φ˜i (ξ)| < cρ− 2 −2β+γi − f or |ξ| > 1 (3.16) α < γi < β, i = 1, . . . , n, where N φ˜i (ξ) = bi (ω)ρ− 2 −2β+γi , ξ = (ρ, ω),
and bi (·) is a bounded measurable function on S N −1 , which is non zero on a set of positive measure. Theorem 3.4. Let ϕ1 , ϕ2 , . . . , ϕn ∈ H β be such that the corresponding Fourier transforms φ1 , φ2 , . . . , φn are well defined and satisfy (3.16). Assume that the functions φ˜1 , φ˜2 , . . . , φ˜n are linearly independent. Let L = span{ϕ1 , ϕ2 , . . . , ϕn }, γ0 := min{γ1 , γ2 , . . . , γn }, and let θ0 = (γ0 − α)/(β − α). Then, h i α β (3.17) H , H θ ,1 ⊂ H α , HLβ , 0
θ0 ,∞
and (3.18)
h
H α , HLβ
i θ,q
= H α , H β θ,q , 0 < θ < θ0 , q ∈ [1, ∞].
Proof. The proof is a direct consequence of Theorem 3.3 and the fact that T : H α → H 0 defined by Tˆu = µα u ˆ, u ∈ H α , is an isometry from H α to H 0 and from β β−α H to H . 4. Shift estimates for the Biharmonic operator on polygonal domains Let Ω be a polygonal domain in R2 with boundary ∂Ω. Let ∂Ω be the polygonal curve P1 P2 · · · Pm P1 . At each point Pj , we denote by ωj the measure of the angle at Pj measured from inside Ω. Let ω := max{ωj : j = 1, 2, . . . , m}. We consider the biharmonic problem: Given f ∈ L2 (Ω), find u such that (1.1). Let V = H02 (Ω) and X Z ∂2v ∂2u dx, u, v, ∈ V. a(u, v) := Ω ∂xi ∂xj ∂xi ∂xj 1≤i,j≤2
The bilinear form a(·, ·) defines a scalar product on V and the induced norm is equivalent to the standard norm on H02 (Ω). The variational form of (1.1) is : Find u ∈ V such that Z (4.1) a(u, v) = f v dx for all v ∈ V. Ω
12
C. BACUTA
Let Sω denote a sector domain defined by Sω = {(r, θ), 0 < r < rω , −ω/2 < θ < ω/2}, where, without loss of generality, we assume that Sω ⊂ Ω for a sufficiently small rω , and that the vertex corresponding to the largest angle of Ω is the vertex of the sector domain Sω . We associate to (1.1), with Ω = Sω , the following characteristic equation sin2 (zω) = z 2 sin2 ω.
(4.2)
In order to simplify the exposition of the proof, we assume that r r ω2 sin ω 2 − 1 6= + 1 − (4.3) sin 2 − sin ω ω2 and Rez 6= 2 for any solution z of (4.2).
(4.4)
The restriction (4.3) assures that the equation (4.2) has only simple roots. Let z1 , z2 , . . . , zn be all the roots of (4.2) such that 0 < Re(zj ) < 2. It is known that, under the assumptions (4.3) and (4.4), the solution u of (4.1) can be written as n X (4.5) u = uR + kj Sj , j=1 4
where uR ∈ H (Sω ) is the regular part of u, the Sj ’s are called the singular functions, and the kj ’s are the coefficients of the singular functions. More precisely, for j = 1, 2, . . . , n, we have Sj (r, θ) = r1+zj uj (θ), with uj smooth func0 0 tion on [−ω/2, R ω/2] satisfying uj (−ω/2) = uj (ω/2) = uj (−ω/2) = uj (ω/2) = 0, and kj = cj Sω f ϕj dx, with cj being nonzero and depending only on ω. The function ϕj is called the dual singular function of the singular function Sj and ϕj (r, θ) = η(r) r1−zj uj (θ) − wj , where wj ∈ V is defined for a smooth truncation function η to be the solution of (4.1) with f = ∆2 (η(r) r1−zj uj (θ)). In addition, we have that the regular part uR satisfies (4.6)
kuR kH 4 (Sω ) ≤ ckf k,
for all f ∈ L2 (Sω ).
For the proof of the expansion (4.5), see, e.g., [14, 16, 17, 21, 25]. Next, we define K = span{ϕ1 , ϕ2 , . . . , ϕn } and 1 (4.7) s0 := min{Re(zj ) | j = 1, 2, . . . , n}. 2 The main theorem of the paper is based on the following embedding result: Lemma 4.1. For Ω = Sω and K defined above, we have (4.8)
[H −2 (Ω), L2 (Ω)]s0 ,1 ⊂ [H −2 (Ω), L2 (Ω)K ]s0 ,∞ .
To prove it, we first reduce the problem to the case Ω = R2 and then apply the interpolation results of the previous section. A detailed proof is given in Appendix 6.1. Now, we are ready to state the main result of the paper. Theorem 4.2. Let Ω be a polygonal domain in R2 with boundary ∂Ω. Assume that all the angles ωj of ∂Ω satisfy (4.3) and (4.4). Let s0 = s0 (ω) be the threshold defined by (4.7), where ω is the largest inner angle of the polygon ∂Ω. If u is the variational solution of (1.1), then the regularity estimate (1.3) holds. Consequently, the classical estimate (1.2) holds also.
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
13
Proof. We consider a covering of Ω with m + 1 subdomains. For each angle of ∂Ω, we associate a sector domain Sωj ⊂ Ω and complete the covering of Ω with a domain Ω0 ⊂ Ω such that Ω0 has smooth boundary and ∂Ω0 does not contain any vertex of ∂Ω. As done in [3], by using a smooth partition of unity subordinated to the described covering of Ω, the problem of deriving a shift estimate on Ω can be reduced to deriving similar estimates on each of the subdomains of the covering. Since the amount of regularity of the solution of (1.1) for a sector domain decreases as the angle of the a sector domain increases, (see Figure 1), to prove (1.3), it will be enough to prove the estimate for Ω = Sω and for Ω = Ω0 . First, let us consider the case Ω = Ω0 . Then, it is known that the solution u of (4.1) satisfies kukH 4 (Ω) ≤ ckf k,
for all f ∈ L2 (Ω),
and for all f ∈ H −2 (Ω).
kukH 2 (Ω) ≤ ckf kH −2 (Ω) ,
Interpolating these two inequalities with 0 < s < 1 and 1 ≤ q ≤ ∞, we obtain (4.9) kuk[H 2 (Ω),H 4 (Ω)]s,q ≤ ckf k[H −2 (Ω),L2 (Ω)]s,q , f ∈ H −2 (Ω), L2 (Ω) s,q . In particular, taking s = s0 and q = ∞ we get (4.10)
kuk[H 2 (Ω),H 4 (Ω)]s
0 ,∞
≤ ckf k[H −2 (Ω),L2 (Ω)]s
0 ,∞
, f ∈ H −2 (Ω), L2 (Ω) s
0 ,∞
.
Using the standard embedding result [H −2 (Ω), L2 (Ω)]s0 ,1 ⊂ [H −2 (Ω), L2 (Ω)]s0 ,∞ , and (4.10), we get , f ∈ H −2 (Ω), L2 (Ω) s ,1 . 0 2 2+s0 4 Thus, by introducing the Besov spaces B∞ (Ω) := H (Ω), H (Ω) s ,∞ and 0 B1−2+s0 (Ω) := H −2 (Ω), L2 (Ω) s ,1 , the estimate (4.11) becomes (1.3). 0 Let us consider now the case Ω = Sω. From the expansion (4.5) and the estimate (4.6), we have (4.11)
(4.12)
kuk[H 2 (Ω),H 4 (Ω)]s
0 ,∞
≤ ckf k[H −2 (Ω),L2 (Ω)]s
kukH 4 (Ω) ≤ ckf k,
0 ,1
for all f ∈ L2 (Ω)K .
Combining (4.12) with the standard estimate (4.13)
kukH 2 (Ω) ≤ ckf kH −2 (Ω) ,
for all f ∈ H −2 (Ω),
by interpolation, we obtain (4.14)
kuk[H 2 (Ω),H 4 (Ω)]s0 ,∞ ≤ ckf k[H −2 (Ω),L2 (Ω)K ]s0 ,∞ .
From (4.14) and the embedding result (4.8) of Lemma 4.1, we conclude that (1.3) holds for Ω = Sω . Here, s0 corresponds to the largest inner angle ω of the polygon. The function ω → 2 + 2s0 (ω) represents the regularity threshold for the biharmonic problem on Sω . From the plot of the function, given in Figure 1, we see that function decreases on the interval (0, 2π). Thus, by involving interpolation and the reiteration theorem, (see part iii) and part iv) of Proposition 6.2), the estimate (1.3) holds for all sector domains Sωj , j = 1, 2, . . . , m.
14
C. BACUTA
6
5
4
3
2
1
0
omega
.7Pi
1.23Pi
1.43Pi
2Pi
Figure1: Regularity for the biharmonic problem Therefore, we have proved (1.3) for any polygonal domain with not self intersecting boundary. Next, we prove the standard estimate (1.2). Let T : H −2 (Ω) → H02 (Ω) be defined by T f = u, where u is the solution of (4.1). Then, (1.3) becomes (4.15)
kT f k[H 2 ,H 4 ]s0 ,∞ ≤ ckf k[H −2 ,H 0 ]s0 ,1 ,
and the standard estimate (4.13) can be written us (4.16)
kT f k[H 2 ,H 4 ]0,2 ≤ ckf k[H −2 ,H 0 ]0,2 .
Let 0 < s < s0 . By Proposition 6.2 part iii) (“By interpolation” Theorem) , applied with λ = ss0 and q = 2, we obtain (4.17)
kT f k[[H 2 ,H 4 ]0,2 ,[H 2 ,H 4 ]s
]
0 ,∞ λ,2
≤ ckf k[[H −2 ,H 0 ]0,2 ,[H −2 ,H 0 ]s
]
0 ,1 λ,2
.
By applying the reiteration theorem, we recover the classical estimate (1.2). The proof is completed. Remark 4.3. Figure 1 gives the graph of the function ω → 2 + 2s0 (ω) which represents also the regularity threshold for the biharmonic problem in terms of the largest inner angle of the polygon ω ∈ (0, π) ∪ (π, 2π). From (4.2) and (4.7), it follows that s0 (ω) → 1/2 for ω → π and s0 (ω) → 1/4 for ω → 2π. On the same graph we represent the number of singular (dual singular) functions as a function of ω ∈ (0, π) ∪ (π, 2π). Note that, if ω0 ∈ (0, 2π) is the solution of tan ω = ω (ω0 ≈ 1.43π) and ω > ω0 , then, the the dimension of the space K is six. Remark 4.4. If the coefficient of the most singular function in the expansion (4.5) is not zero, then the solution u of (1.1) does not belong to Bq2+2s0 (Ω) for any q < ∞. Thus, any possible improvement of the shift estimate (1.3) would be in the form: (4.18)
2+2s0 kukB∞ ≤ ckf kBq−2+2s0 (Ω) , f ∈ Bq−2+2s0 (Ω), q > 1. (Ω)
The validity of (4.18) remains an open problem.
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
15
5. Regularity for the Stokes and Navier-Stokes systems on convex polygonal domains As a consequence of the regularity for the biharmonic problem we prove shift estimates for the Stokes and the Navier-Stokes systems. Let Ω be a convex polygonal domain in R2 with boundary ∂Ω and let ω be the measure of the largest angle of ∂Ω. In this case, we have that s0 = s0 (ω) ∈ (1/2, 1), (see Remark 4.3). Let γ0 := 2s0 − 1. Note that γ0 ∈ (0, 1). Then, according to the estimate (1.2), the solution u of the biharmonic problem (1.1) satisfies (5.1)
kukH 3+γ ≤ ckf kH −1+γ ,
for all f ∈ H −1+γ (Ω), −1 ≤ γ < γ0 .
The parameter γ0 can be viewed as the exact amount of extra regularity due to the convexity of the domain Ω. Next, we consider the steady-state Stokes problem in the velocity-pressure formulation. More precisely we consider the following problem: Given F ∈ (H γ )2 with 0 ≤ γ < γ0 , find the vector-valued function u and the scalar-valued function p satisfying −∆u + ∇p = F in Ω, ∇·u=0 in Ω, (5.2) u = 0 on ∂Ω, R p = 0. Ω The first two equations are considered in the standard weak sense. According to a well known Kellogg-Osborn result, [20], we have that (5.2) has a unique solution (u, p) ∈ ((H 2 (Ω))2 , H 1 (Ω)). Since ∇ · u = 0 in Ω, and u = 0 on ∂Ω, one can find w ∈ H02 , (see for example I.3.1 in [15] or [2]), such that ∂w ∂w ,− . u = (u1 , u2 ) = curl w := ∂x2 ∂x1 If we substitute u = curl w in (5.2), we get ( ∂p ∂w −∆( ∂x ) + ∂x = f1 2 1 (5.3) ∂p ∂w ∆( ∂x1 ) + ∂x2 = f2
in Ω, in Ω,
∂ ∂ and ∂x to the where (f1 , f2 ) = F. Next, we apply the differential operators − ∂x 2 1 first and second equations of (5.3), respectively, and sum up the two new equations. Thus, we have that w ∈ H02 and
∆2 w =
∂f1 ∂f2 − ∂x1 ∂x2
in Ω.
Consequently, for a fixed γ ∈ (0, γ0 ), from (5.1), we have that
∂f2 ∂f1
− , kwkH 3+γ ≤ c ∂x1 ∂x2 −1+γ H
where c is a constant independent of F. It follows that kuk(H 2+γ )2 ≤ ckFk(H γ )2 ,
for all F ∈ (H γ (Ω))2 .
From the first part of (5.2), we have ∇p = ∆u + F. Hence, k∇pk(H γ )2 ≤ k∆uk(H γ )2 + kFk(H γ )2 ≤ kuk(H 2+γ )2 + kFk(H γ )2 ≤ ckFk(H γ )2 . In conclusion we obtain:
16
C. BACUTA
Theorem 5.1. Let Ω be a convex polygonal domain in R2 with ω the measure of the largest angle. Let γ0 = 2s0 (ω) − 1 and let (u, p) be the solution of (5.2). Then for any γ ∈ (0, γ0 ), there exist a constant c such that (5.4)
kuk(H 2+γ )2 + kpkH 1+γ ≤ ckFk(H γ )2
for all F ∈ (H γ )2 .
We conclude this section by a similar regularity result for solutions of the NavierStokes equations
(5.5)
−∆u + (u · ∇)u + ∇p = G in Ω, ∇·u=0 in Ω, u = 0 on ∂Ω, R p = 0. Ω
Theorem 5.2. Let Ω be a convex polygonal domain in R2 with ω the measure of the largest angle. Let γ0 = 2s0 (ω) − 1, G ∈ (H γ )2 with 0 ≤ γ < γ0 and let (u, p) ∈ (H01 )2 × L2 be a solution of (5.5). Then, (5.6)
u ∈ (H 2+γ )2
and p ∈ H 1+γ .
Proof. The main idea of the proof is due to Temam. The proof given here was inspired by the proof presented by Kellogg and Osborn in [20]. Let (u, p) ∈ (H01 )2 × L2 be a solution of (5.5) and denote F = G − (u · ∇)u. According to [20], we have that u ∈ (H 5/3 )2 . In particular we get that ∇u ∈ (H 2/3 )2 and u is bounded. Thus F ∈ (H min{γ,2/3} )2 , and by using (5.4), we have that u ∈ (H 2+min{γ,2/3} )2 and ∇u ∈ (H 1+min{γ,2/3} )2 . Then, we deduce that, in fact, F ∈ (H γ )2 . Using (5.4) again , we conclude that (5.6) holds. Remark 5.3. In the light of Theorem 4.2, we can extend the last two results to hold for the threshold value γ = γ0 by considering the interpolation norms with the second index of interpolation q 6= 2. More precisely, we have to take q = 1 for the norm of F or G and q = ∞ for the norm of (u, p). In other words, the last two theorems hold if the spaces of type H k+γ are replaced by Bqk+γ0 with the above mentioned choice for q. 6. Appendix Here, we present the postponed proofs and some auxiliary results. 6.1. The Postponed Proofs. Proof of Theorem 2.6. Let u be fixed in Y and set (6.1)
−1 −1 w := (I + t2 S −1 )−1 u and wϕ := (I + t2 SK ) u.
Then, according to (2.12), (2.6) and (2.13), we have Z Z ∞ dt kuk2[Y,XK ]s,2 = t2s (wK , u)Y , kuk2[Y,X]s,2 = t 0 and kuk2[Y,XK ]s,∞ = sup t2s (wK , u)Y ,
∞
t2s (w, u)Y 0
t>0
respectively. Thus, (2.18) and (2.19) would follow provided we establish
(6.2) (wK , u)Y = (w, u)Y + t2 Mt−1 yt , yt .
dt , t
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
17
From (6.1), we get that −1 (I + t2 SK )wK = u.
(6.3)
Combining (2.15) and (6.3) we obtain (I + t2 S −1 )wK = u + t2
n X
αk ϕk ,
k=1
where αk are constants determined by QK (S −1 wϕ ) = applying (I + t2 S −1 )−1 to both sides, we have wK = w + t2
(6.4)
n X
Pn
k=1
αk ϕk . Equivalently,
αk (I + t2 S −1 )−1 ϕk .
k=1
We calculate the coefficients αk by taking the (·, ·)X inner product with ϕ on both sides of (6.4), i.e., (wK , ϕj )X = (w, ϕj )X +t2
n X
αk (I + t2 S −1 )−1 ϕk , ϕj
X
= (w, ϕj )X +t2
n X
αk (ϕk , ϕj )X,t .
k=1
k=1
−1 From (6.3), since SK wϕk ∈ XK , one sees that (wK , ϕj )X = (u, ϕj )X . With the notation adopted in (2.16) and (6.1) we obtain the following n × n system: n X
αk (ϕk , ϕj )X,t t = t−2 ((u, ϕj )X − (u, ϕj )X,t ).
k=1
Using (2.4) and a simple manipulation of the operator S we get (6.5) Thus
t2 (u, ϕj )Y,t = (u, ϕj )X − (u, ϕj )X,t . n X
αk (ϕk , ϕj )X,t = (u, ϕj )Y,t = (yt )j .
k=1
Let α ∈ Cn be the the vector with components α1 , α2 , · · · , αn , then α = Mt−1 yt . Now, going back to (6.4), we get (wK , u)Y = (w, y) + t2
n X
αk (ϕk , u)Y,t .
k=1
By substituting the vector α we obtain (6.2) and complete the proof. Next, we present a general subspace interpolation lemma used for the proof of ˜ Y˜ ) be two pairs of Hilbert spaces satisfying (2.1) and Theorem 4.2. Let (X, Y ), (X, ∗ ∗ ∗ ˜∗ ˜ let (Y , X ), (Y , X ) be the corresponding dual pairs. Then, the dual pairs satisfy (2.1) also, see e.g., [18]. We further identify Y ∗ with Y and Y˜ ∗ with Y˜ , and assume that there are linear operators E and R such that (6.6)
˜ are bounded operators, E : Y → Y˜ , E : X → X
(6.7)
˜ → X, are bounded operators, R : Y˜ → Y, R : X
(6.8)
REu = u
for all u ∈ Y.
18
C. BACUTA
e = E(K) ⊂ Y˜ = Y˜ ∗ be closed subspaces of Y and Y˜ , Let K ⊂ Y = Y ∗ , K respectively, and denote the corresponding orthogonal complements by YK and Y˜Ke . Let θ0 ∈ (0, 1) be such that ˜ ∗ , Y˜ ]θ ,1 ⊂ [X ˜ ∗ , Y˜ e ]θ ,∞ . (6.9) [X K
0
0
Lemma 6.1. Assume that (6.6)-(6.9) are satisfied. Then, [X ∗ , Y ]θ0 ,1 ⊂ [X ∗ , YK ]θ0 ,∞
(6.10)
Proof. Using the duality, from (6.6)-(6.8), we obtain linear operators E ∗ , R∗ such that ˜ ∗ → X ∗ , are bounded operators, (6.11) E ∗ : Y˜ → Y, E ∗ : X (6.12)
˜ ∗ are bounded operators, R∗ : Y → Y˜ , R∗ : X ∗ → X E ∗ R∗ u = u
(6.13)
for all u ∈ Y,
E ∗ maps Y˜Ke to YK .
(6.14)
From (6.11) and (6.14), by interpolation, we obtain (6.15)
kE ∗ vk[X ∗ ,YK ]θ0 ,∞ ≤ ckvk[X˜ ∗ ,Y˜ e ]θ K
0 ,∞
for all v ∈ Y˜ .
For any u ∈ Y , we take v := R∗ u in (6.15), and use (6.13) to get (6.16)
kuk[X ∗ ,YK ]θ0 ,∞ ≤ ckR∗ uk[X˜ ∗ ,Y˜ e ]θ K
0 ,∞
for all u ∈ Y.
Also, from the hypothesis (6.9), we deduce that (6.17)
kR∗ uk[X˜ ∗ ,Y˜ e ]θ K
0 ,∞
≤ ckR∗ uk[X˜ ∗ ,Y˜ ]θ
0 ,1
for all u ∈ Y.
From (6.12), again by interpolation, we have in particular (6.18)
kR∗ uk[X˜ ∗ ,Y˜ ]θ
0 ,1
≤ ckuk[X ∗ ,Y ]θ0 ,1
for all u ∈ Y.
Combining (6.16)-(6.18), it follows that (6.19)
kuk[X ∗ ,YK ]θ0 ,∞ ≤ ckuk[X ∗ ,Y ]θ0 ,1
for all u ∈ Y.
Since Y is dense in both [X ∗ , Y ]θ0 ,1 and [X ∗ , YK ]θ0 ,∞ we obtain (6.10).
Proof of Lemma 4.1. In [4], it was proven that that there are bounded operators E and R, E : L2 (Ω) −→ L2 (R), E : H02 (Ω) −→ H 2 (R2 ), R : L2 (R2 ) −→ L2 (Ω), R : H 2 (R2 ), −→ H02 (Ω) such that REu = u, for all u ∈ L2 (Ω). Next, let φj be the Fourier transform of Eϕj , j = 1, . . . , n. Using the asymptotic expansion of integrals estimates presented in [4], (see also [11, 19, 27]), we have that the functions {Eϕj , j = 1, . . . , n} satisfy for some positive constants c and , |φj (ξ) − φ˜j (ξ)| < cρ−1+(−2+sj )− f or |ξ| > 1 (6.20) −2 < −2 + si < 0, i = 1, . . . , n, where sj = Re(zj ) and φ˜j (ξ) = bi (ω)ρ−1+(−2+sj ) , ξ = (ρ, ω) in polar coordinates,
SUBSPACE INTERPOLATION AND REGULARITY ESTIMATES
19
and bj (·) is a bounded measurable function on the unit circle, which is non zero on a set of positive measure. Let s0 = s0 (ω) be defined by (4.7). Thus, we have that the functions {Eϕj , j = 1, . . . , n} satisfy the hypothesis (3.16) of Theorem 3.4 with N = 2, β = 0, α = −2 and γj = −2 + sj , j = 1, . . . , n. Let us define now L := span{Eϕj , j = 1, . . . , n}. By Theorem 3.4 applied with θ0 = s0 , we have that [H −2 (R2 ), L2 (R2 )L ]s0 ,1 ⊂ [H −2 (R2 ), L2 (R2 )L ]s0 ,∞ .
(6.21)
Finally, using (6.21) and Lemma 6.1, we conclude that (4.8) holds. 6.2. Appendix B. Interpolation Results. Using the notation of Section 2, we review the classical interpolation results involved in this paper. The proofs can be found, for example, in [9, 10, 23]. Proposition 6.2. Let (X, Y ) be a pair of Banach spaces satisfying (2.1). The following hold. i) The relative strength of the smoothness index s: [Y, X]s,q ⊂ [Y, X]s˜,˜q , 0 < s˜ < s < 1, and 1 ≤ q, q˜ ≤ ∞. ii) The relative strength of the second index of interpolation: [Y, X]s,q ⊂ [Y, X]s,˜q , 1 ≤ q ≤ q˜ ≤ ∞. ˜ Y˜ ) be another pair of Banach spaces iii) “By Interpolation” Theorem: Let (X, ˜ satisfy satisfying (2.1). If T : Y → Y˜ and T : X → X kT f kY˜ ≤ c0 kf kY ,
and kT f kX˜ ≤ c1 kf kX ,
then, 1−λ λ kT f k[Y˜ ,X] c1 kf k[Y,X]λ,q , 0 < λ < 1, and 1 ≤ q ≤ ∞. ˜ λ,q ≤ c0
iv) The Reiteration Theorem: For any 0 ≤ s0 < s1 ≤ 1, q0 , q1 , q ∈ [1, ∞], and 0 < λ < 1, [[Y, X]s0 ,q0 , [Y, X]s1 ,q1 ]λ,q = [Y, X](1−λ)s0 +λs1 ,q . References [1] R. A. Adams. Sobolev Spaces. Academic Press, New York, 1975. [2] C. Bacuta and J. H. Bramble. Regularity estimates for solutions of the equations of linear elasticity in convex plane polygonal domains. Z. angew. Math. Phys.(ZAMP), 54(2003), 874878. [3] C.Bacuta, J. H. Bramble and J. Pasciak. Using finite element tools in proving shift theorems for elliptic boundary value problems. Numerical Linear Algebra with Application, Volume 10, Issue no 1-2, 33-64, 2003. [4] C.Bacuta, J. H. Bramble and J. Pasciak. Shift theorems for the biharmonic Dirichlet problem, Recent Progress in Computational and Applied PDEs, Kluwer Academic/Plenum Publishers, volume of proceedings for the CAPDE conference held in Zhangjiajie, China, July 2001. [5] C. Bacuta, V. Nistor and L. Zikatanov. Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps. Numerische Mathematik, Vol. 100, No 2, 2005, pp. 165 -184 [6] C. Bacuta, V. Nistor and L. Zikatanov. Improving the rate of convergence of ‘high order finite elements’ on polyhedra I: apriori estimates. Numerical Functional Analysis and Optimization, Vol. 26, No 6, 2005, pp 613-639. [7] C. Bacuta, V. Nistor and L. Zikatanov. Improving the rate of convergence of ‘high order finite elements’ on polyhedra II: Mesh Refinements and Interpolation. Numerical Functional Analysis and Optimization, Vol. 28, No 7-8, 2007, pp 775-824.
20
C. BACUTA
[8] C.Bacuta, J. H. Bramble, and J. Xu. Regularity estimates for elliptic boundary value problems in Besov spaces, Mathematics of Computation, 72 (2003), 1577-1595. [9] C. Bennett and R. Sharpley. Interpolation of Operators. Academic Press, New-York, 1988. [10] J. Bergh and J. Lofstrom. Interpolation Spaces. Springer-Verlag, New York, 1976. [11] N. Bleistein and R. Handelsman. Asymptotic expansions of integrals. Holt, Rinehart and Winston, New York, 1975. [12] H. Blum and R. Rannacher. On the Boundary Value Problem of Biharmonic Operator on Domains with Angular Corners, Math. Meth. in the Appl. Sci., 2(1980) 556-581. [13] S. Brenner and L.R. Scott. The Mathematical Theory of Finite Element Methods. SpringerVerlag, New York, 1994. [14] M. Dauge. Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics 1341. Springer-Verlag, Berlin, 1988. [15] V. Girault and P.A. Raviart. Finite Element Methods for Navier-Stokes Equations. SpringerVerlag, Berlin, 1986. [16] P. Grisvard. Elliptic Problems in Nonsmooth Domains. Pitman, Boston, 1985. [17] P. Grisvard. Singularities in Boundary Value Problems. Masson, Paris, 1992. [18] W. Hackbusch. Elliptic Differential Equations. Springer-Verlag, Berlin Heidelberg, 1992. [19] R. B. Kellogg. Interpolation between subspaces of a Hilbert space, Technical note BN-719, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, 1971. [20] R. B. Kellogg and J. Osborn. A regularity result for Stokes Problem in a convex Polygon. Journal of Functional Analysis, 21, 1976, 397-431. [21] V. Kondratiev. Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc., 16:227-313, 1967. [22] V. A. Kozlov, V. G. Mazya and J. Rossmann. Elliptic Boundary Value Problems in Domains with Point Singularities. American Mathematical Society, Mathematical Surveys and Monographs, vol. 52, 1997. [23] J. L. Lions and E. Magenes. Non-homogeneous Boundary Value Problems and Applications, I. Springer-Verlag, New York, 1972. [24] J. L. Lions and P. Peetre. Sur une classe d’espaces d’interpolation. Institut des Hautes Etudes Scientifique. Publ.Math., 19:5-68, 1964. [25] S. A. Nazarov and B. A. Plamenevsky. Elliptic Problems in Domains with Piecewise Smooth Boundaries. Expositions in Mathematics, vol. 13, de Gruyter, New York, 1994. [26] J. Ne˘ cas. Les Methodes Directes en Theorie des Equations Elliptiques. Academia, Prague, 1967. [27] F. W. Olver. Asymptotics and Special Functions. Academic Press, New York, 1974. [28] F. Riesz and B. Sz-Nagy. Functional Analysis (Second Edition), Frederick Ungar, New York, 1969. [29] K. Yosida Functional Analysis. Springer-Verlag, Berlin Heidelberg, 1978. Mathematical Sciences, University of Delaware, Newark, DE 19716, USA. E-mail address: [email protected]
