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 )
0<t≤δ
t>δ
≤
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 û is defined
by
Z
û(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 û(ξ)v̂(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 û, u ∈ D(S) = H 2β (RN ),
Su
1
where µ(ξ) = (1 + |ξ|2 ) 2 , ξ ∈ RN . For the remaining part of this chapter, H β
denotes the space H β (RN ) and Ĥ β is the space {û |u ∈ H β }. For û, v̂ ∈ Ĥ β , we
define the inner product and the norm by
Z
1/2
(û, v̂)β = µ2β ûv̂ dζ, ||û||β = (û, û)β .
To simplify the notation, we denote the the inner products (·, ·)0 and h·, ·i0 by (·, ·)
and h·, ·i, respectively. The norm || · ||0 on H 0 or Ĥ 0 is simply || · ||. Thus, for
X = Ĥ β and Y = Ĥ 0 , we have
2β
4β
µ û
µ û
(û, φ)Y,t =
,
φ
and
(û,
φ)
=
,
φ
.
X,t
µ2β + t2
µ2β + t2
Let φ ∈ Ĥ β be such that for some constants > 0 and c > 0,
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
(3.2)
Ĥφβ is dense in Ĥ α1
if and only if
α1 ≤ α0 .
Thus, in particular, we have that Ĥφβ is dense in Ĥ 0 and the pair (Ĥφβ , Ĥ 0 ) satisfies
(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
(3.5)
Ĥ 0 , Ĥ β
⊂ Ĥ 0 , Ĥφβ
.
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)
kûk[Ĥ 0 ,Ĥ β ]θ
0 ,1
tθ0
=
Z
0
µ(ξ)2β
|û(ξ)|2 dξ
µ(ξ)2β + t2
1/2
dt
,
t
and due to (2.21) we get
2
(3.7)
kûk2Ĥ 0 ,Ĥ β
[
φ]
θ0 ,∞
= sup t
2θ0
(û, û)Y,t + t
2+2θ0
t>0
|(û, φ)Y,t |
(φ, φ)X,t
!
,
where X = Ĥ β and Y = H 0 . Using (3.1), it is easy to see that, for a large enough
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
|(û, φ)Y,t |
≤ ckûk[Ĥ 0 ,Ĥ β ]θ ,1
0
((φ, φ)X,t )1/2
for all û ∈ [Ĥ 0 , Ĥ β ]θ0 ,1 ,
for some positive constants c = c(δ). From (3.8), we get
Z
µ(ξ)2β
M ≤ c sup t2 |(û, φ)Y,t | ≤ c sup t2
|û(ξ)||φ(ξ)| dξ
µ(ξ)2β + t2
t>δ
t>δ
Z
Z
Z
(3.10)
2β
2β
≤ µ(ξ) |û(ξ)||φ(ξ)| dξ =
µ(ξ) |û||φ| dξ +
µ(ξ)2β |û||φ| dξ.
|ξ|<1
|ξ|>1
The last two integrals can be estimated as follows:
Z
(3.11)
µ(ξ)2β |û||φ| dξ ≤ c kûkkφk ≤ c(φ)kûk[Ĥ 0 ,Ĥ β ]θ
|ξ|<1
0 ,1
,
10
C. BACUTA
and, using again (3.8), we have
(3.12)
Z
Z
Z
2 ∞ µ(ξ)3β
µ(ξ)2β |û||φ| dξ =
|û(ξ)||φ(ξ)| dt dξ
π 0 µ(ξ)2β + t2
|ξ|>1
|ξ|>1
2
=
π
Z
2
π
Z
=
∞
0
µ3β
|û||φ| dξ dt
µ2β + t2
Z
|ξ|>1
∞
µβ |û|
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 |û|

dξ 
µ2β + t2
Z
∞
Z
0
0
2
µ |φ|

dξ 
µ2β + t2
dt
|ξ|>1
µ2β |û|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 
2β
2
µ |û|

dξ 
µ2β + t2
Z
dt
≤ ckûk[Ĥ 0 ,Ĥ 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 ∈
Ĥ β (RN ) be such that for some constants > 0 and c > 0, we have
N
|φi (ξ) − φ̃i (ξ)| < cρ− 2 −2β+αi − f or |ξ| > 1
(3.13)
0 < αi < β, i = 1, . . . , n,
where
N
φ̃i (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 φ̃1 , φ̃2 , . . . , φ̃n 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) − φ̃i (ξ)| < cρ− 2 −2β+γi − f or |ξ| > 1
(3.16)
α < γi < β, i = 1, . . . , n,
where
N
φ̃i (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 φ̃1 , φ̃2 , . . . , φ̃n 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 û, u ∈ H α , is an isometry from H α to H 0 and from
H β to H β−α .
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
Theorem 4.2. Let (X, Y ), (X̃, Ỹ ) be two pairs of Hilbert spaces satisfying (2.1) and
let (Y ∗ , X ∗ ), (Ỹ ∗ , X̃ ∗ ) be the corresponding dual pairs. Then, the dual pairs satisfy
(2.1) also, see e.g., [18]. We further identify Y ∗ with Y and Ỹ ∗ with Ỹ , and assume
that there are linear operators E and R such that
(6.6)
E : Y → Ỹ , E : X → X̃ are bounded operators,
(6.7)
R : Ỹ → Y, R : X̃ → X, are bounded operators,
(6.8)
REu = u
for all u ∈ Y.
18
C. BACUTA
e = E(K) ⊂ Ỹ = Ỹ ∗ be closed subspaces of Y and Ỹ ,
Let K ⊂ Y = Y ∗ , K
respectively, and denote the corresponding orthogonal complements by YK and ỸKe .
Let θ0 ∈ (0, 1) be such that
[X̃ ∗ , Ỹ ]θ0 ,1 ⊂ [X̃ ∗ , ỸKe ]θ0 ,∞ .
(6.9)
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
(6.11)
E ∗ : Ỹ → Y, E ∗ : X̃ ∗ → X ∗ , are bounded operators,
(6.12)
R∗ : Y → Ỹ , R∗ : X ∗ → X̃ ∗ are bounded operators,
E ∗ R∗ u = u
(6.13)
for all u ∈ Y,
E ∗ maps ỸKe to YK .
(6.14)
From (6.11) and (6.14), by interpolation, we obtain
(6.15)
kE ∗ vk[X ∗ ,YK ]θ0 ,∞ ≤ ckvk[X̃ ∗ ,Ỹ e ]θ
K
0 ,∞
for all v ∈ Ỹ .
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[X̃ ∗ ,Ỹ e ]θ
K
0 ,∞
for all u ∈ Y.
Also, from the hypothesis (6.9), we deduce that
(6.17)
kR∗ uk[X̃ ∗ ,Ỹ e ]θ
K
0 ,∞
≤ ckR∗ uk[X̃ ∗ ,Ỹ ]θ
0 ,1
for all u ∈ Y.
From (6.12), again by interpolation, we have in particular
(6.18)
kR∗ uk[X̃ ∗ ,Ỹ ]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]s̃,q̃ , 0 < s̃ < s < 1, and 1 ≤ q, q̃ ≤ ∞.
ii) The relative strength of the second index of interpolation:
[Y, X]s,q ⊂ [Y, X]s,q̃ , 1 ≤ q ≤ q̃ ≤ ∞.
iii) “By Interpolation” Theorem: Let (X̃, Ỹ ) be another pair of Banach spaces
satisfying (2.1). If T : Y → Ỹ and T : X → X̃ satisfy
kT f kỸ ≤ c0 kf kY ,
and kT f kX̃ ≤ c1 kf kX ,
then,
kT f k[Ỹ ,X̃]λ,q ≤ c1−λ
cλ1 kf k[Y,X]λ,q , 0 < λ < 1, and 1 ≤ q ≤ ∞.
0
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. Nec̆as. 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]