Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

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318 R. Haller-Dintelmann and J. Rehberg

iv) Let ζ ∈ L

∞

bearealfunctionwithapositivelowerbound.Then−ζ∇·μ∇+1

has its spectrum in [1, ∞[ and admits a bounded functional calculus on L

v) The operator ∇·μ∇, considered on L

, p ∈ ]1, ∞[, is densely deﬁned and

generates a strongly continuous semigroup of contractions there.

vi) Let ζ ∈ L

∞

be a real function with a positive lower bound. Then, under

Assumption 3.1, the operator ζ∇·μ∇ satisﬁes the estimate

(ζ∇·μ∇−1 −λ)

−1



L(L

)

≤

1+|λ|

, if Re λ ≥−

(3.3)

and, hence, generates a bounded, analytic semigroup on every space L

, p ∈

]1, ∞[.

Proof. Assertions i)–iii) are standard, while iv) follows from ii) and the subsequent

Lemma 3.6. Part v) is proved in [49, Thm. 4.28/Prop. 4.11]. Finally, a proof of

(3.3) is contained in [33, Thm. 5.2/Remark 5.1] and the second part of vi) then

follows from [50, Thm. 7.7] or [45, Ch. IX.1.6]. 

Lemma 3.6. Let w ∈ L

∞

be a real function with a strictly positive lower bound and

let B be a selfadjoint, positive operator on L

. Then the operator wB +1has its

spectrum in [1, ∞[ and admits a bounded functional calculus on L

Proof. We equip L

with the equivalent scalar product (f, g) →



fgw

−1

dx and

name the resulting Hilbert space

.ObservethatC ∈L(L

), iﬀ C ∈L(

). The

equation



(wB +1)f gw

−1

dx =



Bf g dx +



f gw

−1

dx =



f Bg dx +



fgw

−1



f(wB +1)gw

−1

dx for f, g ∈ dom

(B)

shows that wB + 1 is symmetric on

Let λ<1. Then, by the hypotheses on B and w, the operator

−1

(wB +1− λ)=B + w

−1

(1 − λ)

on L

is self-adjoint and has a strictly positive lower form bound. Thus, it is

continuously invertible on L

. This now implies invertibility of wB +1−λ on L

as well as on

. So, every λ ∈ ]−∞, 1[ is in the resolvent set of wB +1 on

and

this, together with the symmetry shown above, implies that the operator wB +1

is selfadjoint in

and has its spectrum in [1, ∞[.

Exploiting now a spectral integral representation wB +1=



∞

λdP (λ), one

obtains a bounded functional calculus of wB +1 on

. This implies a bounded

functional calculus also on L

, since both norms are equivalent. 

One essential instrument for our subsequent considerations are (upper)

Gaussian estimates.

Maximal Parabolic Regularity for Divergence Operators 319

Theorem 3.7. Let ζ,V be real, positive L

∞

functions and let ζ admit a positive

lower bound. Then the semigroup, generated by ζ∇·μ∇−V satisﬁes upper Gaussian

estimates, precisely:

t(ζ∇·μ∇−V )

f)(x) =



(x, y)f (y) dy, x ∈ Ω,f∈ L

forsomemeasurablefunctionK

:Ω× Ω → R

and for every ε>0 there exist

constants c, b > 0, such that

0 ≤ K

(x, y) ≤

d/2

−b

|x−y|

εt

,t>0,a.a.x, y ∈ Ω. (3.4)

If V admits a lower bound V

∗

> 0,thenε may be taken as 0.

Proof. Letusﬁrstconsiderthecaseζ ≡ 1. If V is only nonnegative, then the

estimate in (3.4) follows from [49, Theorem 6.10] (see also [7]). If V admits a

strictly positive lower bound V

∗

, then one may write V = V − V

∗

+ V

∗

and thus

obtains (3.4) with ε =0.

The case of general ζ is implied by the multiplicative perturbation result

in [23]. 

Lemma 3.8. Let ζ ∈ L

∞

be a real function with a strictly positive lower bound and

p ∈ ]1, ∞[. Then, under Assumption 3.1, dom



−ζ∇·μ∇+1



1/2

=dom



−∇·

μ∇+1



1/2

,andthenorms



−ζ∇·μ∇ +1



1/2

·

and 



−∇·μ∇+1



1/2

·

are equivalent.

Proof. Since ζ has positive lower and upper bounds, D

:= dom

(−ζ∇·μ∇+1)

equals D := dom

(−∇·μ∇+1) and the corresponding graph norms are equivalent.

Thus, necessarily also [L

]

1/2

=[L

,D]

1/2

including the equivalence of the

corresponding norms. In order to conclude, we will show that both, −∇ · μ∇ +1

and −ζ∇·μ∇+1 admit bounded imaginary powers on L

. Then by [56, Ch. 1.15.3],

one obtains the identity

dom



−∇·μ∇+1



1/2

=[L

,D]

1/2

=[L

]

1/2

=dom



−ζ∇·μ∇ +1



1/2

including the equivalence of the graph norms.

The operator −∇·μ∇+ 1 has bounded imaginary powers thanks to Proposi-

tion 3.5 v) and [17, Corollary 1]. Concerning −ζ∇·μ∇+1, one can argue as follows:

this operator has a bounded H

∞

calculus on L

by Proposition 3.5 iv) and the

associated semigroup admits Gaussian estimates with ε = 0 by Theorem 3.7. Thus

the bounded H

∞

functional calculus extrapolates to all spaces L

, p ∈ ]1, ∞[, by

[24, Theorem 3.4] and this in particular implies bounded imaginary powers. 

320 R. Haller-Dintelmann and J. Rehberg

4. Mapping properties for (−∇ · μ∇ +1)

1/2

In this chapter we prove that, under certain topological conditions on Ω and Γ,

the mapping

(−∇·μ∇+1)

1/2

: H

1,q

→ L

is a topological isomorphism for q ∈ ]1, 2[. We abbreviate −∇·μ∇ by A

throughout

this chapter. Let us introduce the following

Assumption 4.1. There is a bi-Lipschitz mapping φ from a neighborhood of

Ωinto

such that φ(Ω ∪ Γ) = K

−

or K

−

∪ ΣorK

−

∪ Σ

The main results of this section are the following three theorems.

Theorem 4.2. Under the general assumptions made in Section 2 the following holds

true: For every q ∈ ]1, 2], the operator (A

+1)

−1/2

is a continuous operator from

into H

1,q

. Hence, it continuously maps

−1,q

into L

for any q ∈ [2, ∞[.

Theorem 4.3. Let in addition Assumption 4.1 be fulﬁlled. Then, for every q ∈ ]1, 2],

the operator (A

+1)

1/2

maps H

1,q

continuously into L

.Hence,itcontinuously

maps L

into

−1,q

for any q ∈ [2, ∞[.

Putting these two results together, one immediately gets the following iso-

morphism property of the square root of A

+1.

Theorem 4.4. Under Assumption 4.1, (A

+1)

1/2

provides a topological isomor-

phism between H

1,q

and L

for q ∈ ]1, 2] and a topological isomorphism between

and

−1,q

for any q ∈ [2, ∞[.

Remark 4.5. In all three theorems the second assertion follows from the ﬁrst by

the self-adjointness of A

on L

and duality; thus one may focus on the proof of

the ﬁrst assertions.

Let us ﬁrst prove the continuity of the operator (A

+1)

−1/2

: L

→ H

1,q

In order to do so, we observe that this follows, whenever

1. The Riesz transform ∇(A

+1)

−1/2

is a bounded operator on L

, and, addi-

tionally,

2. (A

+1)

−1/2

maps L

into H

1,q

The ﬁrst item is proved in [49, Thm. 7.26]. It remains to show 2. The ﬁrst point

makes clear that (A

+1)

−1/2

maps L

continuously into H

1,q

, thus one has only

to verify the correct boundary behavior of the images. If f ∈ L

→ L

, then one

has (A

+1)

−1/2

f ∈ H

1,2

→ H

1,q

. Thus, the assertion follows from 1. and the

density of L

in L

Remark 4.6. Theorem 4.2 is not true for other values of q in general, see [8, Ch. 4]

for a further discussion.

We now prove Theorem 4.3. It will be deduced from the subsequent deep

result on divergence operators with Dirichlet boundary conditions and some per-

manence principles.

Maximal Parabolic Regularity for Divergence Operators 321

Proposition 4.7 (Auscher/Tchamitchian, [9]). Let q ∈ ]1, ∞[ and Ω be a strongly

Lipschitz domain. Then the root of the operator A

, combined with a homogeneous

Dirichlet boundary condition, maps H

1,q

(Ω) continuously into L

(Ω).

For further reference we mention the following consequence of Theorem 4.2

and Proposition 4.7.

Corollary 4.8. Under the hypotheses of Proposition 4.7 the operator (A

+1)

−1/2

provides a topological isomorphism between L

and H

1,q

,ifq ∈ ]1, 2].

Proof. The only thing to show is that the continuity of A

1/2

from Proposition 4.7

carries over to (A

+1)

1/2

. For this it suﬃces to show that the mapping (A

1/2

−1/2

=(1+A

−1

)

1/2

: L

→ L

extends to a continuous mapping from

into itself. Since the operator includes a homogeneous Dirichlet condition, the

spectrum of A

iscontainedinaninterval[ε, ∞[forsomeε>0. But the

spectrum of A

,consideredonL

, is independent from q, see [49, Thm. 7.10].

Hence, A

−1

is well deﬁned and continuous on every L

. Moreover, the spectrum

of 1 + A

−1

, considered as an operator on L

, is thus contained in a bounded

interval [1,δ] by the spectral mapping theorem, see [45, Ch. III.6.3]. Consequently,

(1 + A

−1

)

1/2

: L

→ L

is also a continuous operator by classical functional

calculus, see [21, Ch. VII.3]. 

In view of Assumption 4.1 it is a natural idea to reduce our considerations to

the three model constellations mentioned there. In order to do so, we have to show

that the assertion of Theorem 4.3 is invariant under bi-Lipschitz transformations

of the domain. The proof will stem from the following lemma.

Lemma 4.9. Assume that φ is a bi-Lipschitzian mapping from a neighborhood of

Ω into R

.Letφ(Ω) = Ω



and φ(Γ) = Γ



. Deﬁne for any function f ∈ L

(Ω



(Φf)(x) = f(φ(x)) = (f ◦ φ)(x), x ∈ Ω.

Then

i) The restriction of Φ to any L

(Ω



), 1 ≤ p<∞, provides a linear, topological

isomorphism between this space and L

(Ω).

ii) For any p ∈ ]1, ∞[, the mapping Φ induces a linear, topological isomorphism

: H

1,p



(Ω



) → H

1,p

(Ω).

iii) Φ

∗



is a linear, topological isomorphism between

−1,p

(Ω) and

−1,p



(Ω



)

for any p ∈ ]1, ∞[.

iv) One has

∗



= −∇ · μ



∇ (4.1)

with



(y) =

det(Dφ)(φ

−1

(y))

(Dφ)(φ

−1

(y)) μ(φ

−1

(y))



Dφ



(φ

−1

(y))

for almost all y ∈ Ω



. Here, Dφ denotes the derivative of φ and det(Dφ) the

corresponding determinant.

322 R. Haller-Dintelmann and J. Rehberg

v) μ



also is bounded, Lebesgue measurable, elliptic and takes real, symmetric

matrices as values.

vi) The restriction of Φ

∗

Φ to L

(Ω



) equals the multiplication operator which is

given by the function

det(Dφ)(φ

−1

(·))

−1

Remark 4.10. It is well known that

det(Dφ)(φ

−1

(·))

is a function from L

∞

which, additionally, has a positive lower bound, due to the bi-Lipschitz property

of φ, see [27, Ch. 3]. In the sequel we denote this function by ζ.

Lemma 4.11. Let p ∈ ]1, ∞[. Then, in the notation of the preceding lemma, the

operator



−∇·μ



∇+1



1/2

maps H

1,p



(Ω



) continuously into L

(Ω



),if(A

+1)

1/2

maps H

1,p

(Ω) continuously into L

(Ω).

Proof. We will employ the formula

−1/2



∞

−1/2

(B + t)

−1

dt, (4.2)

B being a positive operator on a Banach space X, see [56, Ch. 1.14/1.15] or [50,

Ch. 2.6].

The operators A

+1, −∇·μ



∇+1 and −ζ∇·μ



∇+1 are positive operators

in the sense of [56, Ch. 1.14] on any L

, see Proposition 3.5. From (4.1) and vi) of

the preceding lemma one deduces

∗



+1+t



= −∇ · μ



∇ + ζ

−1

(1 + t)

for every t>0. This leads to

−1



+1+t



−1



∗



−1



−∇·μ



∇+ ζ

−1

(1 + t)



−1



−ζ∇·μ



∇+1+t



−1

ζ.

(4.3)

The H

1,2



(Ω



) ↔

−1,2



(Ω



) duality is the extended L

(Ω



) duality. Thus, when

restricting (4.3) to L

,Φ

∗

may then be viewed as the adjoint with respect to the

duality. Integrating this equation with weight

−1/2

, one obtains, according

to (4.2),

−1

+1)

−1/2

(Φ

∗

)

−1



−ζ∇·μ



∇ +1



−1/2

ζ. (4.4)

Observe that the corresponding integrals converge in L(L

), according to Propo-

sition 3.5. Inverting (4.4), we get the operator equation

∗

+1)

1/2

= ζ

−1



−ζ∇·μ



∇+1



1/2

From this, the continuity of



−ζ∇·μ



∇ +1



1/2

: H

1,p



(Ω



) → L

(Ω



)

follows by our supposition on A

and straightforward continuity arguments on

Φ, see [40] for a detailed discussion. An application of Lemma 3.8 concludes the

proof. 

Maximal Parabolic Regularity for Divergence Operators 323

Lemma 4.11 allows us to reduce the proof of Theorem 4.3 to Ω = K

−

and

the three cases Γ = ∅,Γ=ΣorΓ=Σ

. The ﬁrst case, Γ = ∅,isalready

contained in Proposition 4.7. In order to treat the second one, we use a reﬂection

argument. Let us point out the main ideas for this: First, one deﬁnes the operator

E : L

−

) → L

(K) which assigns to every function from L

−

) its symmetric

extension. Let us further denote by R : L

(K) → L

−

) the restriction operator.

Finally, one deﬁnes −∇·ˆμ∇ : H

1,2

(K) →

−1,2

(K) as the symmetric extension of

−∇·μ∇ to K. Note that this latter operator is then combined with homogeneous

Dirichlet conditions. The deﬁnition of −∇· ˆμ∇ in particular implies for t ≥ 0



+1+t



−1

f = R



−∇· ˆμ∇ +1+t



−1

Ef for all f ∈ L

−

Multiplying this equation by

−1/2

and integrating over t, one obtains, in accor-

dance with (4.2),

+1)

−1/2

f = R



−∇· ˆμ∇ +1



−1/2

Ef, f ∈ L

−

This equation extends to all f ∈ L

−

)withp ∈ ]1, 2[. Now one exploits the

fact that



−∇ · ˆμ∇ +1



−1/2

is a surjection onto the whole H

1,p

(K) by Corol-

lary 4.8. Then some straightforward arguments show that (A

+1)

−1/2

: L

−

) →

1,p

−

) also is a surjection. Since, by Theorem 4.2 (A

+1)

−1/2

: L

−

) →

1,p

−

) is continuous, the continuity of the inverse ﬁnally is implied by the open

mapping theorem.

In order to prove the same for the third model constellation, i.e., Γ = Σ

one shows

Lemma 4.12. There is a bi-Lipschitz mapping φ : R

→ R

that maps K

−

∪ Σ

onto K

−

∪ Σ.

Thus, the proof of Theorem 4.3 in the case Γ = Σ

results from the case

Γ = Σ and Lemmas 4.11 and 4.12.

Remark 4.13. Let us mention that Lemma 4.11, only applied to Ω = K and Γ = ∅

(the pure Dirichlet case) already provides a zoo of geometries which is not covered

by [9]. Notice in this context that the image of a strongly Lipschitz domain under

a bi-Lipschitz transformation need not be a strongly Lipschitz domain at all, cf.

[34, Ch. 1.2].

5. Maximal parabolic regularity for A

In this section we intend to prove the ﬁrst main result of this work announced

in the introduction, i.e., maximal parabolic regularity of A in spaces with nega-

tive diﬀerentiability index. Let us ﬁrst recall the notion of maximal parabolic L

regularity.

324 R. Haller-Dintelmann and J. Rehberg

Deﬁnition 5.1. Let 1 <s<∞,letX be a Banach space and let J := ]T

,T[ ⊆ R

be a bounded interval. Assume that B is a closed operator in X with dense domain

D (in the sequel always equipped with the graph norm). We say that B satisﬁes

maximal parabolic L

(J; X) regularity, if for any f ∈ L

(J; X) there exists a unique

function u ∈ W

1,s

(J; X) ∩L

(J; D) satisfying



+ Bu = f, u(T

)=0,

where the time derivative is taken in the sense of X-valued distributions on J,see

[4, Ch. III.1].

Remark 5.2.

i) It is well known that the property of maximal parabolic regularity of an

operator B is independent of s ∈ ]1, ∞[ and the speciﬁc choice of the interval

J (cf. [20]). Thus, in the following we will say for short that B admits maximal

parabolic regularity on X.

ii) If an operator satisﬁes maximal parabolic regularity on a Banach space X,

then its negative generates an analytic semigroup on X (cf. [20]). In partic-

ular, a suitable left half-plane belongs to its resolvent set.

iii) If X is a Hilbert space, the converse is also true: The negative of every gen-

erator of an analytic semigroup on X satisﬁes maximal parabolic regularity,

cf.[19]or[20].

iv) If −B is a generator of an analytic semigroup on a Banach space X,andS

indicates the space of X-valued step functions on J, then we deﬁne



∂

∂t

+ B



−1

: S

→ C(J; X) → L

(J; X)





∂

∂t

+ B



−1



(t):=B



−(t−s)B

f(s)ds,

compare (5.4) below. It is known that S

is a dense subspace of L

(J; X),

if s ∈ [1, ∞[, see [29, Lemma IV.1.3]. Using this, it is easy to see that B has

maximal parabolic regularity on X, if and only if the operator B



∂

∂t

+ B



−1

continuously extends to an operator from L

(J; X)intoitself.

v) Observe that

1,s

(J; X) ∩L

(J; D) → C(J;(X, D)

1−

). (5.1)

The next lemma, needed below, shows that maximal parabolic regularity is

maintained by interpolation:

Lemma 5.3. Suppose that X, Y are Banach spaces, which are contained in a third

Banach space Z with continuous injections. Let B be a linear operator on Z whose

restrictions to each of the spaces X, Y induce closed, densely deﬁned operators

there. Assume that the induced operators fulﬁll maximal parabolic regularity on X

and Y , respectively. Then B satisﬁes maximal parabolic regularity on each of the

interpolation spaces [X, Y ]

and (X, Y )

θ,s

with θ ∈ ]0, 1[ and s ∈ ]1, ∞[.

Maximal Parabolic Regularity for Divergence Operators 325

The next theorem will be the cornerstone on maximal parabolic regularity

of this work and details of the proof will be pointed out below in Subsections 5.1

and 5.2.

Theorem 5.4. Let Ω, Γ fulﬁll Assumption 3.1 and set q

iso

:= sup M

iso

,where

iso

:= {q ∈ [2, ∞[:−∇·μ∇+1:H

1,q

→

−1,q

is a topological isomorphism}.

Then −∇·μ∇ satisﬁes maximal parabolic regularity on

−1,q

for all q ∈ [2,q

∗

iso

where by r

∗

we denote the Sobolev conjugated index of r, i.e.,

∗



∞, if r ≥ d,



−



−1

, if r ∈ [1,d[ .

Remark 5.5.

i) If Ω, Γ fulﬁll Assumption 3.1, then q

iso

> 2, see [37] and also [35].

ii) It is clear by Lax-Milgram and interpolation (see Proposition 3.4) that M

iso

is the interval [2,q

iso

[or[2,q

iso

]. Moreover, it can be concluded from a deep

theorem of Sneiberg [54] (see also [8, Lemma 4.16]) that the second case

cannot occur.

Proposition 5.6. If Ω is a bounded Lipschitz domain and Γ is any closed subset of

∂Ω,then−∇ · μ∇ satisﬁes maximal parabolic regularity on L

for all p ∈ ]1, ∞[.

In particular, ∇·μ∇ generates an analytic semigroup on each L

,cf.Remark5.2.

Proof. The operator −∇·μ∇ possesses upper Gaussian estimates by Theorem 3.7

and this implies maximal parabolic regularity on L

,ifp ∈ ]1, ∞[, see [42] or [16].

Alternatively, the assertion of Proposition 5.6 may be deduced as follows:

First, the induced semigroup on any L

is contractive, see Proposition, 3.5 v).

Then one applies [47, Cor. 1.1]. 

Lemma 5.7. Let Ω, Γ fulﬁll Assumption 4.1.Then∇·μ∇ generates an analytic

semigroup on

−1,q

for all q ∈ [2, ∞[.

Proof. One has the operator identity



−∇·μ∇+λ



−1



−∇·μ∇+1



1/2



−∇·μ∇+λ



−1



−∇·μ∇+1



−1/2

, Re λ ≥ 0,

(5.2)

on L

. Under Assumption 4.1 (−∇·μ∇+1)

1/2

is a topological isomorphism between

and

−1,q

for every q ∈ [2, ∞[, thanks to Theorem 4.4. Thus, via (5.2), the

corresponding resolvent estimate carries over from L

−1,q

by the density of

−1,q

. 

In the next step we show

Theorem 5.8. Let Ω, Γ fulﬁll Assumption 4.1.Then−∇ · μ∇ satisﬁes maximal

parabolic regularity on

−1,q

for all q ∈ [2, ∞[.

This will be a consequence of Theorem 4.4 and the following two lemmata.

326 R. Haller-Dintelmann and J. Rehberg

Lemma 5.9. Assume that the operator B fulﬁlls maximal parabolic regularity on a

Banach space X and has no spectrum in ]−∞, 0].IfS

again denotes the space

of X-valued step functions on J, then one has, for every α ∈ ]0, 1[,



∂

∂t

+ B



−1

ψ =(B +1)



∂

∂t

+ B



−1

(B +1)

−α

ψ for all ψ ∈ S

. (5.3)

Proof. First, B satisﬁes a resolvent estimate (B + λ)

−1



L(X)

≤

|λ|

for all λ from

a suitable right half-space. Since, additionally, B has no spectrum in ]−∞, 0], the

operators (B +1)

−α

and (B +1)

are well deﬁned on X.

If x ∈ X and χ

denotes the indicator function of an interval I =]a, b[ ⊆ J,

then one calculates, by the deﬁnition of B(

∂

∂t

+ B)

−1



∂

∂t

+ B



−1

(t)=B



−(t−s)B

xχ

(s) ds

⎧

⎪

⎨

⎪

⎩

0, if t<a,



1 − e

(a−t)B



x, if t ∈ [a, b],



(b−t)B

−e

(a−t)B



x, if t>b,

(5.4)

compare [50, Ch. 1.2]. This gives, for every step function

l=1

∈ S



∂

∂t

+ B



−1

(B +1)

−α

l=1



∂

∂t

+ B



−1

(B +1)

−α

=(B +1)

−α



∂

∂t

+ B



−1

l=1

since (B +1)

−α

commutes with the semigroup operators e

−tB

. 

Remark 5.10. By the density of S

in L

(J; X)fors ∈ [1, ∞[, equation (5.3)

extends to the whole of L

(J; X), since the left-hand side is a continuous operator

on L

(J; X) by maximal regularity of B.

Lemma 5.11. Assume that X, Y are Banach spaces, where X continuously and

densely injects into Y .SupposeB to be an operator on Y , whose maximal restric-

tion B|

to X satisﬁes maximal parabolic regularity there. If B|

has no spectrum

in ]−∞, 0] and (B +1)

provides a topological isomorphism from X onto Y ,then

B also satisﬁes maximal parabolic regularity on Y .

Proof. Let S

be the set of step functions on J, taking their values in X.By

the density of X in Y , S

is also dense in L

(J; Y ). Due to Lemma 5.9, we may

Maximal Parabolic Regularity for Divergence Operators 327

estimate for any ψ ∈ S



∂

∂t

+ B



−1

L(L

(J;Y ))

(B +1)



∂

∂t

+ B



−1

(B +1)

−α

L(L

(J;Y ))

≤ c(B +1)



L(L

(J;X);L

(J;Y ))



∂

∂t

+ B



−1

L(L

(J;X))

(B +1)

−α

ψ

(J;X)

≤ c(B +1)



L(X;Y )



∂

∂t

+ B



−1

L(L

(J;X))

(B +1)

−α



L(Y ;X)

ψ

(J;Y )

By density, B



∂

∂t



−1

extends to a continuous operator on the whole of L

(J; Y )

and the assertion follows by Remark 5.2 iv). 

The proof of Theorem 5.8 is now obtained by the isomorphism property



−∇·μ∇+1



1/2

: L

→

−1,q

, assured by Theorem 4.4, and afterwards applying

Proposition 5.6 and Lemma 5.11, putting there X := L

, Y :=

−1,q

, B :=

−∇·μ∇ and α := 1/2.

Now we intend to ‘globalize’ Theorem 5.8, in other words: We prove that

−∇ · μ∇ satisﬁes maximal parabolic regularity on

−1,q

for suitable q if Ω, Γ

satisfy only Assumption 3.1, i.e., if K

−

, K

−

∪ΣandK

−

∪Σ

need only to be model

sets for the constellation around boundary points. Obviously, then the variety of

admissible Ω’s and Γ’s increases considerably; in particular, Γ may have more than

one connected component.

5.1. Auxiliaries

We continue with a result that allows us to ‘localize’ the elliptic operator.

Lemma 5.12. Let Ω, Γ satisfy Assumption 3.1 and let Υ ⊆ R

be open, such that

•

:= Ω ∩ Υ is also a Lipschitz domain. Furthermore, we put Γ

•

:= Γ ∩ Υ and ﬁx

an arbitrary, real-valued function η ∈ C

∞

) with supp(η) ⊆ Υ.Denotebyμ

•

the restriction of the coeﬃcient function μ to Ω

•

and assume v ∈ H

1,2

(Ω) to be

the solution of

−∇·μ∇v = f ∈

−1,2

(Ω).

Then the following holds true:

i) For all q ∈ ]1, ∞[ the anti-linear form

•

: w →f, $ηw

−1,2

(where $ηw again means the extension of ηw by zero to the whole Ω) is well

deﬁned and continuous on H

1,q



•

(Ω

•

), whenever f is an anti-linear form from

−1,q

(Ω). The mapping

−1,q

(Ω)  f → f

•

∈

−1,q

•

(Ω

•

) is continuous.