The square root problem for second order, divergence form operators with mixed boundary conditions on $L^p$

0.46 MB·English

by  Pascal Auscher

THE SQUARE ROOT PROBLEM FOR SECOND ORDER, DIVERGENCE
FORM OPERATORS WITH MIXED BOUNDARY CONDITIONS ON Lp
2
PASCALAUSCHER,NADINEBADR,ROBERTHALLER-DINTELMANN,ANDJOACHIMREHBERG
1
0
2
Abstract. We show that, under very general conditions on the domain Ω and the Dirichlet
t partDoftheboundary,theoperator(cid:0)−∇·µ∇+1(cid:1)1/2withmixedboundaryconditionsprovides
Oc atopologicalisomorphismbetweenWD1,p(Ω)andLp(Ω),ifp∈]1,2].

2

] 1. Introduction
A
C The main purpose of this paper is to identify the domain of the square root of a divergence
. form operator µ +1 on Lp(Ω) as a Sobolev space W1,p(Ω) of differentiability order 1 for
h −∇· ∇ D
p ]1,2]. (The subscript D indicates the subspace of W1,p(Ω) whose elements vanish on the
t
a bo∈undary part D.) Our focus lies on non-smooth geometric situations in Rd. So, we allow for
m
mixed boundary conditions and, additionally, deviate from the Lipschitz property of the domain
[ Ω in the following spirit: the boundary ∂Ω decomposes into a closed subset D (the Dirichlet

1 part) and its complement, which may share a common frontier within ∂Ω. Concerning D, we
v onlydemandthatitsatisfiesthewell-knownAhlfors-Davidcondition(equivalently: isa(d 1)-set
−
0 in the sense of Jonsson/Wallin [35, II.1]), and only for points from the complement we demand
8
bi-Lipschitzian charts around. As special cases, the pure Dirichlet (D =∂Ω) and pure Neumann
7
case (D = ) are also included in our considerations. Finally the coefficient function µ is just
0 ∅
supposed to be real, measurable, bounded and elliptic in general, cf. Assumption 4.2. Together,
.
0
this setting should cover nearly everything that occurs in real-world problems – as long as the
1
domain does not have irregularities like cracks meeting the Neumann boundary part ∂Ω D.
2
\
1 The identification of the domain for fractional powers of elliptic operators, in particular that
: of square roots, has a long history. Concerning Kato’s square root problem – in the Hilbert
v
space L2 – see e.g. [9], [23], [6] (here only the non-selfadjoint case is of interest). Early efforts,
i
X devotedtothe determinationofdomainsforfractionalpowersinthe non-Hilbertspacecaseseem
r to culminate in [47]. In recent years the problem has been investigated in the case of Lp (p=2)
a forinstancein[5],[7],[34],[32],[8]; butonlythelasttwoarededicatedtothecaseΩ=Rd. I6n[8]
6
thedomainisastrongLipschitzdomainandthe boundaryconditionsareeitherpureDirichletor
pure Neumann. Our result generalizes this to a large extent and, at the same time, gives a new
proofforthesespecialcases,usingmore’global’arguments. Since,inthecaseofanon-symmetric
coefficient function µ, for the nonsmooth constellations described above no general condition is
known that assures ( µ +1)1/2 :W1,2(Ω) L2(Ω) to be an isomorphism, this is supposed
−∇· ∇ D →
as one of our assumptions. This serves then as our starting point to show the corresponding
isomorphism property of ( µ +1)1/2 :W1,p(Ω) Lp(Ω) for p ]1,2[.
−∇· ∇ D → ∈
While this is already interesting in itself, our originalmotivation comes from the applications:
having the isomorphism ( µ +1)1/2 :W1,p(Ω) Lp(Ω) at hand, the adjoint isomorphism
( µ +1)1/2 ∗ = ( −∇·µT∇ +1)1/2 : LqD(Ω) →W−1,q(Ω) allows to carry over substantial
−∇· ∇ −∇· ∇ → D
properties of the operators µ on the Lp-scale to the scale of W−1,q-spaces for q [2, [.
(cid:0) (cid:1) −∇· ∇ D ∈ ∞
1991 Mathematics Subject Classification. Primary: 35J15,42B20, 47B44;Secondary: 26D15,46B70,35K20.
Key words and phrases. Kato’ssquarerootproblem,Ellipticoperatorswithbounded measurablecoefficients,
Interpolationincaseofmixedboundaryvalues,Hardy’sinequality,Calderon-Zygmunddecomposition.
1
2 PASCALAUSCHER,NADINEBADR,ROBERTHALLER-DINTELMANN,ANDJOACHIMREHBERG
In particular, this concerns the ∞-calculus and maximal parabolic regularity, see Section 11,
H
whichinturnis apowerfultoolforthe treatmentoflinearandnonlinearparabolicequations,see
e.g. [46] and [30].
The paper is organized as follows: after presenting some notation and general assumptions in
Section 2, in Section 3 we introduce the Sobolev scale W1,p(Ω), 1 p , related to mixed
D ≤ ≤ ∞
boundary conditions and point out some of their properties. In Section 4 we define properly the
elliptic operator under consideration and collect some known facts for it. The main result on
the isomorphism property for the square root of the elliptic operator is precisely formulated in
Section 5. The following sections contain preparatory material for the proof of the main result,
which is finished at the end of Section 10. Some of these results have their own interest, such as
Hardy’s inequality for mixed boundary conditions that is proved in Section 6 and the results on
realand complex interpolationfor the spaces W1,p(Ω), 1 p , from Section 8, so we shortly
D ≤ ≤∞
want to comment on these.
Our proof of Hardy’s inequality heavily rests on two things: first one uses an operator that
extends functions from W1,p(Ω) to W1,p(Ω ), where Ω is a domain containing Ω. Then one is
D 0 • •
in a situation where the deep results of Ancona [2], Lewis [41] and Wannebo [51], combined with
Lehrb¨ack’s [40] ingenious characterizationof p-fatness, may be applied.
The proof of the interpolation results, as well as other steps in the proof of the main result,
arefundamentallybasedonanadaptedCaldero´n-ZygmunddecompositionforSobolevfunctions.
Such a decomposition was first introduced in [5] and has also succesfully been used in [10], see
also [11]. We have to modify it, since the main point here is, that the decomposition has to
respect the boundary conditions. This is accomplished by incorporating Hardy’s inequality into
the controlling maximal operator. This result, which is at the heart of our considerations, is
contained in Section 7.
Allthesepreparations,togetherwithoff-diagonalestimatesforthesemigroupgeneratedbyour
operator, cf. Section 9, lead to the proof of the main result in Section 10. Finally, in Section 11
we draw some consequences, as already sketched above.
Acknowledgments. TheauthorswanttothankA.Ancona,P.Koskela,V.Maz’yaandW.Tre-
bels for valuable discussions and hints on the topic.
2. Notation and general assumptions
Throughout the paper we will use x,y,... for vectors in Rd and the symbol B(x,r) stands for
the ballinRd aroundxwithradiusr. ForE,F Rd wedenote byd(E,F) the distancebetween
⊆
E and F and if E = x , then we write d(x,F) or d (x) instead.
F
{ }
Regarding our geometric setting, we suppose the following assumption throughout this work.
Assumption 2.1. (i) Ω Rd isaboundeddomainandD isaclosedsubsetoftheboundary
⊆
∂Ω (to be understood as the Dirichlet boundary part). For every x ∂Ω D there
∈ \
exists an open neighbourhood U of x and a bi-Lipschitz map φ from U onto the cube
x x x
K :=] 1,1[d, such that the following three conditions are satisfied:
−
φ (x)=0,
x
(2.1) φ (U Ω)= x K :x <0 =:K ,
x x d −
∩ { ∈ }
φ (U ∂Ω)= x K :x =0 =:Σ.
x x d
∩ { ∈ }
(ii) We suppose that D is either empty or satisfies the Ahlfors-David condition: There are
constants c ,c >0 and r >0, such that for all x D and all r ]0,r ]
0 1 AD AD
∈ ∈
(2.2) c rd−1 (D B(x,r)) c rd−1,
0 d−1 1
≤H ∩ ≤
where denotes (here and in the sequel) the (d 1)-dimensional Hausdorff measure.
d−1
H −
SQUARE ROOTS OF DIVERGENCE OPERATORS 3
Remark 2.2. (i) Condition (2.2) means that D is a (d 1)-set in the sense of Jons-
−
son/Wallin [35, Ch. II].
(ii) On the set ∂Ω U the measure equals the surface measure σ which
∩ x∈∂Ω\D x Hd−1
can be constructed via the bi-Lipschitzian charts φ around these boundary points,
(cid:0)S (cid:1) x
compare[24,Section 3.3.4C] or[31, Section3]. Inparticular,(2.2)assuresthe property
σ ∂Ω U >0.
x∈∂Ω\D x
∩ ∪
(iii) We emphasize that the cases D =∂Ω or D = are not excluded.
(cid:0) (cid:0) (cid:1)(cid:1) ∅
Assumption 2.3. In case of D = ∂Ω we additionally require that there is a bounded domain
6
Ω⋆ Ω and constants c⋆,r⋆ >0, such that
⊇
(i) ∂Ω⋆ satisfies the condition
(2.3) c⋆rd−1 d−1 ∂Ω⋆ B(x,r) , x ∂Ω⋆, r ]0,r⋆].
≤H ∩ ∈ ∈
(ii) Ω• :=Ω⋆ D is connected,(cid:0)and, hence, ag(cid:1)ain a domain.
\
(iii) Ω and ∂Ω⋆ D have positive distance to each other.
\
Remark 2.4. Let us comment on Assumption 2.3 since its context becomes clear only in Sec-
tion 6.
(i) It is established in order to get Hardy’s inequality for elements from W1,p(Ω). This
D
is achieved via an extension operator from W1,p(Ω) to W1,p(Ω ), and the validity of
D 0 •
Hardy’s inequality for functions from W1,p(Ω ) – which is to be proved.
0 •
If there is an open ball B Ω, such that B D is connected, then one can put
⊇ \
Ω⋆ :=B,andAssumption2.3issatisfied. Butthisisnotalwaysthecase,asthefollowing
examplesshows: takeΩ= x:1< x <2 andD = x: x =1 x: x =2,x 0 .
1
{ | | } { | | }∪{ | | ≥ }
Obviously,if Ω⋆ is open and contains Ω D, then Ω⋆ D cannot be connected. In this
∪ \
case one can take Ω⋆ := x : 1 < x < 3 instead. This suggest already the general
{ | | }
procedure: ifaconnectedcomponentof∂ΩconsistsonlyofpointsfromD,thenΩshould
not be extended across this boundary part. In the opposite case it should.
(ii) Itisnothardtoseethatcondition(iii)oftheassumptionimplies∂Ω D Ω⋆. Itseems
\ ⊆
that even every connected component of ∂Ω which contains a point from ∂Ω D must
\
in total be contained in Ω⋆. Since this is nowhere needed in our further considerations
we consider this as a heuristics and do not prove it.
(iii) We emphazise that – more or less – all constellations, relevant for applications, are
coveredby Assumptions 2.1 and 2.3.
IfB isaclosedoperatoronaBanachspaceX,thenwedenotebydom (B)thedomainofthis
X
operator. (X,Y) denotes the space of linear, continuous operators from X into Y; if X = Y,
L
thenweabbreviate (X). Furthermore,wewillwrite , forthepairingofelementsofX and
X′
L h· ·i
the dual space X′ of X.
Finally,the letters c andC denote genericconstantsthat maychange value fromoccurenceto
occurence.
3. Sobolev spaces related to boundary conditions
In this section we will introduce the Sobolev spaces related to mixed boundary conditions and
prove some results related to them that will be needed later.
If Υ is an open subset of Rd and F a closed subset of Υ, e.g. the Dirichlet part D of ∂Ω, for
1 q < we define W1,q(Υ) as the completion of
≤ ∞ F
(3.1) C∞(Υ):= ψ :ψ C∞(Rd), supp(ψ) F =
F { |Υ ∈ ∩ ∅}
with respectto the normψ ψ q+ ψ q dx 1/q. For 1<q < the dual ofthis space will
7→ Υ|∇ | | | ∞
be denoted by W−1,q′(Υ) with 1/q+1/q′ = 1. Here, the dual is to be understood with respect
F (cid:0)R (cid:1)
4 PASCALAUSCHER,NADINEBADR,ROBERTHALLER-DINTELMANN,ANDJOACHIMREHBERG
to the extended L2 scalar product, or, in other words: W−1,q′(Υ) is the space of continuous
F
antilinear forms on W1,q(Υ).
F
If misunderstandings are not to be expected, we drop the Ω in the notation of spaces, i.e.
function spaces without an explicitely given domain are to be understood as function spaces on
Ω.
Remark3.1. ThespaceW1,q(Ω)admitsacontinuoustraceoperatorintothespaceLq(D; )
d−1
H
for all 1 q < , cf. [35, Ch. V]. Hence, the functions f W1,q(Ω) satisfy f =0 -a.e.
≤ ∞ ∈ D |D Hd−1
Finally, we define the respective spaces for the case q = . We set W1,∞(Υ) := Lip (Υ)
∞ F ∞,F
with
(3.2) Lip (Υ):= f :f (L∞ Lip)(Rd),f =0 = f (L∞ Lip)(Υ),f =0 .
∞,F |Υ ∈ ∩ |F ∈ ∩ |F
The norm on this spac(cid:8)e is (cid:9) (cid:8) (cid:9)
f(x) f(y)
f + sup | − |.
k kL∞(Υ) x y
x,y∈Υ,x=6 y
| − |
Thelastequalityin(3.2)isaconsequenceoftheWhitneyextensiontheorem. WehaveLip (Υ)
∞,F ⊆
f 1,∞(Υ):f =0 ( 1,∞(Υ) is defined using distributions) and the converse holds iff Ω
F
∈W | W
is uniformly locally convex by [27, Theorem 7].
(cid:8) (cid:9)
Lemma 3.2. Let Υ Rd be a bounded domain and F a (relatively) closed subset of ∂Υ. Then
⊆
W1,∞(Υ) W1,q(Υ) for 1 q < .
F ⊆ F ≤ ∞
Proof. Let (α ) be the sequence of cut-off functions defined on R+ by
n n
0, if 0 t<1/n,
≤
α (t)= nt 1, if 1/n t 2/n,
n 
− ≤ ≤
1, if t>2/n.
Remark that for t = 1 the sequence αn(t) tends to 1 as n . Furthermore, for all t 0 we
6 → ∞ ≥
have 0 tα′ (t) 2 and the sequence (tα′ (t)) tends to 0.
For x≤ Rnd w≤e set w (x) := α (d(x,F)n). Tnhen, by the above considerations, w 1 almost
n n n
∈ →
everywhere as n and
→∞
w (x) = α′ (d(x,F)) d(x,F) α′ (d(x,F))
|∇ n | n |∇ |≤ n |
almost everywhere. Thus d(x,F) (cid:12)wn(x) is b(cid:12)ounded and co(cid:12)nverges to 0 almost everywhere as
|∇(cid:12) | (cid:12) (cid:12)
n .
→Le∞t g W1,∞(Υ), which we consider as defined on Rd. Since Υ is bounded, we may assume
∈ F
thatghascompactsupportinsomelargeballB. Letg :=gw . Theng iscompactlysupported
n n n
in B and in Rd F. We claim that g g in W1,q(Rd). Indeed, g g =g(1 w ) and, by the
n n n
dominated conv\ergence theorem, g(1 →w ) 0 in Lq(Rd), since w− 1. −
n n
− → →
Now, for the gradient, we have
g g =(1 w ) g+g w .
n n n
∇ −∇ − ∇ ∇
Again by the dominated convergence theorem, the first term converges to 0 in Lq(Rd).
It remains to prove that kg∇wnkLq(Rd) converges to 0. We have for x∈Rd
g(x)
(3.3) (g w )(x)= d(x,F) w (x).
n n
∇ d(x,F) ∇
Since g is Lipschitz continuous on the whole of Rd and satisfies g =0 on F, we find
g(x) g(x) g(x )
∗
sup = sup − C,
x∈Rd(cid:12)d(x,F)(cid:12) x∈Rd(cid:12) x−x∗ (cid:12)≤
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
SQUARE ROOTS OF DIVERGENCE OPERATORS 5
wherex F denotesanelementofF thatrealizesthe distanceofxtoF. Sobothfactorsonthe
∗
∈
right hand side in (3.3) are bounded and d(x,F) w (x) goes to 0 almost everywhereas n .
n
∇ →∞
Thus, since g has compact support, the dominated convergence theorem yields g w 0 in
n
Lq(Rd). ∇ →
Finally, it suffices to convolve this approximation with a smooth mollifying function that has
small support to conclude g W1,q(Υ). (cid:3)
∈ F
Next, we establish the following extension property for function spaces on domains, satisfying
just part (i) of Assumption 2.1. This has been proved in [22] for q = 2. For convenience of the
reader we include a proof.
Lemma 3.3. Let Ω and D satisfy Assumption 2.1 (i). Then there is a continuous extension
operator E which maps each space W1,q(Ω) continuously into W1,q(Rd), q [1, ]. Moreover, E
maps Lq(Ω) continuously into Lq(RdD) for q [1, ]. D ∈ ∞
∈ ∞
Proof. Let,foreveryx ∂Ω D thesetU beanopenneighbourhoodthatsatisfiesthecondition
x
∈ \
from Assumption 2.1 (i). Let U ,...,U be a finite subcovering of ∂Ω D and let η C∞(Rd)
x1 xℓ \ ∈ 0
be a function that is identically one in a neighbourhood of ∂Ω D and has its support in U :=
\
ℓ U .
j=1 xj
Assume ψ C∞(Ω); then we can write ψ =ηψ+(1 η)ψ. By the definition of C∞(Ω) and η
S ∈ D − D
it is clear that the support of (1 η)ψ is contained in Ω, thus this function may be extended by
0 to the whole space Rd – while −its W1,q-norm is preserved.
It remains to define the extension of the function ηψ, what we will do now. For this, let
η ,...,η be a partition of unity on supp(η), subordinated to the covering U ,...,U . Then
1 ℓ x1 xℓ
we can write ηψ = ℓ η ηψ and have to define an extension for every function η ηψ. For
r=1 r r
doing so, we first transform the corresponding function under the corresponding mapping φ
P ] xr
from Assumption 2.1 (i) to η ηψ = (η ηψ) φ−1 on the half cube K . Afterwards, by even
r r ◦ xr −
[
reflection, one obtains a function η ηψ W1,q(K) on the cube K. It is clear by construction
r
[ ∈
that supp(η ηψ) has a positive distance to ∂K. Transforming back, one ends up with a function
r
η ηψ W1,q(U ) whose support has a positive distance to ∂U . Thus, this function may also
r ∈ xr xr
be extended by 0 to the whole of Rd, preserving again the W1,q norm.
]
Lastly, one observes that all the mappings W1,q(U Ω) η ηψ η ηψ W1,q(K ),
] [ [xr ∩ ∋ r 7→ r ∈ −
W1,q(K ) η ηψ η ηψ W1,q(K)andW1,q(K) η ηψ η ηψ W1,q(U )arecontinuous.
− ∋ r 7→ r ∈ ∋ r 7→ r ∈ xr
Thus, adding up, one arrives at an extension of ψ whose W1,q(Rd)-norm may be estimated by
c ψ with c independent from ψ. Hence, the mapping E, up to now defined on C∞(Ω),
k kW1,q(Ω) D
continuously and uniquely extends to a mapping from W1,q to W1,q(Rd).
D
ItremainstoshowthattheimagesinfactevenlyinW1,q(Rd). Fordoingso,onefirstobserves
D
that, by constructionof the extensionoperator,for any ψ C∞(Ω), the supportof the extended
functionEψhasapositivedistancetoD–butEψneednot∈beDsmooth. Clearly,onemayconvolve
EψsuitablyinordertoobtainanappropriateapproximationintheW1,q(Rd)-norm–maintaining
apositivedistanceofthesupporttothesetD. Thus,EmapsC∞(Ω)continuouslyintoW1,q(Rd),
D D
what is also true for its continuous extension to the whole space W1,q(Ω).
D
It is not hard to see that the operator E extends to a continuous operator from Lq(Ω) to
Lq(Rd), where q [1, ]. (cid:3)
∈ ∞
Remark 3.4. (i) By construction, all extended functions Ef have their support in Ω
∪
ℓ U , and, hence, in a suitably large ball.
j=1 xj
(ii) Employing Lemma 3.3 in conjunction with (i), one can establish the corresponding
S
Sobolev embeddings W1,p(Ω) ֒ Lq(Ω) (compactness included) in a straightforward
D →
manner.
6 PASCALAUSCHER,NADINEBADR,ROBERTHALLER-DINTELMANN,ANDJOACHIMREHBERG
(iii) When combining E with a multiplication operator that is induced by a function η
0
C∞(Rd), η 1 on Ω, one may achieve that the support of the extended function∈s
0 0 ≡
shrinks to a set which is arbitrarily close to Ω.
Remark 3.5. The geometric setting of Assumption 2.1 still allows for a Poincar´einequality for
functions from W1,p, as soon as D = . This is proved in [31, Thm. 3.5], if Ω is a Lipschitz
D 6 ∅
domain. In fact, the proof only needs that a part of D admits positive boundary measure and
this is guaranteed by Remark 2.2 (ii).
This Poincar´e inequality entails that, whenever D = , the norms given by f and
6 ∅ k kWD1,p
f for f W1,p are equivalent. So, in this case, in all subsequent considerations one may
k∇ kLp ∈ D
freely replace the one by the other.
4. The divergence operator: Definition and elementary properties
We turnnowto the definitionofthe elliptic divergenceoperatorthatwill be investigated. Letus
first introduce the ellipticity supposition on the coefficients.
Assumption 4.1. The coefficient function µ is a Lebesgue measurable, bounded function on Ω
taking its values in the set of real, d d matrices, satisfying for some µ >0 the usual ellipticity
•
×
condition
ξTµ(x)ξ µ ξ 2, for all ξ Rd and almost all x Ω.
•
≥ | | ∈ ∈
The operator A:W1,2 W−1,2 is defined by
D → D
(4.1) Aψ,ϕ :=t(ψ,ϕ):= µ ψ ϕdx, ψ,ϕ W1,2.
h iWD−1,2 ZΩ ∇ ·∇ ∈ D
Often we will write more suggestively µ instead of A.
−∇· ∇
The L2 realization of A, i.e. the maximal restriction of A to the space L2, will be denoted by
the same symbol A; clearly this is identical with the operator that is induced by the sesquilinear
formt. If B is a densely defined, closedoperatoron L2, then by the Lp realizationof B we mean
its restriction to Lp if p > 2 and the Lp closure of B if p [1,2[. (For all operators we have in
∈
mind, this Lp-closure exists.)
As a starting point of our considerations we assume that the square root of our operator is
well-behaved on L2. This is true in many relevant cases, but seems not to be known under our
assumptions in general.
Assumption4.2. Theoperator( µ +1)1/2 :W1,2 L2 providesatopologicalisomorphism;
−∇· ∇ D →
in other words: the domain of ( µ +1)1/2 on L2 is the form domain W1,2.
−∇· ∇ D
Remark 4.3. (i) If this assumption is satisfied for a coefficient function µ, then it is also
true for the adjoint coefficient function, cf. [44, Thm. 8.2].
(ii) Assumption 4.2 is always fulfilled if the coefficient function µ takes its values in the set
of real symmetric d d-matrices.
×
(iii) In view of non-symmetric coefficient functions see [9] and [23].
Finally, we collect some facts on µ as an operator on the L2 and on the Lp scale.
−∇· ∇
Proposition 4.4. Let Ω Rd be a domain and let D ∂Ω (relatively) closed.
⊆ ⊆
(i) The restriction of µ to L2 is a densely defined sectorial operator.
−∇· ∇
(ii) The operator µ generates an analytic semigroup on L2.
∇· ∇
(iii) The form domain W1,2 is invariant under multiplication with functions from W1,q, if
D
q >d.
SQUARE ROOTS OF DIVERGENCE OPERATORS 7
Proof. (i) It is not hard to see that the form t is closed and its numerical range lies in
the sector z C : Imz kµkL∞ Rez . Thus, the assertion follows from a classical
{ ∈ | | ≤ µ• }
representation theorem for forms, see [37, Ch. VI.2.1].
(ii) This follows from (i) and [37, Ch. V.3.2].
(iii) First, for u C∞(Ω) and v C∞(Ω) the product uv is obviously in C∞(Ω) W1,2.
∈ D ∈ D ⊆ D
But, by definition of W1,2, the set C∞(Ω) (see (3.1)) is dense in W1,2 and C∞(Ω) is
D D D
dense in W1,q. Thus, the assertion is implied by the continuity of the mapping
W1,2 W1,q (u,v) uv W1,2,
D × ∋ 7→ ∈
because W1,2 is closed in W1,2. (cid:3)
D
Proposition 4.5. Let Ω and D satisfy Assumption 2.1 (i). Then the semigroup generated by
µ in L2 satisfies upper Gaussian estimates, precisely:
∇· ∇
(et∇·µ∇f)(x)= K (x,y)f(y)dy, for a.a. x Ω, f L2,
t
∈ ∈
ZΩ
for some measurable function K :Ω Ω R and for all ε >0 there exist constants C,c >0,
t +
× →
such that
(4.2) 0≤Kt(x,y)≤ tdC/2 e−c|x−ty|2 eεt, t>0, a.a.x,y∈Ω.
Proof. A proof is given in [22] – heavily resting on [4], compare also [44, Thm. 6.10]. (cid:3)
Proposition 4.6. Let Ω and D satisfy Assumption 2.1 (i).
(i) For every p [1, ], the operator µ generates a semigroup of contractions on Lp.
∈ ∞ ∇· ∇
(ii) For all q ]1, [ the operator µ +1 admits a bounded ∞-calculus on Lq with
∈ ∞ −∇· ∇ H
∞-angle arctankµkL∞. In particular, it admits bounded imaginary powers.
H µ•
Proof. (i) The operator µ generates a semigroup of contractions on L2 (see [44,
∇ · ∇
Thm 1.54]) as well as on L∞ (see [44, Ch. 4.3.1]). By interpolation this carries over
to every Lq with q ]2, [ and, by duality, to q [1,2].
(ii) Since the numerica∈l ran∞ge of µ is conta∈ined in the sector z C : Imz
−∇· ∇ { ∈ | | ≤
kµkL∞ Rez , the assertion holds true for q = 2, see [26, Cor. 7.1.17]. Secondly, the
µ• }
semigroupgeneratedby µ 1 obeys the Gaussianestimate (4.2)with ε=0. Thus,
∇· ∇−
the first assertion follows from [20, Theorem 3.1]. The second claim is a consequence of
the first, see [16, Section 2.4]. (cid:3)
5. The main result: the isomorphism property of the square root
We can now formulate our main goal, that is to prove that the mapping
(A+1)1/2 =( µ +1)1/2 :W1,q Lq
−∇· ∇ D →
is a topologicalisomorphismfor q ]1,2[. We abbreviate µ +1by A throughoutthe rest
0
∈ −∇· ∇
of this work.
More precisely, we want to show the following main result of this paper.
Theorem 5.1. Under Assumptions 2.1, 4.1 and 4.2 the following holds true:
(i) For every q ]1,2] the operator A−1/2 is a continuous operator from Lq into W1,q.
∈ 0 D
Hence, its adjoint continuously maps W−1,q into Lq for any q [2, [.
D ∈ ∞
(ii) If, additionally, Assumption 2.3 holds and q ]1,2], then A1/2 maps W1,q continuously
∈ 0 D
into Lq. Hence, its adjoint continuously maps Lq into W−1,q for any q [2, [.
D ∈ ∞
8 PASCALAUSCHER,NADINEBADR,ROBERTHALLER-DINTELMANN,ANDJOACHIMREHBERG
Wecanimmediatelygivetheproofof(i),i.e.thecontinuityoftheoperatorA−1/2 :Lq W1,q.
0 → D
We observe that this follows, whenever
1. The Riesz transform A−1/2 is a bounded operator on Lq, and, additionally,
∇ 0
2. A−1/2 maps Lq into W1,q.
0 D
The first item is proved in [44, Thm. 7.26], compare also [19]. It remains to show 2. The first
point makes clear that A−1/2 maps Lq continuously into W1,q, thus one only has to verify the
0
correctboundarybehaviorofthe images. Iff L2 ֒ Lq,then onehas A−1/2f W1,2 ֒ W1,q,
∈ → 0 ∈ D → D
due to Assumption 4.2. Thus, the assertion follows from 1. and the density of L2 in Lq.
Remark 5.2. Theorem 5.1 (i) is not true for other values of q in general, see [5, Ch. 4] for a
further discussion.
The hard work is to prove the second part, that is the continuity of A1/2 : W1,q Lq. The
proof is inspired by [5], where this is shown in the case Ω = Rd, and wi0ll be deDvelo→ped in the
following five sections.
6. Hardy’s inequality
A major tool in our considerations is an inequality of Hardy type for functions in W1,p, so
D
functions that vanish only on the part D of the boundary. Here the additional Assumption 2.3
comes into play.
We recallthat, fora setF Rd, the symbold denotes the functiononRd that measuresthe
F
⊆
distance to F. The result we want to show in this section, is the following.
Theorem 6.1. Under Assumption 2.1 and Assumption 2.3, for every p ]1, [ there is a
∈ ∞
constant c , such that
p
p
f
(6.1) dx c f p dx
p
d ≤ |∇ |
ZΩ(cid:12) D(cid:12) ZΩ
(cid:12) (cid:12)
holds for all f W1,p. (cid:12) (cid:12)
∈ D (cid:12) (cid:12)
Since the statement of this theorem is void for D = , we exclude that case for this entire
∅
section.
Let us first quote the deep results on which the proof of Theorem 6.1 will base.
Proposition 6.2 (see [41], [51], see also [38]). Let Ξ Rd be a domain whose complement
K :=Rd Ξ is uniformly p-fat (cf. [41] or [38]). Then Ha⊆rdy’s inequality
\
g p g p
(6.2) dx= dx c g p dx
d d ≤ |∇ |
ZΞ(cid:12) K(cid:12) ZΞ(cid:12) ∂Ξ(cid:12) ZΞ
(cid:12) (cid:12) (cid:12) (cid:12)
holds for all g C∞(Ξ) (and(cid:12)exte(cid:12)nds to all g(cid:12) W(cid:12)1,p(Ξ), p ]1, [ by density).
∈ 0 (cid:12) (cid:12) (cid:12)∈ (cid:12)0 ∈ ∞
Proposition 6.3 ([40, Theorem 1]). Let Ξ Rd be a domain and let again denote the
d−1
⊆ H
(d 1)-dimensional Hausdorff measure. If Ξ satisfies the inner boundary density condition, i.e.
−
(6.3) ∂Ξ B(x,2d (x)) cd (x)d−1, x Ξ,
d−1 ∂Ξ ∂Ξ
H ∩ ≥ ∈
for some constant c>0, the(cid:0)n the complement of(cid:1)Ξ in Rd is uniformly p-fat for all p ]1, [.
∈ ∞
For the proof of Theorem 6.1 let us distinguish two cases, first assuming D = ∂Ω. Then the
Ahlfors-David condition (2.2) on D implies
r d−1
(6.4) D B(y,r) c AD rd−1, y D, r ]0,diam(Ω)]
d−1
H ∩ ≥ diam(Ω) ∈ ∈
(cid:0) (cid:1) (cid:16) (cid:17)
SQUARE ROOTS OF DIVERGENCE OPERATORS 9
with c independent of y and r. But (6.4) implies the inner boundary density condition (6.3),
compare [40]. Thus, one may apply Proposition 6.3 and Proposition 6.2 to obtain the claim of
Theorem 6.1.
It remains to consider the case D = ∂Ω. Let us start with some preparation that brings the
6
assumption 2.3 into play.
Lemma 6.4. Let Ω⋆ and Ω• be defined as in Assumption 2.3. Then the following assertions
hold.
(i) D ∂Ω⋆ =∂Ω•.
∪
(ii) There exists a modified, continuous extension operator E : W1,q(Ω) W1,q(Ω ) con-
• D → 0 •
sistently for all q [1, ].
∈ ∞
Proof. (i) The inclusion ∂Ω• D ∂Ω⋆ is clear; let us show the opposite inclusion. Ob-
⊆ ∪
viously, D Ω = . Moreover, every x D is an accumulation point of Ω since it is
• •
∩ ∅ ∈
an accumulation point of Ω Ω . Together, this shows D ∂Ω . On the other hand,
• •
⊆ ⊆
if x ∂Ω⋆, then x / Ω⋆ Ω•. Finally, if x ∂Ω⋆, then it is an accumulation point of
∈ ∈ ⊇ ∈
the open set Ω⋆, and, hence, also an accumulation point of Ω• = Ω⋆ D, since D has
\
Lebesgue measure 0, thanks to (2.2). Thus, ∂Ω⋆ ∂Ω•.
⊆
(ii) Take a function η from C∞(Rd) which is identically 1 on Ω and identically 0 on a
0 0
neighbourhood of ∂Ω⋆ D – what is possible according to Assumption 2.3 (iii). Then
we define E ψ := η Eψ\ .
Let us sh•ow that0E s|aΩt•isfies the required properties: assume first ψ C∞(Ω). It
is clear by the con(cid:0)stru•ct(cid:1)ion of E that the support of Eψ has a positive d∈istanDce to D.
Thus, η0Eψ has apositivedistance toD ∂Ω⋆ ∂Ω•. Mollifyingwith suitablekernels,
η Eψ canthenberepresentedastheW1,∪p(Rd)-li⊇mitofsmoothfunctionswhosesupports
0
avoid ∂Ω . The continuity of E : C∞(Ω) W1,q(Ω) is obvious; hence E extends
• • D → 0 •
continuously to W1,q(Ω). (cid:3)
D
Assuming for the moment that we have (6.2) in case of Ξ = Ω , we find with the help of
•
Lemma 6.4 and Remark 3.5
f p f p E f p
dx dx • dx c (E f)p dx
•
d ≤ d ≤ d ≤ |∇ |
ZΩ(cid:12) D(cid:12) ZΩ(cid:12) ∂Ω•(cid:12) ZΩ•(cid:12) ∂Ω•(cid:12) ZΩ•
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(6.5) (cid:12)(cid:12) (cid:12)(cid:12) ≤ckf(cid:12)(cid:12)kpW1,p(cid:12)(cid:12)≤c |∇f|p(cid:12)(cid:12)dx. (cid:12)(cid:12)
D ZΩ
Thus Hardy’s inequality (6.1) holds true, once we have shown(6.2) with Ω in the place of Ξ for
•
all g W1,p(Ω ).
∈ 0 •
In order to do so, we first recall that Ω is connected, and, hence, a domain, due to As-
•
sumption 2.3. If c0,c⋆,rAD,r⋆ are the constants from Assumptions 2.1 and 2.3, we put c• :=
min(c0,c⋆) and r• := min(rAD,r⋆). Thus, the condition (2.2) in conjunction with Lemma 6.4
implies for all y D the inequality
∈
∂Ω B(y,r) D B(y,r) c rd−1 c rd−1, r ]0,r ],
d−1 • d−1 0 • •
H ∩ ≥H ∩ ≥ ≥ ∈
what gives for r(cid:0) ]0,diam(Ω )(cid:1)] (cid:0) (cid:1)
•
∈
c rd−1, if r diam(Ω )
• • •
(6.6) Hd−1 ∂Ω•∩B(y,r) ≥c r• d−1rd−1, if r <≥diam(Ω ).
(cid:0) (cid:1)  • diam(Ω•) • •
(cid:16) (cid:17)
Analogously, we obtain from Lemma 6.4 and Assumption 2.3, this time for all y ∂Ω⋆,
∈
d−1 ∂Ω• B(y,r) d−1 ∂Ω⋆ B(y,r) c⋆rd−1 c•rd−1, r ]0,r•].
H ∩ ≥H ∩ ≥ ≥ ∈
(cid:0) (cid:1) (cid:0) (cid:1)
10 PASCALAUSCHER,NADINEBADR,ROBERTHALLER-DINTELMANN,ANDJOACHIMREHBERG
Thus, one obtains (6.6) also for y ∂Ω⋆. Thanks to Lemma 6.4 (i), the estimate (6.6) is
∈
thus fulfilled for all y ∂Ω , what implies the inner boundary density condition (6.3), compare
•
∈
[40]. Applying Proposition 6.3 and Proposition 6.2, we get (6.2) for g W1,p(Ω ). Thus the
∈ 0 •
estimate (6.5) finishes the proof of Theorem 6.1.
Remark6.5. ThereisanotherstrategyofproofforHardy’sinequality(6.2),avoidingtheconcept
of ’uniformly p-fat’. In [40] it is proved that the inner boundary density condition (6.3) implies
the so-called p-pointwise Hardy inequality which implies Hardy’s inequality, compare also [38].
7. An adapted Caldero´n-Zygmund decomposition
TheproofofTheorem5.1heavilyreliesonaCaldero´n-ZygmunddecompositionforW1,pfunctions.
D
The important point, which brings the mixed boundary conditions into play, is that we have to
makesurethatforf dom (A1/2)thegoodandthebadpartofthedecompositionarebothalso
∈ Lp 0
in this space. This is not guaranteed neither by the classical Calder´on-Zygmund decomposition
nor by the version for Sobolev functions in [5, Lemma 4.12]. This problem will be solved, by
incorporating the Hardy inequality into the decomposition.
For the ease of notation, in the whole section we set 1/d =0 and we abbreviate for f W1,1
∅ ∈ D
the extended function Ef by f˜.
We denote by the set of all closed axe-parallel cubes, i.e. all sets of the form x Rd :
x m ℓ/2 fQor some midpoint m Rd and sidelength ℓ > 0. In the following, f{or a∈given
∞
| − | ≤ } ∈
cube Q we will often write sQ for some s>0, meaning the cube with the same midpoint m,
∈Q
but sidelength sℓ instead of ℓ.
Furthermore, for every x Rd we set := Q : x Q◦ . Now we may define the
x
Hardy-Littlewood maximal op∈erator M forQall ϕ {L1(∈RdQ) by ∈ }
∈
1
(7.1) (Mϕ)(x)= sup ϕ, x Rd.
Q | | ∈
Q∈Qx | |ZQ
It is well known(see [48, Ch. 1])that M is of weaktype (1,1), sothere is some K >0, suchthat
for all p 1
≥
K
(7.2) x Rd : [M(ϕp)](x) >αp ϕ p , for all α>0 and ϕ Lp(Rd).
∈ | | | | ≤ αpk kLp(Rd) ∈
Lemma 7(cid:12).(cid:8)1. Let Ω and D satisfy Assum(cid:9)(cid:12)ptions 2.1 and 2.3. Let p ]1, [, f W1,p and α>0
(cid:12) (cid:12) ∈ ∞ ∈ D
be given. Then there exist an at most countable index set I, cubes Q , j I, and measurable
j
functions g,b :Ω R, j I, such that for some constant N 0 ind∈epQenden∈t of α and f
j
→ ∈ ≥
(1) f =g+ b ,
j
j∈I
X
(2) g + g + g/d Nα,
L∞ L∞ D L∞
k∇ k k k k k ≤
b
(3) supp(b ) Q , b W1,1 W1,p and b + b +| j| NαQ for every j I,
j ⊆ j j ∈ D ∩ |∇ j| | j| d ≤ | j| ∈
N ZΩ(cid:16) D(cid:17)
(4) Q f p ,
| j|≤ αpk kW1,p
D
j∈I
X
(5) 1 (x) N for all x Rd,
Qj ≤ ∈
j∈I
X
(6) g N f .
k kWD1,p ≤ k kWD1,p
If D = , all the norms f may be replaced by f .
6 ∅ k kWD1,p k∇ kLp
Inordertoverifythefinalstatement,notethatforD = theAhlfors-Davidconditionguaran-
6 ∅
tees that the surface measure of D is strictly positive, cf. Remark 2.2 (ii). Thus we can conclude
by Remark 3.5.
We will subdivide the proof of Lemma 7.1 into six steps.