The Krein-von Neumann Realization of Perturbed Laplacians on Bounded Lipschitz Domains

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by  Jussi Behrndt

THE KREIN-VON NEUMANN REALIZATION OF PERTURBED
LAPLACIANS ON BOUNDED LIPSCHITZ DOMAINS
5
1 JUSSIBEHRNDT,FRITZGESZTESY,TILLMICHELER,ANDMARIUSMITREA
0
2
Abstract. Inthispaperwestudytheself-adjointKrein–vonNeumannreal-
n izationAK oftheperturbedLaplacian−∆+V inaboundedLipschitzdomain
a Ω⊂Rn. Weprovideanexplicitandself-containeddescriptionof thedomain
J of AK interms of Dirichletand Neumann boundary traces, and weestablish
3 aWeyl asymptoticformulafortheeigenvalues ofAK.
1

]
P
S 1. Introduction
.
h
The mainobjective ofthis note is to investigatethe self-adjointKrein–vonNeu-
t
a mann realization associatedto the differential expression−∆+V in L2(Ω), where
m Ω ⊂ Rn, n > 1, is assumed to be a bounded Lipschitz domain and V is a non-
[ negative bounded potential. In particular, we obtain an explicit description of the

domain of A in terms of Dirichlet and Neumann boundary traces, and we prove
1 K
v the Weyl asymptotic formula
6 N(λ,A ) = (2π)−nv |Ω|λn/2+O λ(n−(1/2))/2 . (1.1)
9 K λ→∞ n
8
(cid:0) (cid:1)
2 Here N(λ,AK) denotes the number of nonzero eigenvalues of AK not exceeding λ,
0 v is the volume of the unit ball in Rn, and |Ω| is the (n-dimensional) Lebesgue
n
. measure of Ω.
1
0 Let us first recallthe definition and some properties of the Krein–vonNeumann
5 extension in the abstract setting. Let S be a closed, densely defined, symmetric
1 operator in a Hilbert space H and assume that S is strictly positive, that is, for
v: some c > 0, (Sf,f)H > ckfk2H for all f ∈ dom(S). The Krein–von Neumann
i extension S of S is then given by
X K
r SKf =S∗f, f ∈dom(SK)=dom(S)+˙ ker(S∗), (1.2)
a
see the original papers Krein [47] and von Neumann [61]. It follows that S is
K
a nonnegative self-adjoint extension of S and that for all other nonnegative self-
adjoint extensions S of S the operator inequality S 6 S holds in the sense of
Θ K Θ
quadratic forms. As ker(S ) = ker(S∗), it is clear that 0 is an eigenvalue of S
K K
Date:January14,2015.
2010 Mathematics Subject Classification. Primary 35J25, 35J40, 35P15; Secondary 35P05,
46E35,47A10,47F05.
Keywordsandphrases. Lipschitzdomains,KreinLaplacian,tracemaps,eigenvalues,spectral
analysis,Weylasymptotics.
J.B.gratefullyacknowledges financial support bythe AustrianScience Fund (FWF), project
P25162-N26.
Work of M.M. was partially supported by the Simons Foundation Grant #281566 and by a
UniversityofMissouriResearchLeavegrant.
1
2 J.BEHRNDT,F.GESZTESY,T.MICHELER,ANDM.MITREA
(except if S is self-adjoint, in which case S = S∗ = S). Furthermore, if the self-
K
adjointFriedrichs extension S of S has purely discrete spectrum then the same is
F
true for the spectrum of S with the possible exception of the eigenvalue 0, which
K
may have infinite multiplicity. For further developments, extensive references, and
a more detailed discussion of the properties of the Krein–von Neumann extension
of a symmetric operatorwe refer the reader to [2, Sect. 109],[3], [4]–[6], [7, Chs. 9,
10], [8]–[13], [14], [15], [16], [17], [19], [27], [28], [31, Sect. 15], [32, Sect. 3.3], [35],
[37], [38, Sect. 13.2],[39], [40], [49], [54], [56], [57, Ch. 13], [58], [59], [60], [63], [64],
[65], and the references cited therein.
In the concrete case considered in this paper, the symmetric operator S above
is given by the minimal operator A associated to the differential expression
min
−∆+V in the Hilbert space L2(Ω), that is,
A =−∆+V, dom(A )=H˚2(Ω), (1.3)
min min
where H˚2(Ω) denotes the closure of C∞(Ω) in the Sobolev space H2(Ω), and 0 6
0
V ∈ L∞(Ω). It can be shown that A is the closure of the symmetric operator
min
−∆+V definedonC∞(Ω). WepointoutthathereΩisaboundedLipschitzdomain
0
and no further regularity assumptions on ∂Ω are imposed, thus it is remarkable
that the functions in dom(A ) possess H2-regularity. The adjoint A∗ of A
min min min
coincides with the maximal operator
A =−∆+V, dom(A )= f ∈L2(Ω) −∆f +Vf ∈L2(Ω) , (1.4)
max max
where ∆f is understood in the sense(cid:8)of distributi(cid:12)ons. From (1.2) and(cid:9)(1.3) it is
(cid:12)
clear that the Krein–von Neumann extension A of A is then given by
K min
A =−∆+V, dom(A )=H˚2(Ω)+˙ ker(A ). (1.5)
K K max
In the present situation A is a symmetric operator with infinite defect indices
min
and therefore ker(A∗ ) = ker(A ) is infinite-dimensional. In particular, 0 is
min max
an eigenvalue of A with infinite multiplicity, and hence belongs to the essential
K
spectrum. It is also important to note that in general the functions in ker(A )
K
do not possess any Sobolev regularity, that is, ker(A ) 6⊂ Hs(Ω) for every s >
K
0. Moreover, since Ω is a bounded set, the Friedrichs extension of A (which
min
coincideswiththeself-adjointDirichletoperatorassociatedto−∆+V)hascompact
resolventandhenceitsspectrumisdiscrete. Theabstractconsiderationsabovethen
yield that with the exception of the eigenvalue 0 the spectrum of A consists of a
K
sequence of positive eigenvalues with finite multiplicity which tends to +∞.
ThedescriptionofthedomainoftheKrein–vonNeumannextensionA in(1.5)
K
is not satisfactory for applications involving boundary value problems. Instead, a
moreexplicitdescriptionofdom(A ) viaboundary conditionsseems to be natural
K
and desirable. In the case of a bounded C∞-smooth domain Ω, it is known that
dom(A )= f ∈dom(A ) γ f +M(0)γ f =0 (1.6)
K max N D
holds,where γ andγ den(cid:8)ote the Dirichlet(cid:12)andNeumann traceop(cid:9)erator,respec-
D N (cid:12)
tively, defined on the maximal domain dom(A ), and M(0) is the Dirichlet-to-
max
Neumann map or Weyl–Titchmarsh operator for −∆+V. The description (1.6)
goesback to Grubb [36], where certainclassesofelliptic differentialoperatorswith
smooth coefficients are discussed in great detail. Note that in contrast to the
Dirichlet and Neumann boundary conditions the boundary condition in (1.6) is
THE KREIN-VON NEUMANN REALIZATION OF −∆+V ON LIPSCHITZ DOMAINS 3
nonlocal, as it involves M(0) which, when Ω is smooth, is a boundary pseudodif-
ferential operator of order 1. It is essential for the boundary condition (1.6) that
both trace operators γ and γ are defined on dom(A ). Even in the case of
D N max
a smooth boundary ∂Ω, the elements in dom(A ), in general, do not possess any
K
Hs-regularityfors>0,andhencespecialattentionhastobepaidtothedefinition
and the properties of the trace operators. In the smooth setting the classical anal-
ysis due to Lions and Magenes [48] ensures that γ : dom(A ) → H−1/2(∂Ω)
D max
and γ : dom(A ) → H−3/2(∂Ω) are well-defined continuous mappings when
N max
dom(A ) is equipped with the graph norm.
max
Let us now turn again to the present situation, where Ω is assumed to be a
bounded Lipschitz domain. Our first main objective is to extend the description
of dom(A ) in (1.6) to the nonsmooth setting. The main difficulty here is to
K
define appropriate trace operators on the domain of the maximal operator. We
brieflysketchthestrategyfrom[18],whichismainlybasedandinspiredbyabstract
extension theory of symmetric operators. For this denote by A and A the self-
D N
adjoint realizationsof −∆+V correspondingto Dirichlet and Neumann boundary
conditions, respectively. Recall that by [42] and [30] their domains dom(A ) and
D
dom(A ) are both contained in H3/2(Ω). Now consider the boundary spaces
N
G (∂Ω):= γ f f ∈dom(A ) , G (∂Ω):= γ f f ∈dom(A ) , (1.7)
D D N N N D
equipped with(cid:8)suitab(cid:12)le inner produ(cid:9)cts induced by t(cid:8)he Ne(cid:12)umann-to-Diric(cid:9)hlet map
(cid:12) (cid:12)
andDirichlet-to-Neumannmapfor−∆+V−i,seeSection3forthedetails. Itturns
outthatG (∂Ω)andG (∂Ω)arebothHilbertspaceswhicharedenselyembedded
D N
inL2(∂Ω). Itwasshownin[18]thatthe Dirichlettraceoperatorγ andNeumann
D
trace operator γ can be extended by continuity to surjective mappings
N
γ :dom(A )→G (∂Ω)∗ and γ :dom(A )→G (∂Ω)∗, (1.8)
D max N N max D
where G (∂Ω)∗ and G (∂Ω)∗ denote the adjoint (i.e., conjugate dual) spaces of
D N
G (∂Ωe) and G (∂Ω), respectively. Withinethe same process also the Dirichlet-to-
D N
Neumann map M(0) of −∆+V (originally defined as a mapping from H1(∂Ω) to
L2(∂Ω))admitsanextensiontoamappingM(0)fromG (∂Ω)∗ toG (∂Ω)∗. With
N D
the help of the trace maps γ and γ , and the extended Dirichlet-to-Neumann
D N
operator M(0) we are then able to extendfthe description of the domain of the
Krein–vonNeumannextensioenforsmeoothdomainsin(1.6)to thecaseofLipschitz
domains. fMore precisely, we show in Theorem 3.3 that the Krein–von Neumann
extension A of A is defined on
K min
dom(A )= f ∈dom(A ) γ f +M(0)γ f =0 . (1.9)
K max N D
For an exhaustive treatme(cid:8)nt of boundary t(cid:12)race operators on bou(cid:9)nded Lipschitz
domains in Rn and applications to Schr¨oding(cid:12)eer operaftors wee refer to [17].
Our second main objective in this paper is to prove the Weyl asymptotic for-
mula (1.1) for the nonzero eigenvalues of A . We mention that the study of the
K
asymptoticbehaviorofthe spectraldistributionfunctionofthe DirichletLaplacian
originates in the work by Weyl (cf. [67], [66], and the references in [68]), and that
generalizationsoftheclassicalWeylasymptoticformulawereobtainedinnumerous
papers - we refer the reader to [20], [21], [22], [23], [24], [25], [26], [55], [62], and
the introduction in [16] for more details. There are relatively few papers available
that treat the spectral asymptotics of the Krein Laplacian or the perturbed Krein
Laplacian A . Essentially these considerations are inspired by Alonso and Simon
K
4 J.BEHRNDT,F.GESZTESY,T.MICHELER,ANDM.MITREA
who, at the end of their paper [3] posed the question if the asymptotics of the
nonzeroeigenvaluesofthe KreinLaplacianis givenby Weyl’s formula? In the case
where Ω is bounded and C∞-smooth, and V ∈ C∞(Ω), this has been shown to
be the case three years later by Grubb [37], see also the more recent contributions
[51], [52], and [39]. Following the ideas in [37] it was shown in [14] that for so-
called quasi-convex domains (a nonsmooth subclass of bounded Lipschitz domains
with the key feature that dom(A ) and dom(A ) are both contained in H2(Ω))
D N
the Krein–von Neumann extension A is spectrally equivalent to the buckling of
K
a clamped plate problem, which in turn can be reformulated with the help of the
quadratic forms
a[f,g]:= A f,A g and t[f,g]:= f,A g , (1.10)
min min L2(Ω) min L2(Ω)
defined on dom(A(cid:0) ) = H˚2(Ω(cid:1)). In the Hilbert spa(cid:0)ce (H˚2(Ω(cid:1)),a[·,·]) the form t
min
canbeexpressedwiththehelpofanonnegativecompactoperatorT,anditfollows
that
λ∈σ (A )\{0} if and only if λ−1 ∈σ (T), (1.11)
p K p
counting multiplicities. These considerations can be extended from quasi-convex
domains to the more general setting of Lipschitz domains, see, for instance, Sec-
tion 4 and Lemma 4.2. Finally, the main ingredient in the proof of the Weyl
asymptotic formula (1.1) for the Krein–von Neumann extension A of −∆+V
K
on a bounded Lipschitz domain Ω is then a more general Weyl type asymptotic
formula due to Kozlov [45] (see also [44], [46]) which yields the asymptotics of the
spectral distribution of function of the compact operator T, and hence via (1.11)
the asymptotics of the spectral distribution function of A . This reasoning in the
K
proofofoursecondmainresultTheorem4.1isalongthelinesof[14,15],wherethe
special case of quasi-convex domains was treated. For perturbed Krein Laplacians
this result completes work that started with Grubb more than 30 years ago and
demonstrates that the question posed by Alonso and Simon in [3] regarding the
validityoftheWeylasymptoticformulacontinuestohaveanaffirmativeanswerfor
bounded Lipschitz domains – the natural end of the line in the development from
smooth domains all the way to minimally smooth ones.
2. Schro¨dinger Operators on Bounded Lipschitz Domains
This section is devoted to a study of self-adjoint Schr¨odinger operators on a
nonempty, bounded Lipschitz domain in Rn (automatically assumed to be open).
We shall make the following general assumption.
Hypothesis 2.1. Let n ∈ N\{1}, assume that Ω ⊂ Rn is a bounded Lipschitz
domain, and suppose that 06V ∈L∞(Ω).
We consider operator realizations of the differential expression −∆+V in the
Hilbert space L2(Ω). For this we define the preminimal realization A of −∆+V
p
by
∞
A :=−∆+V, dom(A ):=C (Ω). (2.1)
p p 0
It is clear that A is a densely defined, symmetric operator in L2(Ω), and hence
p
closable. The minimal realization A of −∆+V is defined as the closure of A
min p
in L2(Ω),
A :=A . (2.2)
min p
THE KREIN-VON NEUMANN REALIZATION OF −∆+V ON LIPSCHITZ DOMAINS 5
ItfollowsthatA isadenselydefined,closed,symmetricoperatorinL2(Ω). The
min
maximal realization A of −∆+V is given by
max
A :=−∆+V, dom(A ):= f ∈L2(Ω) −∆f +Vf ∈L2(Ω) , (2.3)
max max
where the expression ∆f, f ∈L2(Ω), is(cid:8)understood(cid:12)in the sense of distrib(cid:9)utions.
(cid:12)
In the next lemma we collect some properties of the operators A , A , and
p min
A . The standard L2-based Sobolev spaces of order s > 0 will be denoted by
max
Hs(Ω); for the closure of C∞(Ω) in Hs(Ω) we write H˚s(Ω).
0
Lemma 2.2. Assume Hypothesis 2.1 and let A , A , and A be as introduced
p min max
above. Then the following assertions hold:
(i) A and A are adjoints of each other, that is,
min max
A∗ =A∗ =A and A =A =A∗ . (2.4)
min p max min p max
(ii) A is defined on H˚2(Ω), that is,
min
dom(A )=H˚2(Ω), (2.5)
min
andthegraphnormofA andtheH2-normareequivalentondom(A ).
min min
(iii) A is strictly positive, that is, for some C >0 we have
min
(Aminf,f)L2(Ω) >Ckfk2L2(Ω), f ∈H˚2(Ω). (2.6)
(iv) A has infinite deficiency indices.
min
One recalls that the Friedrichs extension A of A is defined by
F min
A :=−∆+V, dom(A ):= f ∈H˚1(Ω) ∆f ∈L2(Ω) . (2.7)
F F
It is well-known that AF is a strictly posit(cid:8)ive self-adjoi(cid:12)(cid:12)nt operator i(cid:9)n L2(Ω) with
compact resolvent (see, e.g. [29, Sect. VI.1]).
In this note we are particularly interested in the Krein–von Neumann extension
A of A . According to (1.2), A is given by
K min K
A :=−∆+V, dom(A ):=dom(A ) ∔ ker(A ). (2.8)
K K min max
In the following theorem we briefly collect some well-known properties of the
Krein–vonNeumannextensionA inthepresentsetting. Formoredetailswerefer
K
the reader to the celebratedpaper [47] by Krein and to [3], [4], [11], [14], [15], [16],
[39], and [40] for further developments and references.
Theorem 2.3. Assume Hypothesis 2.1 and let A be the Krein–von Neumann
K
extension of A . Then the following assertions hold:
min
(i) A is a nonnegative self-adjoint operator in L2(Ω) and σ(A ) consists of
K K
eigenvalues only. The eigenvalue 0 has infinite multiplicity,
dim(ker(A ))=∞,
K
and the restriction AK|(ker(AK))⊥ is a strictly positive self-adjoint operator
in the Hilbert space (ker(A ))⊥ with compact resolvent.
K
(ii) dom(A )6⊂Hs(Ω) for every s>0.
K
(iii) A nonnegative self-adjoint operator B in L2(Ω) is a self-adjoint extension
of A if and only if for some (and, hence for all) µ<0,
min
(A −µ)−1 6(B−µ)−1 6(A −µ)−1. (2.9)
F K
6 J.BEHRNDT,F.GESZTESY,T.MICHELER,ANDM.MITREA
We also mention that the Friedrich extension A and the Krein–von Neumann
F
extension A are relatively prime (or disjoint), that is,
K
dom(A )∩dom(A )=dom(A )=H˚2(Ω). (2.10)
F K min
For later purposes we briefly recall some properties of the Dirichlet and Neu-
mann trace operator and the corresponding self-adjoint Dirichlet and Neumann
realizations of −∆+V in L2(Ω). We consider the space
H3/2(Ω):= f ∈H3/2(Ω) ∆f ∈L2(Ω) , (2.11)
∆
equipped with the inner product(cid:8) (cid:12) (cid:9)
(cid:12)
(f,g)H∆3/2(Ω) =(f,g)H3/2(Ω)+(∆f,∆g)L2(Ω), f,g ∈H∆3/2(Ω). (2.12)
One recalls that the Dirichlet and Neumann trace operators γ and γ defined by
D N
γ f :=f ↾ and γ f :=n·∇f ↾ , f ∈C∞(Ω), (2.13)
D ∂Ω N ∂Ω
admit continuous extensions to operators
γ :H3/2(Ω)→H1(∂Ω) and γ :H3/2(Ω)→L2(∂Ω). (2.14)
D ∆ N ∆
Here H1(∂Ω) denotes the usual L2-based Sobolev space of order 1 on ∂Ω; cf. [50,
Chapter 3] and [53]. It is important to note that the extensions in (2.14) are both
surjective, see [35, Lemma 3.1 and Lemma 3.2].
In the next theorem we collect some properties of the Dirichlet realization A
D
and Neumann realization A of −∆+V in L2(Ω). We recall that the operators
N
A and A are defined as the unique self-adjoint operators corresponding to the
D N
closed nonnegative forms
aD[f,g]:=(∇f,∇g)(L2(Ω))n +(Vf,g)L2(Ω), dom(aD):=H˚1(Ω),
(2.15)
aN[f,g]:=(∇f,∇g)(L2(Ω))n +(Vf,g)L2(Ω), dom(aN):=H1(Ω).
In particular, one has A = A by (2.7). In the next theorem we collect some
F D
well-knownfactsabouttheself-adjointoperatorsA andA . TheH3/2-regularity
D N
of the functions in their domains is remarkable, and a consequence of Ω being a
boundedLipschitz domain. We referthe readerto[34,Lemma3.4andLemma4.8]
for more details, see also [41, 42] and [30].
Theorem 2.4. Assume Hypothesis 2.1 and let A and A be the self-adjoint
D N
Dirichlet and Neumann realization of −∆+V in L2(Ω), respectively. Then the
following assertions hold:
(i) The operator A coincides with the Friedrichs extension A and is given
D F
by
A =−∆+V, dom(A )= f ∈H3/2(Ω) γ f =0 . (2.16)
D D ∆ D
The resolvent of AD is compact, and(cid:8)the spectrum(cid:12)of AD is(cid:9)purely discrete
(cid:12)
and contained in (0,∞).
(ii) The operator A is given by
N
A =−∆+V, dom(A )= f ∈H3/2(Ω) γ f =0 . (2.17)
N N ∆ N
The resolvent of AN is compact, and(cid:8)the spectrum(cid:12)of AN is(cid:9)purely discrete
(cid:12)
and contained in [0,∞).
THE KREIN-VON NEUMANN REALIZATION OF −∆+V ON LIPSCHITZ DOMAINS 7
3. Boundary conditions for the Krein–von Neumann realization
Our goal in this section is to obtain an explicit description of the domain of the
Krein–von Neumann extension A in terms of Dirichlet and Neumann boundary
K
traces. Forthiswedescribeanextensionprocedureofthetracemapsγ andγ in
D N
(2.14) onto dom(A ) from [18]. We recall that for ϕ∈ H1(∂Ω) and z ∈ ρ(A ),
max D
the boundary value problem
−∆f +Vf =zf, γ f =ϕ, (3.1)
D
admits a unique solution f (ϕ) ∈ H3/2(Ω). Making use of this fact and the trace
z ∆
operators(2.14)we define the Dirichlet-to-NeumannoperatorM(z), z ∈ρ(A ), as
D
follows:
M(z):L2(∂Ω)⊃H1(∂Ω)→L2(∂Ω), ϕ7→−γ f (ϕ), (3.2)
N z
where f (ϕ)∈H3/2(Ω) is the unique solution of (3.1). It can be shown that M(z)
z ∆
is an unbounded operator in L2(∂Ω). Moreover, if z ∈ ρ(A )∩ρ(A ) then M(z)
D N
is invertible and the inverse M(z)−1 is a bounded operator defined on L2(∂Ω).
Considering z =i, we set
Σ:=Im(−M(i)−1). (3.3)
The imaginary part ImM(i) of M(i) is a densely defined bounded operator in
L2(∂Ω) and hence it admits a bounded closure
Λ:=Im(M(i)) (3.4)
inL2(∂Ω). BothoperatorsΣ andΛ areself-adjointandinvertiblewith unbounded
inverses. Introducing the boundary spaces
G (∂Ω):= γ f f ∈dom(A ) (3.5)
D D N
and (cid:8) (cid:12) (cid:9)
G (∂Ω):= γ f (cid:12)f ∈dom(A ) , (3.6)
N N D
we equip G (∂Ω) and G (∂Ω) with the inner products
D N (cid:8) (cid:12) (cid:9)
(cid:12)
(ϕ,ψ)GD(∂Ω) := Σ−1/2ϕ,Σ−1/2ψ L2(∂Ω), ϕ,ψ ∈GD(∂Ω), (3.7)
and (cid:0) (cid:1)
(ϕ,ψ)GN(∂Ω) := Λ−1/2ϕ,Λ−1/2ψ L2(∂Ω), ϕ,ψ ∈GN(∂Ω), (3.8)
respectively. Then GD(∂Ω)(cid:0)and GN(∂Ω) bo(cid:1)th become Hilbert spaces which are
dense in L2(∂Ω). The corresponding adjoint (i.e., conjugate dual) spaces will be
denotedby G (∂Ω)∗ andG (∂Ω)∗,respectively. The followingresultcanbe found
D N
in [18, Section 4.1].
Theorem 3.1. Assume Hypothesis 2.1. Then the Dirichlet trace operator γ and
D
Neumann trace operator γ in (2.14) can be extended by continuity to surjective
N
mappings
γ :dom(A )→G (∂Ω)∗ and γ :dom(A )→G (∂Ω)∗ (3.9)
D max N N max D
such that ker(γ )=ker(γ )=dom(A ) and ker(γ )=ker(γ )=dom(A ).
D D D N N N
e e
In a similar manner the boundary value problem (3.1) can be considered for
all ϕ ∈ G (∂eΩ)∗ and the Dirichlet-to-Neumanneoperator M(·) in (3.2) can be
N
extended. More precisely, the following statement holds.
Theorem 3.2. AssumeHypothesis 2.1andlet γ and γ betheextendedDirichlet
D N
and Neumann trace operator from Theorem 3.1. Then the following are true:
e e
8 J.BEHRNDT,F.GESZTESY,T.MICHELER,ANDM.MITREA
(i) For ϕ∈G (∂Ω)∗ and z ∈ρ(A ) the boundary value problem
N D
−∆f +Vf =zf, γ f =ϕ, (3.10)
D
admits a unique solution f (ϕ)∈dom(A ).
z max
(ii) For z ∈ρ(A ) the Dirichlet-to-Neumanen operator M(z) in (3.2) admits a
D
continuous extension
M(z):G (∂Ω)∗ →G (∂Ω)∗, ϕ7→−γ f (ϕ), (3.11)
N D N z
where f (ϕ)∈dom(A ) is the unique solution of (3.10).
z max
f e
Nowweareabletostateourmainresultinthissection: Adescriptionofthedo-
mainofthe Krein–vonNeumann extensionA interms ofDirichletandNeumann
K
boundary traces. The extended Dirichlet-to-Neumannmap at z =0 will enter as a
regularizationparameter in the boundary condition. For C∞-smooth domains this
result goes back to Grubb [36], where a certain class of elliptic differential opera-
tors with smooth coefficients is discussed systematically. For the special case of a
so-called quasi-convex domains Theorem 3.3 reduces to [15, Theorem 5.5] and [35,
Theorem 13.1]. In an abstract setting the Krein–von Neumann extension appears
in a similar form in [18, Example 3.9].
Theorem 3.3. Assume Hypothesis 2.1 and let γ , γ and M(0) be as in Theo-
D N
rem 3.1 and Theorem 3.2. Then the Krein–von Neumann extension A of A is
K min
given by e e f
A =−∆+V, dom(A )= f ∈dom(A ) γ f +M(0)γ f =0 . (3.12)
K K max N D
Proof. We recallthat the Krein–v(cid:8)onNeumannext(cid:12)ensionA ofA is d(cid:9)efined on
(cid:12)e fK emin
dom(A )=dom(A ) ∔ ker(A ). (3.13)
K min max
Thus, from Lemma 2.2(ii) one concludes
dom(A )=H˚2(Ω) ∔ ker(A ). (3.14)
K max,Ω
Next, we show the inclusion
dom(A )⊆ f ∈dom(A ) γ f +M(0)γ f =0 . (3.15)
K max N D
Fixf ∈dom(A )anddeco(cid:8)mposef inthefor(cid:12)mf =f +f ,wher(cid:9)ef ∈H˚2(Ω)
K (cid:12)e mfin e0 min
and f ∈ ker(A ) (cf. (3.14)). Thus, γ f = γ f = 0 and γ f =
0 max D min D min N min
γ f =0, and hence it follows from Theorem 3.2(ii) that
N min
e
M(0)γ f =M(0)γ (f +f )=M(0)γ f =−γ f =−γ f. (3.16)
D D min 0 D 0 N 0 N
e
Thus, γ f +M(0)γ f =0 and the inclusion (3.15) holds.
fN e fD e f e e e
Next we verify the opposite inclusion
e f e
dom(A )⊇ f ∈dom(A ) γ f +M(0)γ f =0 . (3.17)
K max N D
We use the direct sum deco(cid:8)mposition (cid:12) (cid:9)
(cid:12)e f e
dom(A )=dom(A )+˙ ker(A ), (3.18)
max D max
which is not difficult to check. Assuming that f ∈ dom(A ) satisfies the
max,Ω
boundary condition
M(0)γ f +γ f =0, (3.19)
D N
f e e
THE KREIN-VON NEUMANN REALIZATION OF −∆+V ON LIPSCHITZ DOMAINS 9
according to the decomposition (3.18) we write f in the form f = f +f , where
D 0
f ∈dom(A )andf ∈ker(A ). Thus, γ f =γ f =0byTheorem3.1and
D D 0 max D D D D
with the help of Theorem 3.2(ii) one obtains
e
M(0)γ f =M(0)γ (f +f )=M(0)γ f =−γ f . (3.20)
D D D 0 D 0 N 0
Taking into account the boundary condition (3.19) one concludes
f e f e f e e
−γ f =M(0)γ f =−γ f , (3.21)
N D N 0
and hence
e f e e
0=γ (f −f )=γ f . (3.22)
N 0 N D
Together with Theorem 3.1 this implies f ∈ ker(γ ) = ker(γ ) = dom(A ).
D N N N
Thus, one arrives at e e
f ∈dom(A )∩dom(A )=dom(Ae )=H˚2(Ω), (3.23)
D D N min
where (2.10) and A =A was used (cf. Theorem 2.4(i)). Summing up, one has
D F
f =f +f ∈H˚2(Ω) ∔ ker(A )=dom(A ), (3.24)
D 0 max K
which shows (3.17) and completes the proof of Theorem 3.3. (cid:3)
4. Spectral asymptotics of the Krein–von Neumann extension
As the main result in this section we derive the following Weyl-type spectral
asymptotics for the Krein–vonNeumann extension A of A .
K min
Theorem 4.1. Assume Hypothesis 2.1. Let {λj}j∈N ⊂ (0,∞) be the strictly pos-
itive eigenvalues of the Krein–von Neumann extension A enumerated in nonde-
K
creasing order counting multiplicity, and let
N(λ,A ):=# j ∈N:0<λ 6λ , λ>0, (4.1)
K j
betheeigenvalue distribution func(cid:8)tion for A . Then t(cid:9)hefollowing Weylasymptotic
K
formula holds,
v |Ω|
N(λ,A ) = n λn/2+O λ(n−(1/2))/2 , (4.2)
K λ→∞ (2π)n
(cid:0) (cid:1)
where v = πn/2/Γ((n/2)+1) denotes the (Euclidean) volume of the unit ball in
n
Rn (with Γ(·) the classical Gamma function [1, Sect. 6.1]) and |Ω| represents the
(n-dimensional) Lebesgue measure of Ω.
The proof of Theorem 4.1 follows along the lines of [14, 15], where the case of
quasi-convex domains was investigated. The main ingredients are a general Weyl
type asymptotic formula due to Kozlov [45] (see also [44], [46] for related results)
and the connection between the eigenvalues of the so-called buckling operator and
the positive eigenvalues of the Krein–von Neumann extension A (cf. [15], [16]).
K
We first consider the quadratic forms a and t on dom(A )=H˚2(Ω) defined by
min
a[f,g]:= A f,A g , f,g ∈dom(a):=H˚2(Ω), (4.3)
min min L2(Ω)
t[f,g]:= (cid:0)f,A g (cid:1), f,g ∈dom(t):=H˚2(Ω). (4.4)
min L2(Ω)
(cid:0) (cid:1)
10 J.BEHRNDT,F.GESZTESY,T.MICHELER,ANDM.MITREA
Since the graph norm of A and the H2-norm are equivalent on domA =
min min
H˚2(Ω) by Lemma 2.2(ii), it follows that W := (dom(a);(·,·)W), where the inner
product is defined by
(f,g)W :=a[f,g]= Aminf,Aming L2(Ω), f,g ∈dom(a), (4.5)
is a Hilbert space. One observe(cid:0)s that the emb(cid:1)edding ι : W → L2(Ω) is compact;
this is a consequence of Ω being bounded. Next, we consider for fixed g ∈ W the
functional
W ∋f 7→t[ιf,ιg], (4.6)
whichis continuousonthe Hilbert space W andhence canbe representedwith the
help of a bounded operator T in W in the form
(f,Tg)W =t[ιf,ιg], f,g ∈W. (4.7)
The nonnegativity of the form t implies that T is a self-adjoint and nonnegative
operator in W. Furthermore, one obtains from (4.4) that
(f,Tg)W =t[ιf,ιg]= ιf,Aminιg L2(Ω) = f,ι∗Aminιg W, f,g ∈W, (4.8)
and hence, (cid:0) (cid:1) (cid:0) (cid:1)
∗
T =ι A ι. (4.9)
min
In particular, since A ι : W → L2(Ω) is defined on the whole space W and
min
is closed as an operator from W to L2(Ω), it follows that A ι is bounded and
min
hence the compactness of ι∗ : L2(Ω) → W implies that T = ι∗A ι is a compact
min
operator in the Hilbert space W.
The next useful lemma shows that the eigenvalues of T are precisely the recip-
rocals of the nonzero eigenvalues of A . Lemma 4.2 is inspired by the connection
K
of the Krein–von Neumann extension to the buckling of a clamped plate problem
(cf. [15, Theorem 6.2] and [14, 16, 37]).
Lemma4.2. AssumeHypothesis2.1andletT bethenonnegativecompactoperator
in W defined by (4.7). Then
λ∈σ (A )\{0} if and only if λ−1 ∈σ (T), (4.10)
p K p
counting multiplicities.
Proof. Assumefirstthatλ6=0isaneigenvalueofA andletg be acorresponding
K
eigenfunction. We decompose g in the form
g =g +g , g ∈dom(A ), g ∈ker(A ) (4.11)
min 0 min min 0 max
(cf. (2.8)), where g 6=0 as λ6=0. Then one concludes
min
A g =A (g +g )=A g, (4.12)
min min K min 0 K
and hence,
A g −λg =A g−λg =λg−λg =λg ∈ker(A ), (4.13)
min min min K min min 0 max
so that
A A g =λA g =λA g . (4.14)
max min min max min min min