Table Of Content7513_ot164titelei 1 2.12.2005, 10:55 Uhr
Operator Theory: Advances and
Applications
Vol. 164
Editor:
I. Gohberg
Editorial Office: S. T. Kuroda (Tokyo)
School of Mathematical P. Lancaster (Calgary)
Sciences L. E. Lerer (Haifa)
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Editorial Board: L. Rodman (Williamsburg)
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7513_ot164titelei 2 2.12.2005, 10:55 Uhr
Pseudo-Differential Operators
and Related Topics
Paolo Boggiatto
Luigi Rodino
Joachim Toft
M.W. Wong
Editors
Birkhäuser Verlag
. .
Basel Boston Berlin
7513_ot164titelei 3 2.12.2005, 10:55 Uhr
Editors:
Paolo Boggiatto Joachim Toft
Dipartimento di Matematica School of Mathematics and Systems
Università di Torino Engineering
Via Carlo Alberto, 10 Växjö University
10123 Torino SE-351 95 Växjö
Italy Sweden
e-mail: [email protected] e-mail: [email protected]
Luigi Rodino M. W. Wong
Dipartimento di Matematica Department of Mathematics and Statistics
Università di Torino York University
Via Carlo Alberto, 10 4700 Keele Street
10123 Torino Toronto, Ontario M3J 1P3
Italy Canada
e-mail: [email protected] e-mail: [email protected]
2000 Mathematics Subject Classification: Primary 35A17, 35L15, 35L40, 35L45, 35L60,
35L80, 35P05, 35S05, 35S30, 42C15, 43A80, 46F05, 47G30, 53D12, 58J40, 60G12,
60G15, 81R30; Secondary 17B45, 35B65, 35C20, 35J50, 35L30, 42C40, 44A05, 45C05,
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ISBN 3-7643-7513-2 Birkhäuser Verlag, Basel – Boston – Berlin
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7513_ot164titelei 4 2.12.2005, 10:55 Uhr
Contents
Preface ......................................................................vii
M.S. Agranovich
Strongly Elliptic Second Order Systems with Spectral Parameter
in Transmission Conditions on a Nonclosed Surface ......................1
A. Ascanelli and M. Cicognani
Well-Posedness of the Cauchy Problem for Some Degenerate
Hyperbolic Operators ...................................................23
M. Cappiello and L. Zanghirati
Quasilinear Hyperbolic Equations with SG-Coefficients .................43
I. Kamotski and M. Ruzhansky
Representation of Solutions and Regularity Properties for Weakly
Hyperbolic Systems ....................................................53
M. Ruzhansky and M. Sugimoto
Global Calculus of Fourier Integral Operators,Weighted Estimates,
and Applications to Global Analysis of Hyperbolic Equations ...........65
G. Garello and A. Morando
Lp-Continuity for Pseudo-Differential Operators ........................79
V.S. Rabinovich
Fredholm Property of Pseudo-Differential Operators on Weighted
H¨older-Zygmund Spaces ................................................95
M.W. Wong
Weyl Transforms and Convolution Operators on the Heisenberg Group 115
M. de Gosson
Uncertainty Principle, Phase Space Ellipsoids and Weyl Calculus ......121
K. Furutani
Pseudo-Differential Operator and Reproducing Kernels Arising in
Geometric Quantization ...............................................133
J. Toft
Hudson’s Theorem and Rank One Operators in Weyl Calculus .........153
A. Khrennikov
Distributions and Pseudo-Differential Operators on Infinite-Dimensional
Spaces with Applications in Quantum Physics .........................161
N. Teofanov
Ultradistributions and Time-Frequency Analysis .......................173
vi Contents
O. Christensen
Frames and Generalized Shift-Invariant Systems .......................193
P. Wahlberg
The Wigner Distribution of Gaussian Weakly Harmonizable
Stochastic Processes ...................................................211
E. Cordero, F. De Mari, K. Nowak, and A. Tabacco
Reproducing Groups for the Metaplectic Representation ...............227
Preface
Asasatelliteconferencetothe FourthCongressofEuropeanMathematicsheldat
StockholmUniversityin2004,theInternationalConferenceonPseudo-Differential
Operators and Related Topics was held at V¨axjo¨ University in Sweden from June
22toJune25,2004.TheconferencewassupportedbyVa¨xjo¨University,theFIRB
Research Group on Microlocal Analysis of Universit`a di Torino, and the Inter-
national Society for Analysis, its Applications and Computation (ISAAC). The
conferencewaswellattendedbyabout50mathematiciansfromBulgaria,Canada,
Denmark,England,Finland,Germany,Italy,Japan,Mexico,SerbiaandMontene-
gro, Russia and Sweden.
The conference covered a broad spectrum of topics related to pseudo-differential
operators such as partial differential equations, quantization, Wigner transforms
andWeyltransformsonLiegroups,mathematicalphysics,time-frequencyanalysis
and stochastic processes. The speakers were enthusiastic about the prospect of
publishing articles based on their presentations in a volume to be published in
ProfessorIsraelGohberg’sprestigiousseriesentitled “OperatorTheory:Advances
and Applications”.
All contributions from speakers have been carefully refereed and the articles col-
lected in this volume give a representative flavour of the mathematics presented
at the conference. This volume is a permanent record of the conference and a
valuable complement to the volume “Advances in Pseudo-Differential Operators”
published in the same series in 2004, which is devoted to the Special Session on
Pseudo-Differential Operators at the Fourth ISAAC Congress held at York Uni-
versity in August 2003.
OperatorTheory:
Advances andApplications,Vol.164,1–21
(cid:1)c 2006Birkh¨auserVerlagBasel/Switzerland
Strongly Elliptic Second Order Systems
with Spectral Parameter in Transmission
Conditions on a Nonclosed Surface
M.S. Agranovich
Abstract. WeconsideraclassofsecondorderstronglyellipticsystemsinRn,
n≥3,outsideaboundednonclosedsurfaceSwithtransmissionconditionson
S containing a spectral parameter. Assuming that S and its boundary γ are
Lipschitz, we reduce the problems to spectral equations on S for operators
of potential type. We prove the invertibility of these operators in suitable
Sobolev type spaces and indicate spectral consequences. Simultaneously, we
prove the unique solvability of the Dirichlet and Neumann problems with
boundary data on S.
Mathematics SubjectClassification (2000). Primary 35P05; Secondary 35J50,
45C05.
Keywords. Strong ellipticity, transmission condition, spectral equation, Lip-
schitz surface, surface potential, variational approach, Wiener–Hopf method.
1. Introduction
1.1. Statement of the Problems
We consider the second order system of partial differential equations
L (∂)u(x):=L (∂)u(x)+ω2u(x)=0 (1.1)
ω 0
in Rn \S, where S is an (n−1)-dimensional surface with (n −2)-dimensional
boundaryγ.More precisely,we assumethat S is a partofa closedsurface1 Γ;the
latter consists of two open parts S = Γ+ and Γ− without common points and of
theircommonboundaryγ.By∂ wedenotethe“vector”ofpartialderivatives∂ =
j
∂/∂ ,j =1,...,n.TheoperatorL ishomogeneouswithrespecttodifferentiations
j 0
and has constant coefficients. The numerical parameter ω = ω +iω belongs to
1 2
TheworkwassupportedbythegrantofRFFINo.04-01-00914.
1Aclosedsurfaceisacompactsurfacewithoutboundary.
2 M.S. Agranovich
the closed upper half-plane (ω ≥ 0) and is nonnegative if it is real. At infinity,
2
the solutions are subjected to natural radiation or decay conditions depending
on ω. The assumptions are formulated more precisely below in Subsection 1.2.
We assume that n ≥ 3 to avoid the consideration of logarithmic potentials. The
surfaceΓdividesthecomplementofitselfintoaboundedsimplyconnecteddomain
Ω+ and an unbounded domain Ω−. The superscripts + and − will also be used
to denote the boundary values of functions on the inner and outer sides of Γ,
respectively. By ν =ν(x) we denote the unit outward normal at points x∈Γ.
Ourmaingoalistoconsidertwospectralproblemsfor(1.1)withtransmission
conditions on S containing a spectral parameter λ. The correspondingspaces will
be specified later.
Problem I.
u+ =u−, Tu−−Tu+ =λu± on S. (1.2)
Problem II.
Tu+ =Tu−, Tu± =λ[u−−u+] on S. (1.3)
Here Tu is the conormal derivative, see (1.8) below. In the simplest case
of the Helmholtz equation, Tu is the normal derivative ∂ u. We wish to describe
ν
somespectralpropertiesoftheseproblems.Wewillseeshortlythattheyareclosely
connectedwiththenon-spectralDirichletandNeumannproblemsforsystem(1.1):
The Dirichlet problem.
u± =f on S. (1.4)
The Neumann problem.
Tu± =g on S. (1.5)
The surfaces Γ and γ are assumed to be either C∞ or Lipschitz; Γ is con-
nected, while γ can consist of several components. The normal ν(x) is defined
almost everywhere in the Lipschitz case.
In the case of a closed surface S = Γ, Problems I and II and some other
problemsfortheHelmholtzequationwereposedbythephysicistsKatsenelenbaum
andhiscollaboratorsSivovandVoitovichinthe60s.Seethebook[1]anditsrevised
English edition [2]. In [1], a mathematical supplement [3] is contained, written by
the author of the present paper, with the analysis of these and similar problems
in acoustics andelectrodynamics by tools of the theories of surface potentials and
pseudo-differential operators. The surface was assumed to be smooth. The initial
results were obtained by the author in collaboration with his graduate student
Golubeva; see also her note [4]. Conditions (1.2) and (1.3) can be interpreted as
related to a half-transparentscreen.
Similar problems with boundary and transmission conditions on a closed
Lipschitz surface for the Helmholtz equation were considered in [5] and for the
Lam´e system in elasticity theory (and n = 3) in [6]. The general case of systems
(1.1) was considered in [7] and [8]. In [8], systems with variable coefficients were
included into consideration. More precisely, in the last three papers the surface is
Strongly Elliptic Second Order Systems 3
assumed to be either smooth or Lipschitz. Of course,no theory of elliptic pseudo-
differential operators exists in the case of a Lipschitz surface, but there is an
extensivetheoryofclassicalsurfacepotentialsandnon-spectralproblems;see[7,8]
and numerous references therein.
InSection2,werecallsomedefinitionsandtechnicaltoolsfrom[7,8]related
to the case of a closed surface. We also add some supplementary material. In
particular, we introduce the hypersingular operator and Caldero´n projections for
general systems (1.1) following, e.g., the paper [9] on the Laplace and Helmholtz
equations.
Non-spectralproblems(Dirichlet,Neumann,andmoregeneral)withdataon
a nonclosed surface for the Helmholtz equation and the Lam´e system were first
consideredbyStephan[10,11]andthenbyCostabelandStephan[12].Inelasticity
theory, a non-closed surface has the meaning of a crack. The Lam´e system mod-
els an isotropic medium. Cracks in anisotropic elastic media were considered by
Duduchava, Natroshvili and Shargorodski [13] and by some other authors. More-
over, non-spectral problems in elasticity were considered with much more general
conditions on S and in much more general spaces than in the present paper, see
also, e.g., [14] and references therein.
These authors followed Vishik and Eskin (e.g., see [15], [16]) and Eskin [17]
andusedtheWiener–HopfmethodassumingthatΓandγ aresufficientlysmooth.
In Section 3, the main in the present paper, we will show that it is possible
to consider Lipschitz surfaces Γ and γ using the simplest Sobolev type spaces
H±1/2(S) and H(cid:1)±1/2(S). (The spaces are defined in Subsection 2.5.) Instead of
the Wiener–Hopf method, a modification of the classical variational approach is
used(seeourPropositions3.2and3.4forthecaseofpureimaginaryω).We prove
the unique solvability ofthe Dirichlet andNeumann problems for generalsystems
(1.1) and the invertibility of the potential type operators on S corresponding to
these problems. It seems to us that these non-spectral results are of interest even
for the Helmholtz and Lam´e equations.
Exactly the same invertible operators occur in the spectral equations on S
correspondingtoourspectralproblemsIandII.Thusthesimplestspectralresults
become available in the Lipschitz case; see Subsection 3.4.
In Section 4, we briefly mention some further results; they will be published
elsewhere.
1.2. Exact Statement of the Assumptions
(See [7] for details.) The operator L (∂) is an m × m matrix. Replacing ∂ by
0
ξ =(ξ ,...,ξ ), we obtain the principal symbol of the operator −L (∂):
1 n 0
(cid:2)(cid:2)
L (ξ)= A ξ ξ , A =(ars). (1.6)
0 jk j k jk jk
Here the A are real matrices satisfying the symmetry condition ars =asr. The
jk jk kj
matrix L (ξ) is assumed to be positive definite for ξ (cid:3)= 0, which is the strong
0
ellipticity condition for the operator −L(∂). Besides, as in [7, 8], we impose the
