Modern Analysis and Applications: The Mark Krein Centenary Conference Volume 2: Differential Operators and Mechanics

517 Pages·2009·4.869 MB·English

by  Yu. Berezansky, V. Gorbachuk (auth.), Vadim M. Adamyan, Israel Gohberg, Anatoly Kochubei, Gennadiy Popov, Yurij Berezansky, Myroslav Gorbachuk, Valentyna Gorbachuk, Heinz Langer (eds.)

Operator Theory: Advances and Applications
Vol. 191
Editor:
I. Gohberg
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School of Mathematical Sciences V. Olshevski (Storrs, CT, USA)
Tel Aviv University M. Putinar (Santa Barbara, CA, USA)
Ramat Aviv A.C.M. Ran (Amsterdam, The Netherlands)
Israel L. Rodman (Williamsburg, VA, USA)
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Modern Analysis
and Applications
The Mark Krein Centenary Conference
Volume 2: Differential Operators and Mechanics
Vadim Adamyan
Yurij Berezansky
Israel Gohberg
Myroslav Gorbachuk
Valentyna Gorbachuk
Anatoly Kochubei
Heinz Langer
Gennadiy Popov
Editors
Birkhäuser
Basel · Boston · Berlin
Editors:
Vadim M. Adamyan Yurij Berezansky
Department of Theoretical Physics Institute of Mathematics
Odessa National I.I. Mechnikov University Ukrainian National Academy of Sciences
Dvoryanska st. 2 Tereshchenkivska st.
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e-mail: [email protected] e-mail: [email protected]
Israel Gohberg Myroslav Gorbachuk
Department of Mathematical Sciences Valentyna Gorbachuk
Raymond and Beverly Sackler Insitute of Mathematics
Faculty of Exact Sciences Ukrainian National Academy of Sciences
Tel Aviv University Tereshchenkivska st.
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e-mail: [email protected] e-mail: [email protected]
Anatoly Kochubei Heinz Langer
Insitute of Mathematics Institute of Analysis and
Ukrainian National Academy of Sciences (cid:54)(cid:70)(cid:76)(cid:72)(cid:81)(cid:87)(cid:76)(cid:189)(cid:70)(cid:3)(cid:38)(cid:82)(cid:80)(cid:83)(cid:88)(cid:87)(cid:76)(cid:81)(cid:74)
Tereshchenkivska st. Technical University of Vienna
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e-mail: [email protected]
Gennadiy Popov
Department of Mathematical Physics
Odessa National I.I. Mechnikov University
Dvoryanska st. 2
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Contents
Preface ................................................................... ix
Yu. Berezansky and V. Gorbachuk
The World Dimension of the Heritage
of a Ukrainian Mathematician ....................................... xi
Part 1: Plenary Talks
S. Aizikovich, V. Alexandrov and I. Trubchik
Bilateral Asymptotic Solution of One Class of Dual
Integral Equations of the Static Contact Problems for
the Foundations Inhomogeneous in Depth ............................ 3
D. Alpay, I. Gohberg, M. Kaashoek, L. Lerer and A. Sakhnovich
Krein Systems ....................................................... 19
Yu.M. Berezansky
Spectral Theory of the Infinite Block Jacobi Type Normal
Matrices, OrthogonalPolynomials on a Complex Domain,
and the Complex Moment Problem .................................. 37
G.L. Brovko, O.A. Ivanova and A.S. Finoshkina
On Geometrical and Analytical Aspects in Formulations of
Problems of Classic and Non-classic Continuum Mechanics ........... 51
F. Gesztesy and M. Mitrea
Robin-to-Robin Maps and Krein-Type Resolvent Formulas for
Schro¨dinger Operators on Bounded Lipschitz Domains ............... 81
M.L. Gorbachuk and V.I. Gorbachuk
On Behavior of Weak Solutions of Operator Differential
Equations on (0,∞) ................................................. 115
M.A. Grekov and N.F. Morozov
Some Modern Methods in Mechanics of Cracks ...................... 127
vi Contents
R. Kushnir and B. Protsiuk
A Method of the Green’s Functions for Quasistatic
Thermoelasticity Problems in Layered Thermosensitive
Bodies under Complex Heat Exchange ............................... 143
S.M. Mkhitaryan
On the Application of the M.G. Krein Method for the Solution
of Integral Equations in Contact Problems in Elasticity Theory ...... 155
G. Popov and N. Vaysfel’d
The Stress Concentration in the Neighborhood of the Spherical
Crack Inside the Infinite Elastic Cone ................................ 173
J. Rovnyak and L.A. Sakhnovich
Pseudospectral Functions for Canonical Differential Systems ......... 187
I. Selezov
On Wave Hyperbolic Model for Disturbance Propagation
in Magnetic Fluid ................................................... 221
Part 2: Research Papers
A. Ashyralyev
High-accuracy Stable Difference Schemes for Well-posed NBVP ...... 229
A. Belyaev
The Factorization of the Flow, Defined by the
Euler-Poisson’sEquations ........................................... 253
V.P. Burskii
On a Moment Problem on a Curve Connected with Ill-posed
Boundary Value Problems for a PDE and Some Other Problems ..... 273
F. Chaban and H. Shynkarenko
The Construction and Analysis of a PosterioriError Estimators
for Piezoelectricity Stationary Problems ............................. 291
H.O. Cordes
Remarks about Observables for the Quantum Mechanical
Harmonic Oscillator ................................................. 305
G. Eskin and J. Ralston
Remark on Spectral Rigidity for Magnetic Schro¨dinger Operators .... 323
S. Gefter and T. Stulova
On Holomorphic Solutions of Some Implicit Linear
Differential Equations in a Banach Space ............................ 331
Contents vii
V.I. Gerasimenko
Groups of Operators for Evolution Equations of Quantum
Many-particle Systems ............................................... 341
V.A. Grishin, V.V. Reut and E.V. Reut
Box-like Shells with Longitudinal Cracks ............................. 357
O.Ye. Hentosh
Lax Integrable Supersymmetric Hierarchies on Extended
Phase Spaces of Two Anticommuting Variables ...................... 365
N.D. Kopachevsky and V.I. Voytitsky
On the Modified Spectral Stefan Problem and
Its Abstract Generalizations ......................................... 381
O. Kryvyy
The Discontinuous Solution for the Piece-homogeneous
TransversalIsotropic Medium ........................................ 395
M.V. Markin
On the Carleman Ultradifferentiability of Weak Solutions
of an Abstract Evolution Equation ................................... 407
J. Michor and G. Teschl
On the Equivalence of Different Lax Pairs for the
Kac–van Moerbeke Hierarchy ........................................ 445
V.A. Mikhailets and A.A. Murach
Elliptic Problems and Ho¨rmander Spaces ............................ 455
G. Poletaev
Connection of Solutions of Abstract Paired Equations
in Rings with Factorization Pairs .................................... 479
V.G. Popov, O.V. Litvin and A.P. Moysyeyenok
The Dynamic Problems About the Definition of Stress State
Near Thin Elastic Inclusions Under the Conditions
of Perfect Coupling .................................................. 485
R.M. Trigub
Fourier Multipliers and Comparison of Linear Operators ............. 499
V.M. Vorobel and V.V. Reut
Forced Vibrations of the Infinite Shell of the Square Cross Section ... 515
Preface
This is the second of two volumes containing peer-reviewed research and survey
papersbasedoninvitedtalksatthe InternationalConferenceonModernAnalysis
and Applications. The conference, which was dedicated to the 100th anniversary
ofthebirthofMarkKrein,oneofthegreatestmathematiciansofthe20thcentury,
was held in Odessa, Ukraine, on April 9–14, 2007. The conference focused on the
main ideas, methods, results, and achievements of M.G. Krein.
This second volume is devoted to the theory of differential operators and
mechanics.Itopenswiththe descriptionoftheconferenceandanumberofsurvey
papers about the work of M.G. Krein. The main part of the book consists of
original research papers presenting the state of the art in the area of differential
operators.
Thefirstvolumeofthese proceedings,entitledOperatorTheoryandRelated
Topics, concerns other aspects of the conference. The two volumes will be of in-
teresttoa wide-rangeofreadershipinpureandappliedmathematics,physicsand
engineering sciences.
OperatorTheory:
Advances andApplications,Vol.191, xi–xv
(cid:2)c 2009Birkh¨auserVerlagBasel/Switzerland
The World Dimension of the Heritage
of a Ukrainian Mathematician
International Conference “Modern Analysis and Applications”
(MAA – 2007) (April 9–14, 2007, Odessa)
Yu. Berezansky and V. Gorbachuk
This forum has been dedicated to the centennial birthday anniversary of one
of the most prominent mathematicians of the twentieth century Mark Grig-
orievich Krein, a corresponding member of the Academy of Sciences of the
Ukr. SSR (1907–1989).
TheorganizersoftheconferencearetheNationalAcademyofSciencesofUkraine,
theMinistryofEducationandScienceofUkraine,theOdessaCityCouncil,theIn-
stituteofMathematicsoftheNationalAcademyofSciencesofUkraine,theOdessa
National I.I. Mechnikov University, the Odessa National Maritime University,the
South-UkrainianStateK.UshinskyPedagogicalUniversity,theInstitute ofMath-
ematics, Economics, and Mechanics, and Faculties of Mathematical Physics and
Theoretical Physics of the Odessa National University.
A substantial assistance for conducting the conference came from the fol-
lowing sponsors: Swedish Institute, European Science Foundation, Odessa City
Council, Manufactured Goods Market Limited, JV Dipolos Limited, Imexbank
(Joint Stock Commercial Bank), Southern Bank, Porto–FrancoBank.
There were 252 scientists who participated in the work of the conference,
coming from 29 countries: Algeria, Armenia, Austria, Belgium, Brazil, Canada,
Denmark, Finland, France, Germany, Great Britain, Israel, Japan, Mexico, the
Netherlands, New Zealand, Norway, Poland, Portugal, Qatar, Russia, Slovakia,
SouthAfricanRepublic,Spain,Sweden,Switzerland,Turkey,Ukraine,USA.Many
of the participating scientists are well known to the broad mathematical commu-
nity.TheyareH.Langer(Austria),P.Sobolevskii(Brazil),P.Lancaster(Canada),
O. Staffans (Finland), V. Zagrebnov (France), H. Dym, I. Gohberg, Yu. Lyubich,
V. Matsaev,V. Milman (Israel),T. Ando (Japan),A. Dijksma (the Netherlands),
B. Pavlov (the New Zealand), M. Mo¨ller (South African Republic), A. Laptev,
A. Lindquist (Sweden), C. Tretter (Switzerland), M. Agranovich, S. Kislyakov,
A. Shkalikov (Russia), V. Adamyan, D. Arov, Yu. Berezansky, M. Gorbachuk,
xii Yu. Berezansky and V. Gorbachuk
I. Kats, E. Khruslov, A. Kochubei, M. Kopachevsky, V. Marchenko, L. Pastur,
G.Popov,F.Rofe-Beketov,A.Samoilenko,Yu.Samoilenko(Ukraine),F.Gesztesy,
Yu. Latushkin, L. Nirenberg, J. Ralston, J. Rovnyak (USA), and others.
TheconferencemeetingswereheldinthebuildingoftheNationalAcademyof
Public Administration under the President of Ukraine. The topics discussed were
concerned with problems in both modern Mathematical Analysis and Mechanics.
Therewere8MathematicalAnalysissectionsincludingthefollowing:thegeometry
of Banach and Krein spaces; evolution equations in Banach spaces; the theory of
operatorson Hilbert and Kreinspaces and its applications; non-selfadjointopera-
tors and algebraic problems of Functional Analysis; spectral theory of differential
anddifferenceoperators;directandinverseproblems;applicationsofspectralthe-
ory to problems in analysis;controltheory and the theory of stochastic processes;
nonlinear analysis; harmonic analysis, the analysis of functions of infinitely many
variables,generalizedfunctions. The problems in Mechanics were discussed in the
corresponding section.
Opening talks containing recollections about the life and the work of
M.G.Krein,importanceofhisscientificheritageforthedevelopmentofthemodern
mathematicsandforthe applicationsweregivenbythe Headofthe SouthernSci-
entific Center, academicianof the NationalAcademy of Sciences, S.A. Andronati,
DirectoroftheInstitute ofMathematics,academicianoftheNationalAcademyof
Sciences of Ukraine, A.M. Samoilenko, Rector of the Odessa National I.I. Mech-
nikov University, academician of the Higher School Academy, V.A. Smintina, as
wellasthestudentsandcollaboratorsofM.G.Krein,–V.M.Adamyan,D.Z.Arov,
Yu.M. Berezansky, I. Gohberg, M.L. Gorbachuk, H. Langer. All 25 plenary talks
discussed the influence of ideas and scientific results of M.G. Kreinon subsequent
developments of different directions in the modern analysis, and applications of
the ideas and the results in various branches of science and technology.
M.G. Krein was a phenomenally gifted mathematician. His way to science
was equally extraordinary. His life as a scientist and a person reflects dramatic
events of Soviet totalitarian times.
Let us shortly describe milestones of the life and work of this brilliant scien-
tist, a world figure in mathematics.
He was born in Kiev on April 3, 1907, in a modest income family that was
raising seven children. M.G. Krein had exhibited his extraordinary mathematical
abilities in his teens. Already when he was 14, he attended lectures and scien-
tific seminars conducted by D.O. Grave and B.M. Delone at Kiev University and
Kiev Polytechnic Institute. At the age of 17, influenced by the work of M. Gorky
“My Universities”, he decided that it was the time to start his own “universities”
and, together with his friend, he went to Odessa to join one of circus troupes,
for he had a dream of becoming an acrobat. However, the fate had its way, and
saved to the world, in the person of M.G. Krein, not an acrobat but a promi-
nent mathematician whose influence on the development of mathematics can not
be overestimated. In Odessa, he met N.G. Chebotarev, a famous algebraistand a