Topics in Analysis and Its Applicationsproceedings of the NATO Advanced Research Workshop, Yerevan, Armenia, 22-25 September 2002

482 Pages·2004·3.13 MB·English

by  Grigor A. Barsegian

Topics in Analysis and its Applications
NATO Science Series
A Series presenting the results of scientific meetings supported under the NATO Science
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Series II:Mathematics,Physics and Chemistry – Vol.147
Topics in Analysis and its
Applications
edited by
G.A.Barsegian
Institute of Mathematics of National Academy of Sciences of Armenia,
Yerevan, Armenia
and
H.G.W.Begehr
Mathematical Institute,
Freie Universität Berlin,
Berlin, Germany
KLUWER ACADEMIC PUBLISHERS
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Table of Contents
Preface ix
Shilov boundary for normed algebras 1
A. Escassut and N.M. Netti
BMO-mappings in the plane 11
C. Andreian Cazacu and V. Stanciu
Harmonic forms on non-orientable surfaces 31
A. Alonso, A. Fern´andez and J. P´erez
Periodic Fatou components and singularities of the inverse
function 47
W. Bergweiler
On the normality of topological target manifolds for Riemann-
Hilbert problems 61
E. Wegert and G. Semmler
Geometric aspects of generalized analytic functions 69
G. Giorgadze
The Riemann-Hilbert boundary value problem on a cut plane 83
N. Manjavidze
On the logarithmic derivative of meromorphic functions 91
A.A. Mokhon’ko and A.Z. Mokhon’ko
Methods for studying level sets of smooth enough functions 105
G.A. Barsegian and G.A. Sukiasyan
Gamma-lines of polynomials and a problem by Erdo¨s-Herzog-
Piranian 119
G.A. Barsegian
A method for studying oscillations of nonlinear differential equa-
tions. Applications to some equations in biology and economics 123
K. Barseghyan and G.A. Barsegian
vi
Counting points of semi-algebraic subsets 149
T. Aliashvili
Behaviour at infinity of polynomials of two variables 163
H.G. Ghazaryan and V.N. Margaryan
On some properties of degenerate elliptic systems of partial
differential equations 191
A. Dzhuraev
Formulas for derivatives of solutions of the ∂ equation in the
−
ball 229
A.I. Petrosyan
On some complex differential and singular integral operators 235
H. Begehr and A. Dzhuraev
Boundary and initial value problems for higher order PDEs in
Clifford analysis 271
E. Obolashvili
On unique solvability of the Dirichlet problem for one class of
properly elliptic equations 287
A.O. Babayan
Dirichlet problems with nonsmooth boundary 295
A.O. C¸elebi
Dirichlet problem in the half-plane for RO varying weight
−
functions 311
H. Hairapetian
About one class of Volterra type linear integral equations with
an interior fixed singular or super-singular point 317
N. Rajabov
The method of discrete singularities of solutions of singular
integral equations with unmoved singularity 327
A. Sahakyan
Localization operators, Wigner transforms and paraproducts 333
M.W. Wong
The flight of an aircraft along a given trajectory and optimal
flight control 347
N.E. Tovmasyan
On mathematical problems of two-dimensional tomography 365
A. Vagarshakyan
vii
On a mixed problem for a composite plane weakened by
arc-type cracks 377
V.N. Hakobyan and L.L. Dashtoyan
Solution of the two-dimensional magnetoelastic Lamb problem 385
G.Y. Baghdasaryan, Z.N. Danoyan and M.A. Mikilyan
On an eigenvalue problem for the anisotropic strip 397
M. Aghalovyan
On singular perturbed equations of thin bodies 403
L. Aghalovyan
Exactly solvable models of stochastic quantum mechanics within
the framework of Langevin-Schro¨dinger type equations 415
A.S. Gevorkyan
Generating functions and wavelet-like decompositions 443
A. Nersessian
Index of Names 453
Index of Subjects 457
Preface
Subsequent to an International ISAAC Conference on “Complex Anal-
ysis, Differential Equations and Related Topics”, the NATO ARW
Topics in Analysis and its Applications
took place in Yerevan, Armenia with participants from Armenia, Aus-
tria,Belarus,Canada,Finland,Georgia,Germany,Italy,Marocco,Poland,
Portugal, Romania, Russia, Spain, Sweden, Tajikistan, Turkey, Ukraine,
USA.Thepresentvolumereflectsthemaincontributionsbutneitherthe
stimulating atmosphere nor the highly friendly relations among the par-
ticipants. Thanks are due to many members of the family of Grigor
Barsegian, the Armenian co-director of this ARW. They have helped a
lottomakethefourdayspleasantandeasyforallparticipants. Without
the support from the Scientific and Environmental Affairs Devision of
the NATO the event would not have taken place. All participants were
very grateful to Dr. F. Pedrazzini, the Programme Director of Phys-
ical and Engineering Science and Technology and his staff for having
arranged the support in time although it was very tough before the be-
ginning.
Most topics in the workshop were devoted to complex analysis as well
of one as of several complex variables. Several contributions came from
elasticity theory.
Onegeneralizationofcomplexanalysisisthetheoryofp-adicanalysis,
a field enjoying a lot of attention in recent years. Multiplicative semi-
norms in an ultrametric normed algebra have many applications in the
study of analytic elements in p-adic analysis. Here the existence of
a Shilov boundary for a semi-multiplicative semi-norm defined on an
algebra is found inside a set of multiplicative semi-norms.
In geometric function theory quasiconformal mappings are studied a
lot since almost 80 years. This concept is called quasiregular in higher
dimensions. More recent research involves mappings of bounded mean
oscillations. Thisresearchissurveyedonaswellforplanedomainsasfor
x
RiemannandKleinsurfaces. ClassicalsubjectsinthetheoryofRiemann
surfaces are harmonic and analytic differentials. Spaces of differential
forms are connected with the topological and the conformal structure of
the surface. Here these connections are investigated for non-orientable
i.e. for Klein surfaces.
A classical but still very modern subject is complex dynamics. This
topic has started mainly with the works of P. Fatou and G. Julia some
eighty years ago and later has lead to the very important areas of frac-
tals and of dynamical systems. One contribution surveys on classical
and recent results on complex dynamics related to inverse functions of
rational or transcendental entire functions.
Boundaryvalueproblemsforanalyticfunctionswereinitiatedalready
in the famous thesis of B. Riemann just 150 years ago. They have influ-
enced not only mathematical analysis a lot but even algebraic geometry
and topology. And still new insights are gotten. Here the nonlinear
Riemann–Hilbert problem for analytic functions with nonsmooth target
mainfolds are treated. The solutions can be attained as extremal func-
tions in certain function classes. Disproving a conjecture an example is
given of a nonsmooth topological target manifold for which the solution
set of the Riemann-Hilbert problem is bigger than in the smooth case.
While complex function theory from the viewpoint of partial differen-
tial equations is nothing else but the theory of solutions to the Cauchy-
Riemann equation generalized analytic functions are related in the same
way to a more general first order elliptic system, the Carleman-Bers-
Vekuasystem. Analyticandtopologicalaspectsofthissystemisstudied
and the Riemann-Hilbert monodromy problem solved for a special class
of regular systems. Moreover, a holomorphic vector bundle on the Rie-
mann sphere is constructed together with an L -connection on it using
p
these systems. Another classical boundary value problem is the Rie-
mann jump problem extensively studied in the former SU in particular
in Belarus and in Georgia. Here this problem is reported on for analytic
vectors in the cut plane.
Value distribution theory is the completion of the theory of analytic
and meromorphic function theory. The logarithmic derivative is a main
tool in this theory. It is here studied for meromorphic functions in
angular domains, one important case of domains investigated in value
distribution theory. Recently in Armenia an entirely new approach to
value distribution theory is made. Instead of considering preimages of
single points, preimages of more general sets e.g. lines are investigated
leading to Γ-lines. This new generalization seems to open a wide field of
applicationsofthemethodsofvaluedistributiontheoryastheconceptof
level sets appears in many fields of mathematics, mathematical physics
PREFACE xi
and natural sciences. The lengths of level sets of smooth functions are
estimated, in particular those of Γ-lines, of monic polynomials for large
classes of curves Γ. In case when Γ is closed this is related to the Erd¨os-
Herzog-Piranian problem. Through Γ-lines also oscillations of solutions
to ordinary differential equations including the Riccati, the Schro¨dinger
and the Painlev´e equations can be treated leading to applications in
population dynamics and economics.
Also in real algebraic and real analytic geometry the question of zeros
has a long history. Rather than localizing the roots of a real polynomial
it is important just to effectively find their number e.g. in the left half
plane. A general algorithm is described for computing the number of
points of an arbitrary finite semi-algebraic subset and the complexity of
the algorithm is estimated.
Thebehaviorofpolynomialsofonevariableatinfinityisquiteobvious.
The situation is different for several variables. Necessary and sufficient
conditions are given such that the polynomial tends to infinity when the
variables do. This property is important for linear differential operators
determining its typ.
One motivation to develop the theory of several complex variables
was once to solve partial differential equations. But the theory became
very abstract and far-reaching and not too many equations were solved
explicitly. In recent years purely analytic methods were used to solve
boundary value problems for mainly first and second order overdeter-
mined systems either in the unit ball or in polydomains. Here first and
second order systemsin bounded domainsdegenerating attheboundary
are solved explicitly. In order to explain the differences one and several
variables are both treated.
Abasictoolintreatingpartialdifferentialequationsincomplexspaces
are Pompeiu operators. For several variables they are known either for
polydomains or for the unit ball. Using the weighted version of the
Cauchy-Pompeiu representation explicit formulas for the derivatives of
the solution to the inhomogeneous Cauchy-Riemann system in the unit
ball having minimal norm in a particular L space are given.
2
−
ThroughhigherorderGreenfunctionsmodifiedhigherorderPompeiu
operatorsareusedaswellforoneasforseveralvariablestoexpressall n-
th order partial derivatives of a complex function by just one particular
n-th order partial derivative via strongly singular integral operators of
Calderon-Zygmund type. In case of several variables both polydomains
and the unit ball are treated.
Complex methods to some extend are also available to problems in
several real variables even in odd number. Quaternionic, octonionic
and Clifford analysis are the proper frameworks. Many different kinds