23/09/2005, 15:15 — 16:15 — Sala P3.10, Pavilhão de Matemática
Bernd Silbermann, Technische Universität Chemnitz, Germany
Asymptotic behavior of variable Toeplitz matrices
Variable Toeplitz matrices are generated by functions of two
variables and share with common Toeplitz matrices a few important
properties which come on light considering the algebra generated by
sequences of such matrices. These considerations allow to deduce
Szegö type theorems which asymptotically describe the distribution
of the eigenvalues.
23/09/2005, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Eugene Shargorodsky, King's College, London, England
Complex methods for Bernoulli free-boundary problems
A Bernoulli free-boundary problem is one of finding domains in the
plane on which a harmonic function simultaneously satisfies
homogeneous linear Dirichlet and inhomogeneous linear Neumann
boundary conditions. The boundary of such a domain (called the free
boundary because it is not prescribed a priori) is the essential
ingredient of a solution. The classical Stokes waves provide an
important example of a Bernoulli free-boundary problem. Existence,
multiplicity or uniqueness, and smoothness of boundaries are
important questions and, despite appearances, the problem of
determining free boundaries is nonlinear. The talk, based on a
joint work with J.F. Toland, will examine an equivalence between
these free-boundary problems and a set of nonlinear
pseudo-differential equations, for one real-valued function of one
real variable, which have the gradient structure of an
Euler-Lagrange equation and can be formulated in terms of
Riemann-Hilbert theory. The equivalence is global in the sense that
it involves no restriction on the amplitudes of solutions, nor on
their smoothness. Non-existence and regularity results will be
described and some important unresolved questions about precisely
how irregular a Bernoulli free boundary can be will be formulated.
16/09/2005, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Alexei Karlovich, Universidade do Minho
Algebras of singular integral operators on Nakano spaces with
Khvedelidze weights over Carleson curves with logarithmic whirl
points
We establish a Fredholm criterion for an arbitrary operator in the
Banach algebra of singular integral operators with piecewise
continuous coefficients on Nakano spaces (generalized Lebesgue
spaces with variable exponent) with Khvedelidze weights over
Carleson curves with logarithmic whirl points. The proofs are based
on the boundedness result for the Cauchy singular integral operator
over arbitrary Carleson curves by Kokilashvili and Samko (presented
on OTFUSA 2005) and on the theory of submltiplicative functions
associated with curves, weights, and spaces developed by
Boettcher-Yu. Karlovich and further by the author.
13/09/2005, 16:00 — 17:00 — Sala P3.10, Pavilhão de Matemática
Steffen Roch, Technische Universität Darmstadt, Germany
Finite sections of band-dominated operators
The goal of this talk is to review recent advances and to present
new results in the numerical analysis of the finite sections method
for general band and band-dominated operators. The main topics are
the stability of the finite sections method and the asymptotic
behavior of singular values. The latter topic is closely related
with compactness and Fredholm properties of approximation
sequences. Special emphasis is paid to band-dominated operators
with coefficients in the classes of slowly oscillating functions
and almost periodic functions, respectively.
15/07/2005, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
A. V. Balakrishnan, The Flight Systems Research Center, University of California, LosAngeles, USA
Mathematical theory of aeroelasticity
The central problem of aeroelasticity involves an endemic safety
issue - the determination of the 'Flutter Boundary' - the speed at
which the wing structure becomes unstable at any given altitude.
Currently all the theoretical work is computational - wedding the
Lagrangian FEM structure codes to the Euler CFD codes to produce a
'time-marching' solution. While they can handle 'real life'
nonlinear - complex geometry - structures and viscous flows, they
are based on approximation by ordinary differential equations, and
limited to specific numerical parameters. In turn this limits the
generality of the results and understanding of phenomena involved;
and of course inadequate for control design for possible
stabilization ('Flutter Suppression'). In this presentation we show
that the problem can be formulated retaining the full continuum
models without approximation, as a boundary value problem for
coupled nonlinear partial differential equations. The flutter speed
can then be characterized as a Hopf bifurcation point for a
nonlinear convolution-evolution equation in the time-domain, which
- and this is the crucial point - is then determined completely by
the linearized equations - linearized about the equilibrium state.
A key step in this approach is a singular integral equation with a
difference kernel, discovered by Camillo Possio in 1938, and
bearing his name, linking the aerodynamics to the structure
dynamics. A challenge here is to choose models which are amenable
to analysis, taking advantage of recent advances in boundary value
problems, and yet can display the phenomena of interest. The
presentation will emphasize problem formulation but will include
recent results both analytical and experimental (flight-tests).
28/06/2005, 16:00 — 17:00 — Sala P3.31, Pavilhão de Matemática
Sergei V. Rogosin, Belarussian State University, Minsk, Belarus
A nonlinear Riemann-Hilbert boundary value problem and its relation to a matrix factorization problem
We consider the relation of a particular Riemann-Hilbert boundary value problem for vector functions with matrix coefficients to a matrix factorization. Main attention is paid to the constructive solution of a matrix -linear conjugation problem with piece-wise constant matrix on the plane cut along a set that consists of a finite number of intervals. The approach is based on the method of functional equations developed for the scalar case in the monograph by V.V. Mityushev and S.V. Rogosin entitled Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions. Theory and Applications, Chapman and Hall / CRC Press, 1999. The work will be published in a forthcoming paper with V.V. Mityushev.
13/05/2005, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Luís Pessoa, Instituto Superior Técnico, U.T. Lisboa
Projecções poly e anti-poly Bergman e operadores integraissingulares
Introduzem-se espaços e projecções poly e anti-poly Bergman, conjuntamente com algumas propriedades elementares. No caso do disco unitário estabelecem-se igualdades entre as projecções referidas e operadores integrais singulares. O uso de mudança de variável, permitirá fixar determinada decomposição de $L^2$ do semi-plano superior na soma directa contável de espaços de funções n-analíticas e n-anti-analíticas. A decomposição é originalmente devida a N. Vasilevski e tentar-se-á apresentar uma prova alternativa. No semi-plano superior calculam-se contra-domínios de particulares OIS restritos aos espaços poly e anti-poly Bergman. A exposição basear-se-á em trabalho conjunto com Yu. I. Karlovich.
22/04/2005, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Natasha Samko, Universidade do Algarve, Faro
Generalized Hölder spaces with non-equilibrated characteristics andFredholmness of singular integral operators
We consider quasi-monotonic functions of the Zygmund-Bary-Stechkin class $\Phi$ with the main emphasis on properties of the index numbers of functions in this class (of Boyd type indices). A special attention is paid to functions whose lower and upper index numbers do not coincide with each other (non-equilibrated functions). It is proved that the bounds for functions in $\Phi$ known in terms of these indices, are exact in a certain sense. We also single out some special family of non-equilibrated functions in $\Phi$ which oscillate in a certain way between two power functions. Given two numbers $0 \lt \alpha , \beta \lt 1$ we explicitly construct examples of functions in $\Phi$ for which $\alpha$ and $\beta$ serve as the index numbers. The developed properties of functions in this class are applied to an investigation of the normal solvability of some singular integral operators in weighted spaces with prescribed modulus of continuity.
15/04/2005, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Frank-Olme Speck, Instituto Superior Técnico, U.T. Lisboa
Diffraction by rectangular wedges involving oblique derivatives
The main objective is the study of a class of boundary value
problems in weak formulation where two boundary conditions are
given on the half-lines bordering the first quadrant that contain
impedance data and oblique derivatives. The associated operators
are reduced by matricial coupling relations to certain boundary
pseudodifferential operators which can be analyzed in detail.
Results are: Fredholm criteria, explicit construction of
generalized inverses in Bessel potential spaces, eventually after
normalization, and regularity results. Particular interest is
devoted to the oblique derivative problem. The lecture is based
upon recent work with L.P. Castro and F.S. Teixeira.
08/04/2005, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Lars Diening, Universität Freiburg, Alemanha
Lebesgue and Sobolev spaces with variable exponent
Lebesgue and Sobolev spaces with variable exponents appear in problems of elasticity, fluid dynamics, calculus of variations, and differential equations with -growth conditions. Unfortunately, these spaces lack some important properties, e.g. translation and convolution are not continuous. Nevertheless, under certain regularity assumptions on the Hardy-Littlewood maximal operator in continuous on . This is the key step in the study of numerous results such as Sobolev embeddings, continuity of singular integrals, extension theorems, and the characterization of the trace spaces. In the talk we summarize the recent developments. The main attention will be on the continuity of the Hardy-Littlewood maximal operator and its applications. We will provide different criteria for the necessary regularity of the exponent .
01/04/2005, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Alexandre Almeida, Universidade de Aveiro
Characterization of function spaces of Riesz and Bessel potentials in case of variable exponent
We consider Riesz and Bessel potential spaces within the framework of the Lebesgue spaces with variable exponent. It is shown that the spaces of these potentials can be characterized in terms of convergence of hypersingular integrals, under natural regularity conditions on the exponent. We also describe a relation between these spaces and the variable Sobolev spaces.
25/02/2005, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Roland Duduchava, A. Razmadze Mathematical Institute, Academy of Sciences, Tbilisi, Georgia
Interface Cracks in Anisotropic Composite Materials
The linear model equations of elasticity give rise to oscillatory solutions in some vicinity of interface crack fronts. In this presentation we apply the Wiener-Hopf method which yields the asymptotic behaviour of the elastic fields and, in addition, criteria to prevent oscillatory solutions. The exponents of the asymptotic expansions are found as eigenvalues of the symbol of corresponding boundary pseudodifferential equations. The method works for three-dimensional anisotropic bodies and we demonstrate it for the example of two anisotropic bodies, one of which is bounded and the other one is its exterior complement. The common boundary is a smooth surface. On one part of this surface, called the interface, the bodies are bounded, while on the complementary part there occurs a crack. By applying the potential method, the problem is reduced to an equivalent system of boundary pseudo-differential equations (BPE) on the interface with the stress vector as unknown. The BPEs are defined via Poincaré-Steklov operators. We prove the unique solvability of these BPEs and write a full asymptotic expansion of the solution near the crack front. The resulting asymptotic expansion for the stress field has a singularity of order $-1/2$ at the boundary if written as a function of the distance to the boundary and the parameter along the boundary. In the general case such solution has logarithmic (so called "shadow") singularities and oscillate. Both of them deteriorate the solution and prevent the decoupling of three important modes. If these singularities eliminate, the asymptotic simplifies significantly and all three modes decouple automatically. In the joint work with M. Costabel and M. Dauge there were found conditions preventing the appearance of logarithmic terms in the asymptotic. In the joint work with A.M. Sändig and W.L. Wendland there was found a criterion preventing oscillation of the solutions. We investigate more detailed the interface crack problem for isotropic bodies. We present a simple criterion in terms of shear moduli and the Poisson ratios and give a rigorous justification for the three-dimensional case. The same result was already formulated by Williams and Ting for two-dimensional bodies without rigorous justification. Some explicit results are available for transversally isotropic bodies as well (O. Chkadua).
14/01/2005, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Yuri Karlovich, Universidad Autónoma del Estado de Morelos, México
Pseudodifferential Operators with Non-Regular Symbols
The talk is devoted to studying pseudodifferential operators of
zero order with non-regular symbols that satisfy a Hölder condition
with respect to the spatial variable and are uniformly bounded
continuous functions of bounded total variation on dyadic intervals
with respect to the dual variable. Applying the Littlewood-Paley
theory and previous results on pseudodifferential operators, we
obtain conditions for the boundedness and compactness of such
pseudodifferential operators on Lebesgue spaces over the real line.
We construct a symbol calculus and a Fredholm theory for
pseudodifferential operators with non-regular symbols that
additionally slowly oscillate at infinity with respect to both
variables.
10/12/2004, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Giuseppe Rosario Mingione, Universita degli Studi di Parma, Italy
Functionals with Non-Standard Growth, Lavrentev Phenomenon andRegularity
Functionals with non standard growth are coercive in a space which is strictly larger than the one where they are bounded and/or a priori finite. The regularity theory of minimizers differs from the usual one available for the standard functionals with polynomial growth, naturally defined in standard Sobolev spaces. I shall outline some regularity results and an approach to the regularity via Lavrentev phenomenon, suggested by a few constructions by Zhikov. Then, as a particular and enlightning case, I will describe the borderline situation of functionals with $p(x)$ growth, where a satisfying theory is available under certain optimal assumptions and I will suggest generalizations for functionals defined in more general spaces, today popular with the name of Orlicz-Musielak spaces.
01/10/2004, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Bernd Silbermann, Technische Universität Chemnitz, Germany
Approximation of Spectra and Numerical Linear Algebra
The talk is devoted to some features around Fredholm sequences, finite splitting property, and asymptotic spectral theory. As examples finite sections of the almost Mathieu operator and collocation matrices of singular integral operators are considered.
24/09/2004, 15:15 — 16:15 — Sala P3.10, Pavilhão de Matemática
Vladimir Rabinovich, Instituto Politecnico Nacional, Mexico
Mellin Pseudodifferential Operator Techniques in the Theory of Singular Integral Operators on Carleson Curves
We will present an approach to the calculation of the local and essential spectra of singular integral operators (SIOs) acting in -spaces with Muckenhoupt weights on a class of Carleson curves which is based on the Mellin pseudodifferential operators technique. From the 70's the local representation of SIOs acting on -spaces with power weights on piece-wise Lyapunov curves are well-known as to be Mellin convolutions. Such representation together with the local principle have allowed to construct a complete Fredholm theory for SIOs with piece-wise continuous coefficients acting on -spaces with power weights on piece-wise Lyapunov curves. As an extension of this approach we will show that SIOs with discontinuous coefficients acting on -spaces with Muckenhoupt weights on a class of Carleson curves have local representations as general Mellin pseudodifferential operators. By means of the limit operators method we obtain the complete description of the local spectra of SIOs which leads then to the description of the essential spectra of SIOs. Also, we are going to discuss some applications of this method to SIOs on Carleson curves acting in Hölder spaces with general weights.
24/09/2004, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Nikolai Karapetyants, Rostov State University, Russia
Integral Operators in the Localized Hölder Spaces with Variable
Exponent
As is known, the operator of fractional integration of order
establishes an isomorphism between the Hölder spaces
and
. We give a survey of some results on integral operators in weighted Hölder and generalized Hölder spaces, including the case of complex and imaginary order
. We also consider problems in the Hölder spaces of variable order. We pay special attention to the case where the order
at the fixed point
of
is higher than in other points. It is known that the singular integral operator does not preserve such a class. At the same time, for fractional integration the problem has a positive solution in the sense that the fractional integral at the point
has higher order
. Besides this, the Riesz potential on a ball and the singular integral operator on a closed smooth curve are also considered in the Hölder spaces of variable order.
17/09/2004, 15:15 — 16:15 — Sala P3.10, Pavilhão de Matemática
Stefan Samko, Universidade do Algarve, Faro
Boundary Value Problems for Analytic Functions and Singular
Operators in the Variable Exponent Spaces with General Weights
We consider the Riemann boundary value problem for analytic
functions in the class of analytic functions represented by the
Cauchy type integral with density in the generalized Lebesgue
spaces with variable exponent. We consider both the cases when the
coefficient G is piecewise continuous or it may be of a more
general nature, admitting its oscillation. The solvability
conditions are derived and in all the cases of solvability the
explicit formulas are given. Following the approach of I.Simonenko,
we make use of the results on the explicit solution of the boundary
value problem to obtain the weight results for Cauchy singular
integral operator in Lebesgue spaces with variable exponent, among
them some extension of the well known Helson-Szego theorem.
17/09/2004, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Vakhtang M. Kokilashvili, A. Razmadze Institute, Tbilisi, Georgia
Weighted Boundedness of Integral Operators in the Variable Exponent
Spaces of Homogeneous Type
The talk deals with the boundedness (compactness) criteria for
various classical integral operators (and their generalizations) in
weighted Banach spaces with non-standard growth. The study of these
spaces and behaviour of integral transforms there have been
stimulated by various problems of elasticity theory, fluid
mechanics, calculus of variations and differential equations with
non-standard growth. The talk focuses on weighted estimates in
variable Lebesgue and Lorentz spaces for integral transforms
defined both on the Euclidean space with Lebesgue measure and
general measure spaces with quasimetrics. We present boundedness
criteria for maximal functions, singular operator and potentials in
weighted variable spaces with weights of power-exponential type.
The solution of two weighted problems for fractional integrals with
variable fractional order is presented. The trace inequality for
the generalized potentials defined on spaces of homogeneous type is
also treated in the variable Lebesgue spaces. We also give a
Sobolev type theorem and its weighted version for fractional
integrals on Carleson curves (the recent result jointly with
S.Samko). An application to the Dirichlet problem for harmonic
functions in "bad" domains within the framework of the variable
Lebesgue spaces is given. The explicit formulas for the solution
are given together with the complete picture of the influence of
the geometry of the domain to the solvability of the problem.
23/07/2004, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Yuri Karlovich, Universidad Autónoma del Estado de Morelos, México
On the Invertibility of Functional Operators on Banach Spaces
The talk is a survey of some results on the invertibility of
functional operators with bijective and surjective shifts on Banach
spaces.