Functional Analysis and Applications Seminar 

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.

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.

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.

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.

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).

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 $C$-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.

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.

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.

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.

Lebesgue and Sobolev spaces with variable exponents appear in problems of elasticity, fluid dynamics, calculus of variations, and differential equations with $p(x)$-growth conditions. Unfortunately, these spaces lack some important properties, e.g. translation and convolution are not continuous. Nevertheless, under certain regularity assumptions on $p$ the Hardy-Littlewood maximal operator in continuous on $L_p(·)$. 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 $p$.

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.

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).

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.

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.

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.

We will present an approach to the calculation of the local and essential spectra of singular integral operators (SIOs) acting in $L^p$-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 $L^p$-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 $L^p$-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 $L^p$-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.

As is known, the operator of fractional integration of order $\alpha$ establishes an isomorphism between the Hölder spaces $H^{\mu }$ and $H^{\mu +\alpha }$. 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 $\alpha$. We also consider problems in the Hölder spaces of variable order. We pay special attention to the case where the order $\mu$ at the fixed point $x_0$ of $[0,1]$ 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 $x_0$ has higher order $\mu +\alpha$. 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.

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.

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.

The talk is a survey of some results on the invertibility of functional operators with bijective and surjective shifts on Banach spaces.