Abstracts

QUASILINEAR HYPERBOLIC SYSTEMS AND DISSIPATIVE MECHANISMS (Hsiao Ling)

The talk is given to introduce recent developments on the subject of quasilinear hyperbolic systems with dissipation,particularly the systems of hyperpolic conservation laws with either lower order dissipation such as frictional damping and relaxation, or higher order dissipation such as viscosity and heat diffution. The emphasis is, on the one hand to the damping mechanism or higher order dissipation, to understand the influence of dissipation on the qualitative behavior of solutions, such as the nonlinear diffusive phenomena, phase transitions, etc., and on the other hand to relaxation effects, to employ dissipation as a device for approximating solutions of hyperbolic systems by passing to the zero relaxation limit.

APROXIMAÇÃO POR FUNÇÕES DE SUPORTE COMPACTO EM DOMÍNIOS IRREGULARES (António Caetano)

Comparam-se os espaços gerais de Besov e de Triebel-Lizorkin sobre domínios com fronteira irregular com o completamento, naqueles espaços, do subconjunto das funções infinitamente continuamente diferenciáveis com suporte compacto nos mesmos domínios. Verifica-se que o conjunto dos parâmetros para os quais aqueles espaços coincidem está fortemente relacionado com a dimensão fractal da fronteira dos domínios.

A UNIQUENESS RESULT FOR A DEGENERATE PARABOLIC PROBLEM (Noureddine Igbida)

We prove uniqueness of a weak solution for a degenerate parabolic problem arising in various physical models. The technique is based in showing that every weak solution is also an entropy solution in the sense of Carrillo and then in adapting the known techniques that give the uniqueness for this type of solutions. We treat equations with lower order terms that have a particular structure and show with a counterexample that for general lower order terms the uniqueness does not hold.

IDENTIFICAÇÃO DE OBSTÁCULOS ATRAVÉS DA DIFRACÇÃO DE ONDAS ACÚSTICAS (Carlos Alves)

A detecção de obstáculos a partir do conhecimento do campo incidente e do campo difractado é um problema inverso com diversas aplicações ao nível da imagiologia. Do ponto de vista matemático trata-se de um problema não-linear e mal-posto. Resultados recentes permitiram estabelecer unicidade sob certas condições , mas permanecem questões em aberto relativas ao enfraquecimento dessas hipóteses e possibilidade de critérios simples de caracterização do obstáculo, por exemplo no que diz respeito à sua conexidade. Serão apresentados critérios para a caracterização de fissuras planas e alguns resultados numéricos.

POLINÓMIOS q-CLÁSSICOS: UM ESTUDO UNIFICADO (Renato Alvarez-Nodarse)

O objectivo do seminário é estabelecer várias caracterizações dos chamados polinómios ortogonais q-clássicos (segundo Hahn), identificando as distintas famílias q-clássicas com certas famílias que figuram nos famosos esquemas de Askey e de Nikiforov et al.; em particular, apresentam-se duas novas famílias de q-polinómios que não se encontram na chamada q-tabela de Askey.

RESULTADOS DE EXTRAPOLAÇÃO EM ESPAÇOS DE FUNÇÕES (Júlio Neves)

Apresentaremos resultados de extrapolação em espaços de Besov-Hölder-Lipschitz e em espaços de Leopold. Veremos como estes resultados nos permitem provar certas imersões, motivadas pelo resultado de Brézis-Wainger (1980) acerca da "quase" continuidade à Lipschitz dos elementos do espaço de Sobolev fraccionário $ H_p^{1+n/p}(\mathbb{R}^n)$, $ 1< p < +\infty$.

FREE BOUNDARY PROBLEMS FOR PARABOLIC EQUATIONS (Galina Bizhanova)

Many-dimensional free boundary problems for second order parabolic equations are considered. Two questions are studied: transformation of the unknown boundaries into prescribed ones and existence and uniqueness of the solutions to the problems in Hölder spaces and coercive estimates for them.

RECENT RESULTS ON REGULARITY FOR THE NAVIER-STOKES EQUATIONS (Namkwon Kim)

Local in time smooth solutions for the Navier-Stokes equations have not been known to persist for a long time. There have been several conditions which assure the persistence, for example, Serrin's conditions, etc... In this talk, we deal with some of the variants of Serrin's conditions and its axisymmetric version. Furthermore, we deal with the existence in a weighted space.

ASYMPTOTIC CONDITIONS AT INFINITY FOR THE STOKES PROBLEM IN A SYSTEM OF PIPES (Konstantin Pileckas)

In the theoretical hydromechanics special attention is given to the investigation of problems in unbounded domains, in particular, in domains with semi-infinite outlets to infinity. The boundary value problems in domains with cylindrical outlets (pipes) belong to this class. This is self-understood, there are no infinite volumes of liquids in the nature and, hence, these problems should be considered only as certain model ones. At the same time, exactly such problems are used by engineers while solving the practical problems. Performing the computer simulation of the flow in long thin pipes, engineers, at first, replace them by semi-infinite cylinders and basing on the intuition, experiments and the engineering know how, suggest some sensible conditions in each of semi-cylinders at infinity. After this, the semi-infinite cylinders must be cut again in order to allow the numerical simulation of the flow. In this connection the proper formulation of asymptotic boundary conditions at infinity gain the decisive significance for the adequate description of real situations. In the talk I discuss mathematical tools which allow to realise the correct setting of asymptotic conditions at infinity in a system of pipes and show how to apply the general theory for concrete physically sensible problems, e.g. for open drains, for pipes occupied by pumps or check valves and for connected pipes with check valve inside. The results were obtained jointly with S. A. Nazarov.

UM PROBLEMA NÃO LINEAR DE DUAS MEMBRANAS (Lisa Santos)

Demonstra-se existência e regularidade de solução para um problema de duas membranas com diferentes coeficientes de elasticidade. O resultado para o problema linearizado está estabelecido na literatura. No caso não linear, para garantir existência de solução, é necessrio impôr uma condição de compatibilidade relacionando a curvatura média da fronteira do domínio e os dados. Pretende-se ainda estudar o comportamento assimptótico da solução, pelo menos no problema linearizado, quando um dos coeficientes de elasticidade é tornado arbitrariamente grande.

EXTENSÃO DE UM RESULTADO DE GOMES TEIXEIRA (Amílcar Branquinho)

Vamos apresentar um resultado de Gomes Teixeira que permite obter de uma forma unificada a função geradora das famílias de polinómios ortogonais clássicas. Da mesma forma veremos como estender este resultado obtendo funções geradoras de algumas famílias de polinómios ortogonais de várias variáveis.

APROXIMAÇÃO RACIONAL E POLINÓMIOS ORTOGONAIS (Guillermo López Lagomasino)

Existe uma relação estreita entre propriedades assimptóticas de sucessões de polinómios ortogonais e a convergência de aproximantes racionais a funções meromorfas. Veremos como esta relação é mutuamente proveitosa às duas teorias.

ESPAÇOS DE FUNÇÕES DE PARÂMETRO DE DERIVAÇÃO GENERALIZADO (Susana Domingues de Moura)

Os espaços de funções que abordaremos generalizam os espaços de Besov e Triebel-Lizorkin. Apresentaremos algumas propriedades dos referidos espaços, nomeadamente caracterizações via decomposições atómicas e subatómicas.

SOLUTIONS OF FULLY NONLINEAR EQUATIONS WITH PATCHES OF ZERO GRADIENT: EXISTENCE, REGULARITY AND CONVEXITY OF LEVEL CURVES (Jorge Salazar)

In this talk (a joint work with Luis Caffarelli), we are going to report on the construction of viscosity solutions (in the Crandall-Lions sense) of fully nonlinear equations of the form $F(D^2 u,x) = g(x,u)$ on $\{|\nabla u| \ne 0$. By an appropiate use of the Alexandroff-Backelmann technique, we prove existence, regularity and, in two dimensions, for $F=\Delta$, $g = cu$ ($c>0$) and constant boundary conditions on a convex domain, we prove that there is only one convex patch.

THE POLYNOMIAL PROPERTY OF ELLIPTIC PROBLEMS AND AN ALGEBRAIC DESCRIPTION OF THEIR ATTRIBUTES (Serguei Nazarov)

The polynomial property of a self-adjoint system of differential operators means that the corresponding sesquilinear form degenerates on a finite-dimensional subspace of polynomials only. Almost all stationary problems in mathematical physics provide an example of such a system. In the lecture it will be shown that this simple property implies that the boundary value problem is elliptic while the kernel and co-kernel of the corresponding operator in Sobolev and Hlder spaces can be described in terms of the polynomial subspace for different kind of domains such as bounded domains, with smooth or piecewise smooth boundaries, and unbounded ones, with conical, cylindrical and quasi-cylindrical outlets to infinity.

WEAK SOLUTIONS FOR A CLASS OF NON-NEWTONIAN FLUIDS WITH ENERGY TRANSFER (Luísa Consiglieri)

We shall consider a nonlinear thermoconvection problem consisting of a coupled system of nonlinear partial differential equations due to temperature dependent coefficients. We prove that weak solutions exist in appropriate Sobolev spaces under mild hypothesis on the regularity of the data. This result is established through a fixed point theorem for multivalued functions, which requires a detailed analysis of the continuous dependence of auxiliary problems, including the associated Lagrange multipliers of the generalized Navier-Stokes system.

NONLINEAR ELLIPTIC EQUATIONS ON NON-SMOOTH DOMAINS UNDER MIXED BOUNDARY VALUR CONDITIONS (Carsten Ebmeyer)

Nonlinear elliptic problems are considered under mixed Dirichlet-Neumann boundary conditions. It is assumed that the domain $\Omega$ has a piecewise smooth boundary (e.g. the domain is a polyhedron). Using a difference quotient technique, we get regularity results for weak solutions in fractional order Sobolev spaces. These results generalize the known results for linear problems.

UNIFORM AND HÖLDER APPROXIMATION (Miguel A. Jimenez)

Many results that appear in dealing with best Chebyshev approximation by algebraic or trigonometric polynomials have been extended to Hölder polynomial approximation. However it is clear that uniform convergence is a necessary but not a sufficient condition for Hlder approximation. We present different requirements that assert Hölder convergence from the uniform one. Applications of these new concepts are presented.

A SENSITIVITY AND ADJOINT CALCULUS FOR THE OPTIMAL CONTROL OF FLOWS WITH SHOCKS (Stefan Ulbrich)

Optimal control or design problems for hyperbolic conservation laws arise for example in the design of ducts and airfoils or in the identification of friction parameters in shallow water flows after neglecting viscous effects. A key point for an efficient solution of the resulting optimization problems is the question of (Frechet-)differentiability of the objective functional. It is well known that entropy solution of conservation laws develop in general discontinuities (shocks) after a finite time. Since a variation of the control leads to a variation of the shock locations, the control-to-state mapping is a priori at best differentiable for a very weak topology on the state space. This causes difficulties for the analysis as well as for an efficient numerical solution of the considered control problems. We present a sensitivity calculus in L^1 for scalar conservation laws with source terms that is based on a first order approximation by nonlinear shift-variations. This sensitivity calculus implies the Frechet-differentiability of objective functionals of tracking type. Furthermore, we derive an adjoint-based gradient representation for tracking type functionals. A key ingredient is an adjoint formula for shock sensitivities and a careful analysis of the adjoint equation. The adjoint state has to be defined as reversible solution of the in general not uniquely solvable adjoint equation. We discuss important stability properties of reversible solutions which can in turn be used to define measure solutions of the sensitivity equations by a duality relation. Moreover, we discuss which discretization schemes of the state equation lead to convergent numerical adjoints and sensitivities. For this schemes, the discrete gradients converge to the gradient of the continuous problem. The discrete gradient can for example be computed by using automatic differentiation tools where the forward mode yields a sensitivity-based, the backward mode an adjoint-based gradient computation. We show by numerical results for the optimal control of flows with shocks in 1- and 2-D, in particular the optimal design of an airfoil, that the consideration of our analytical results leads to efficient optimization methods.

ALGEBRAIC AND ORDER COMPLETION METHODS IN SOLVING NONLINEAR PDEs, PARAMETRIC GLOBAL LIE GROUP ACTIONS AND SOLUTIONS TO HILBERT's FIFTH PROBLEM (Elemér E. Rosinger)

The talk is a survey of the work of the speaker and a few collaborators published in a number of books and papers. The presentation is focused on the basic ideas and results, without going into technical detail, thus it is accessible to a general mathematical audience. 1. The algebraic method constructs a nonlinear theory which gives infinitely many classes of differential algebras of generalized functions that overcome the clelebrated 1954 impossibility result of Laurent Schwartz on the multiplication of distributions. The construction of these algebras - each of which contains the Schwartz distributions - was started in the 1960s and has been much developed since then in the work of several dozen authors. Among the results obtained are global solutions of large classes of linear and nonlinear PDEs. One of the particular cases of such algebras was introduced in the early 1980s by Colombeau. Last year the AMS included the general algebraic nonlinear theory in its Subject Classification 2000 under the heading 46F30, see details at www.ams.org/index/msc/46Fxx.html. Recent results give generalized solutions with dense singularities in which the cardinal of singular points can be larger than that of nonsingular ones. These results have applications in space-time foam structures in General Relativity, as well as in Quantum Gravity. 2. The order completion method, started in 1990, proves to be far more powerful, as it can solve globally all nonlinear PDEs given by continuous expressions. The solutions obtained can be assimilated with usual measurable functions. This overcomes for the first time the 1957 impossibility result of Hans Lewy. 3. Since the work of Chevalley half a century ago, there was a significant shift of interest from local to global Lie groups and their actions. However, with respect to Lie group symmetries of solutions of nonlinear PDEs much of the existing theory has until the recent work of the speaker been limited to local actions and symmetries. The only exception was that of a rather small class of nonlinear PDEs whose Lie group symmetries are "projectable" or "fibre preserving". However, such equations do not contain even such basic ones as the one dimensional nonlinear shock wave equation. In the speaker's 1998 Kluwer book arbitrary global Lie group actions and symmetries are constructed and they can act as well on generalized functions and solutions. In this way Hilbert's fifth problem obtains for the first time its solution, when it is considered in its origianl full generality. The mentioned global Lie group actions can be constructed owing to an extension of the classical parametric method which has in fact for long been used in the case of non-projectable or non-fibre preserving Lie groups. An unexpected benefit of this global parametric method is that one can start a general theory of genuine Lie semigroup actions, that is, of semigroups of actions which are not invertible. An important example is given by the smooth mappings of a manifold into itself, most of which mappings are not invertible.

PADÉ APPROXIMANTS AND ORTHOGONAL POLYNOMIALS (Bernardo de la Calle Ysern)

Padé approximants are rational functions which may be seen as a generalization of Taylor polynomials. In this lecture, basic facts and main classical theorems on convergence theory of Padé approximants to analytic functions are briefly reviewed. We show the important role played by Logarithmic Potential Theory in the development of this topic. We also consider the relationship between Padé approximants and asymptotic properties of polynomials orthogonal with respect to fixed or varying measures. Finally, some original results about Hermite-Padé approximants to a Nikishin system of functions are presented. Such approximants arise in the study of the transcendence of the number $e$ and are related, as Nikishin systems themselves, to number theory.

q-ANÁLOGOS DE FUNÇÕES TRIGONOMÉTRICAS NUMA REDE q-LINEAR. PROPRIEDADES DOS ZEROS. CONSTRUÇÃO DE q-SÉRIES DE FOURIER (José Luís Cardoso)

Breve construção de q-análogos de funções trigonométricas como soluções de equações de diferenças. Definição de redes q-lineares e de redes q-quadráticas. Escrita daquelas soluções na forma de séries hipergeométricas básicas. Definição de q-Integral e $\delta$-Diferenciação. q-Ortogonalidade. Propriedades dos zeros dos q-análogos acima referidos, construção e estudo de q-séries de Fourier a eles associados.

THE POROUS MEDIUM EQUATION: A MODEL OF A FREE BOUNDARY WITH A FINITE SPEED OF PROPAGATION (Carsten Ebmeyer)

The porous medium equation is a degenerate parabolic differential equation. It is used in modelling free boundaries with a finite speed of propagation. At the free boundary the solution may be singular. The aim of this talk is to discuss the regularity of weak solutions and its behaviour near the free boundary. Further, an application of these results to the finite element analysis is given.

EXISTENCE OF SOLUTIONS OF A NON LINEAR CROSS DIFFUSION SYSTEM ARISING IN POPULATION DYNAMICS (Gonzalo Galiano)

We present results on existence of solutions for a strongly coupled cross-diffusion model for two competing species based on a semidiscretization in time. The variables are the population densities of the species. Existence of strictly positive weak solutions to the semidiscrete problem is proved. Moreover, it is shown that the semidiscrete solutions converge to a solution of the continuous problem. The proofs are based on a symmetrization of the problem via an exponential transformation of variables and the use of an entropy functional.

IMERSÕES LIMITE DOS ESPAÇOS DE SOBOLEV (FRACCIONÁRIOS) MODELIZADOS EM ESPAÇOS DE LORENTZ-KARAMATA (Júlio Neves)

Vamos considerar espaços de Sobolev (fraccionários), modelizados em espaços de Lorentz-Karamata e apresentaremos imersões limite destes espaços em espaços de Lorentz-Karamata e em espaços de Orlicz. Estes resultados estão relacionados com as imersões de Peetre-Strichartz-Trudinger e Hansson-Brézis-Wainger (caso limite) e estendem e melhoram as imersões de Edmunds, Gurka e Opic.

ANÁLISE DE SENSIBILIDADES DE PLACA SUJEITA A OBSTÁCULO (Carlos Leal)

We analyse the sensitivity of the solution of a nonlinear obstacle plate problem, with respect to small perturbations of the middle plane of the plate. This analysis, which generalizes the results for the linear case, is done by application of an abstract variational result, where the sensitivity of parameterized variational inequalities in Banach spaces, without uniqueness of solution, is quantified in terms of a generalized derivative, that is the proto-derivative. We prove that the hypotheses required by this abstract sensitivity result are verified for the nonlinear obstacle plate problem. Namely, the constraint set defined by the obstacle is polyhedric and the mapping involved in the definition of the plate problem, considered as a function of the middle plane of the plate, is semi-differentiable. The verification of these two conditions enable to conclude that the sensitivity is characterized by the proto-derivative of the solution mapping associated with the nonlinear obstacle plate problem, in terms of the solution of a variational inequality.

DINÂMICA CAÓTICA: O PONTO DE VISTA PROBABILÍSTICO (José Ferreira Alves)

Em Sistemas Dinâmicos procura-se entender a evolução de sistemas cujo estado se altera ao longo do tempo, tentando prever a sua evolução. Tem-se constatado que um grande número de sistemas se comportam de forma caótica: os estados futuros do sistema são muito afectados por pequenas variações do estado inicial. Isto significa que de um ponto de vista determinístico a evoluçao do sistema é imprevisível. Contudo, resultados recentes mostram que muitos sistemas caóticos admitem uma descrição bastante satisfatória em termos probabilísticos.

ON FUNCTION SPACES OF GENERALIZED SMOOTHNESS (Hans-Gerd Leopold)

Function spaces of generalized smoothness have been introduced and considered from different points of view - characterisation by approximation, interpolation, operators, differences. This have been done by a variety of authors particularly since the middle of the Seventies up to the end of the Eighties. The last years have again witnessed great interest in this question, especially in connection with embeddings and limiting embeddings. We consider here the Fourier-analytical approach for such spaces. This gives, in particular, most of the spaces defined by the methods described above. Furthermore, we derive some consequences for embedding results.

DIFRACÇÃO DE ONDAS SOB UMA PERSPECTIVA DA TEORIA DE OPERADORES (Luís Filipe Castro)

Uma classe de problemas da teoria da difracção, possuindo ligações aos problemas clássicos de Sommerfeld, é analisada sob a perspectiva da teoria de operadores. Tal significa que após uma formulação inicial do problema por via da equação de Helmholtz e de condições de fronteira, este é traduzido por uma única equação caracterizada por um operador. Posteriormente, este operador relacionado (após extensão) com operadores de Wiener-Hopf. A construção de relações explícitas entre os operadores descobertos facilita a factorização destes o que, em última análise, leva àcaracterização de condições para que os problemas considerados estejam (ou não) bem postos. Todo o estudo é apresentado num quadro de espaços de potenciais de Bessel (ou particularizações destes).

ON SOME FRACTIONAL EVOLUTION EQUATIONS (Stig-Olof Londen)

In this talk, we review some recent results on fractional evolution equations of type $$ D_{t}^{\alpha} u+A u=f, \quad t\geq 0, $$ where $\alpha \in (0,2)$, $\alpha \neq 1$. First, as a typical example, we consider the fractional Burgers equation. Next, we formulate a general result with $A$ m-accretive. Some comments on maximal regularity follow; finally we present an unsolved problem concerning fractional nonlinear hyperbolic equations.

VALORES PRÓPRIOS MINIMAIS PARA OPERADORES DE SCRÖDINGER EM VARIEDADES COMPACTAS (Pedro Freitas)

Considera-se o problema de minimizar os valores próprios do operador de Scrödinger $H=-\Delta+\alpha \ka^{2}$ ($\alpha>0$) numa variedade compacta quando a média de $\ka$ é fixa. Em dimensão um, prova-se que o potencial constante deixa de ser minimizante a partir de um valor crítico do parâmetro $\alpha$, e os resultados obtidos permitem obter um minorante para o primeiro valor próprio do operador de Hill. Para dimensões superiores, prova-se que restrições integrais deste tipo não limitam inferiormente os valores possíveis para os valores próprios de $H$.

GAMMA, DIGAMMA, POLYGAMMA FUNCTIONS - AND AN INEQUALITY OF J.S. MARTINS (Horst Alzer)

We give a survey talk on some recently published mean value inequalities for the classical gamma function, the digamma function and the polygamma functions. Further, we provide inequalities for the arithmetic, geometric, and harmonic means of the first n positive integers. These mean values can be expressed in terms of the gamma and digamma functions. Finally, we present an open problem concerning a converse of a mean value inequality of J.S. Martins, published in 1988.

A NEW CHARACTERIZATION OF CLASSICAL FORMS (Pascal Maroni)

We show that an orthogonal polynomial sequence, whose m-th associated sequence of k-th derivative sequence is orthogonal, is necessarily a classical one. When m=0, it is the Hahn's theorem.

INTEGRAL TRANSFORMATIONS OF THE KONTOROVICH-LEBEDEV TYPE (Semyon Yakubovich)

A survey talk about recently published results in the Lp-theory of integral operators associated with special functions of the hypergeometric type as kernels is presented. The theory of the Mellin and the Kontorovich-Lebedev transformations is applied in order to prove related results for other integral operators of this type. The integration process in reciprocal transformations is realized with respect to the index (parameter) and an argument of the kernel function. Different examples are demonstrated.

INTEGRAL TRANSFORMS AND q-POLYNOMIALS (Renato Alvarez-Nodarse)

It is well known the importance of the integral transforms and special functions in several areas of Mathematical and Physical Sciences. In fact, a deep knowledge of the Special Function Theory allows us to solve several problems in quantum and classic Mechanics, non-linear equations, etc. In the last years a special emphasis was given to the so-called q-special functions, and q-hypergeometric (basic) polynomials. In this talk we will show some results concerning the Mellin transforms of certain families of q-polynomials. In fact we will show how to obtain the Mellin transform of the q-polynomials of the so-called q-Hahn tableau. Similar results for the Fourier transform are also known.

ON THE SAPONJYAN PARADOX IN THE THEORY OF THIN PLATES (Serguei Nazarov)

Taking a problem on bending of simply-supported polygonal Kirchoff plate as an example, it is shown that a reformulation of the boundary-value problem in terms of potentials may lead to a complete change in the solution. An approach based on self-adjoint extensions of operators can improve the situation.

Envelopes in function spaces (Dorothee D. Haroske)

We present our recently developed concept of envelopes in function spaces - a relatively simple tool for the study of rather complicated spaces, say, of Besov type $ B^s_{p,q}$, in 'limiting' situations. It is, for instance, well-known that $ B^{n/p}_{p,q} \hookrightarrow L_\infty$ if, and only if, $ 0<p<\infty$, $ 0<q\leq 1$ - but what can be said about the growth of functions $ f\in B^{n/p}_{p,q}$ otherwise, i.e. when $ B^{n/p}_{p,q}$ contains essentially unbounded functions ? Edmunds and Triebel proved that one can characterise such spaces by sharp inequalities involving the non-increasing rearrangement $ f^\ast$ of a function $ f$. This led us to the introduction of the growth envelope function of a function space $ X$, $ {\mathcal E}_{G}^X(t) := \sup\limits_{\Vert f\vert X\Vert \leq 1} \, f^\ast(t)$, $ 0<t<1$. It turns out that in rearrangement-invariant spaces there is a connection between $ {\mathcal E}_{G}^X $ and the fundamental function $ \varphi_X$; we derive further properties and give some examples. The pair $ {\mathbb{E}}_{G}( X ) =\left( {\mathcal E}_{G}^X(t), u_X\right)$ is called growth envelope of $ X$, where $ u_X$, $ 0<u_X\leq\infty$, is the smallest number satisfying

$\displaystyle \left(\int\limits_0^\varepsilon \left[\frac{f^\ast(t)}{{\mathcal......}\right]^v\mu_{G}({\mathrm d} t)\right)^{1/v} \leq \; c\; \Vert f\vert X\Vert$

for some $ c>0$ and all $ f\in X$, and $ \mu_{G}$ is the Borel measure associated with $ -\log {\mathcal E}_{G}^X$. One verifies for the Lorentz spaces $ {\mathbb{E}}_{G}\left(L_{pq}\right) = (t^{-1/p}, q)$, but we also obtain characterisations for spaces of type $ B^s_{p,q}$, where $ n (\frac1p-1)_+ < s \leq \frac{n}{p}$. Instead of investigating the growth of functions one can also focus on their smoothness, i.e. when $ X\hookrightarrow C$ it makes sense to replace $ f^\ast(t)$ by $ \frac{\omega(f,t)}{t}$, where $ \omega(f,t)$ is the modulus of continuity. Now the continuity envelope function $ {\mathcal E}_{C}^X$ and the continuity envelope $ {\mathbb{E}}_{C}$ are introduced completely parallel to $ {\mathcal E}_{G}^X $ and $ {\mathbb{E}}_{G}$, respectively, and similar questions are studied. Naturally these sharp assertions imply a lot of interesting (new) inequalities; in our opinion, however, the essential advantage of this new approach rather results from its simplicity when establishing (so far final) answers to relatively difficult questions.

HARMONIC BERGMAN SPACES IN THE UNIT BALL (Guangbin Ren)

Some basic techniques are established for the theory of harmonic Bergman space. For example, the hypergeometric functions are expressed as the integral of generating functions of Gegenbauer polynomials, so that we can obtain the precise form of the Forelli-Rudin type estimation for the power of harmonic Cauchy kernels. The Mobius transformations are shown to play a central role in the function theory in the real unit ball. The weighted Berman projections are considered in the limit case. As an application, the duality of weighted harmonic Bergman space with small exponents in the real unit ball is shown to be harmonic Bloch space.

INTERPOLAÇÃO DE IDEAIS DE OPERADORES (Pedro Matos )

- Definições e resultados básicos de Teoria da Interpolação
-  Resultados sobre a interpolação de operadores compactos
- Definição de medidas para ideais de operadores
- Resultados sobre interpolação de ideais de operadores

ESPAÇOS DE POTENCIAIS DE BESSEL MODELIZADOS EM ESPAÇOS DE LORENTZ-KARAMATA E IMERSÕES (CASO SUPER-LIMITE) (Júlio S. Neves)

Vamos considerar espaços do tipo de Sobolev $ H^sX$ (fraccionários), $ s>0$, modelizados em espaços de Lorentz-Karamata $ X=Lp,q;b(\mathbb{R}^n)$, com $ p\in]1,+\infty[$, $ q\in[1,+\infty[$ e $ b$ uma função positiva de variação lenta, e apresentaremos imersões destes espaços no caso super-limite, i.e., quando $ s>n/p$. Os espaços de chegada podem ser espaços de funções contínuas e limitadas, espaços de Lipschitz ou espaços quasi-Lipschitz, consoante o valor dos parâmetros. Estes resultados refinam um resultado de Triebel (2001) (no contexto dos espaços de potenciais de Bessel) já por sua vez uma melhoria do importante resultado de Brézis-Wainger (1980) acerca da "quase" continuidade à Lipschitz dos elementos do espaço de Sobolev fraccionário $ H_p^{1+n/p}(\mathbb{R}^n)$, $ 1< p < +\infty$. Os resultados apresentados melhoram e estendem as imersões de Edmunds, Gurka e Opic (1997) (no contexto dos espaços de Sobolev logarítmicos).

NÚMEROS-$s$ DE IMERSÕES DE ESPAÇOS LOGARÍTMICOS EM ESPAÇOS DE ZYGMUND GENERALIZADOS (António Bento)

Serão apresentadas estimativas de números-$s$ das imersões compactas

\begin{displaymath}W^k L_{p, \alpha_1, \ldots, \alpha_m}(\mathbb{R}^n)\hookrightarrow L_{q, \alpha_1, \ldots, \alpha_m}(Q),\end{displaymath}



onde $Q$ é um paralelepípedo em $\mathbb{R}^n$, $L_{p, \alpha_1, \ldots, \alpha_m}(Q)$ são os espaços de Zygmund generalizados, $W^k L_{p, \alpha_1,\ldots, \alpha_m}(\mathbb{R}^n)$ são os espaços logarítmicos de Sobolev, i.e. espaços de Sobolev modelados em $L_{p, \alpha_1, \ldots, \alpha_m}(\mathbb{R}^n)$, $p, q \in ]1, \infty[$, $\alpha_1,\ldots, \alpha_m \in \mathbb{R}$ e $k/n > \max \{0, 1/p -1/q\}$.

SUBSPACE OPTIMIZATION MODELS AND THE GRAPH PARTITIONING PROBLEM (Suely Oliveira)

Parallel data distribution can be modeled as a Graph Partitioning Problem where we wish to divide a graph into subgraphs with roughly equal number of nodes with a mininum number of edges crossing between the subgraphs. This is a well-known NP-complete problem so heuristics need to be used.
Spectral graph partitioning is a relaxation of the discrete optimization problem. The discrete optimization problem can also be solved approximately by using a Semidefinite Programming (SDP) relaxation. In recursive Graph Partitioning an edge crossing between two sets can not affect the later partitioning of either set.
This problem can be addressed by constructing an optimization problem which includes preferences associated with each vertex. The straightforward relaxation of this model corresponds to an extended eigenvalue problem.
Recently we developed a SDP relaxation for this model and have designed a subspace algorithm for the SDP relaxation.

CONVOLUTION COMPLEMENTARITY PROBLEMS (David Stewart)

Convolution complementarity problems (CCP's) combine Linear Complementarity Problems with convolutions and differential equations. CCP's have the form

$\displaystyle 0 \leq u(t) \perp (k*u)(t) + q(t) \geq 0$   for all $\displaystyle t$

They are closely related to the Linear Complementarity Systems of Heemels, Schumacher, van der Schaft, & Weiland. Existence of solutions to CCP's is shown via a numerical discretization provided $ k(0^+)$ is a P-matrix and $ k(t)$ satisfies some other mild regularity conditions. Uniqueness can also be shown under some further mild conditions. These results can be applied to a simplified model of impact for the wave equation where the Signorini contact conditions are replaced by boundary integrated contact conditions.
ON THE INTERPOLATION CONSTANT FOR (SUB)LINEAR OPERATORS IN ORLICZ SPACES (Alexei Karlovich - joint work with Lech Maligranda)

In this talk we deal with the interpolation of bounded sublinear operators from Lebesgue spaces $ L^p$ and $ L^q$ into an Orlicz space $ L^\varphi$, where $ 1\le p<q<\infty$ and $ \varphi^{-1}(t)=t^{1/p}\rho(t^{1/q-1/p})$ for some concave function $ \rho:{\bf R}_+\to{\bf R}_+$. Our main aim is to get estimates for the interpolation constant $ C$.

Theorem

If a sublinear operator $ T$ is bounded in Lebesgue spaces $ L^p$ and $ L^q$, then it is bounded in the Orlicz space $ L^\varphi$ (with both, the Luxemburg and the Orlicz norm) and

$\displaystyle \Vert T\Vert _{L^\varphi\to L^\varphi} \le C\max \Big\{ \Vert T\Vert _{L^p\to L^p}, \Vert T\Vert _{L^q\to L^q} \Big\}, $

with the interpolation constant $ C$ satisfying
(a)
$ C\le 2^{1-1/p}<2$ for $ 1\le p<q= \infty$;
(b)
$ C\le (2\gamma_{p,q})^{1/p} \le 2^{(2-1/q)/p}<4$ for $ 1\le p<q<\infty$, where $ \gamma_{p,q}$ is the constant introduced by G.Sparr.

ENVELOPE FUNCTIONS AND APPROXIMATION NUMBERS: TWO EXAMPLES (Dorothee D. Haroske)

We present some new results on envelope functions in function spaces of type B(s,Y)p,q and explain possible applications to approximation numbers. In particular, we obtain for 0 < s < 1, 0 < q £ ¥, 0 < p £ oo , and s = n/p+s,

ECB(s,Y)p,q(t)   ~   t-(1-s) Y(t)-1 ,

where the continuity envelope function of some function space X is defined by

ECX(t)  ~  


sup
||f|X|| £ 1 

   w(f,t)

t

   ,    t >0,

w(f,t) being the usual modulus of continuity. This result is used to estimate the approximation numbers of the compact embedding id : B(s,Y)p,q(U) --> C(U), 2 £ p £ oo , 0 < q £ oo , 0 < s-n/p < 1, Y  admissible, U the unit ball in Rn, asymptotically by

ak(id : B(s,Y)p,q(U) --> C(U)) ~ k -s/n+1/p  Y(k -1/n)-1  ,     k (- N .

Including an additional lift argument we can similarly determine

ak( id : Lp(logL)a(U) --> B-1 oo , oo (U)) ~ k -(1/n-1/p) (1 + logk)-a ,     k (- N ,

where n < p < oo , a > 0.

pdf version

Polinomios Ortogonales Polares (Héctor E. Pijeira Cabrera)

Sea \mu una medida finita, positiva y de Borel donde  supp \mu = \Delta \subset  [-1,1] contiene un número infinito de puntos. Diremos que {P_n } es  la sucesión de polinomios ortogonales polares mónicos con respecto a la medida \mu y el polo \zeta \in C si:

De la familia de polinomios { P_n } se estudian:

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