Topology and Nonlinear Problems

Transcription

1 Topology and Nonlinear Problems Conference in honour of Kazimierz Gęba August, Warszawa A. Granas, B. Bojarski, H. Żołądek, M. Starostka ABSTRACTS S.Bauer - On refined Seiberg - Witten invariants Abstract: TBA V. Benci - Generalized functions and generalized solutions beyond distributions Abstract: The theory of distributions provides generalized solutions for problems which do not have a classical solution. However, there are problems which do not have solutions, not even in the space of distributions. As model problem you may think of u = u p 1, u > 0, p 2N N 2 with Dirichlet boundary conditions in a bounded open star-shaped set. Having this problem in mind, we construct a new class of functions called ultrafunctions in which the above problem has a (generalized) solution. In this construction, we apply the general ideas of Non Archimedean Mathematics and some techniques of Non Standard Analysis. Ultrafunctions are a particular class of functions defined on a non-archimedean field R R. More exactly, to any continuous function f : R N R, we associate in a canonical way an ultrafunction f : (R ) N R which extends f; but the ultrafunctions are much more than the functions and among them we can find solutions of functional equations which do not have any solutions among the real functions or the distributions. We will present some problems to which the theory of ultrafunctions can be applied. B.Bojarski - Sobolev spaces and averaging Abstract: TBA 1

2 K. Gęba - Equivariant gradient maps and the Conley index Abstract: Let E be a real infinite-dimensional separable Hilbert space. Consider the space LS(E) of all continuous maps f : E E, f(x) = x F (x) such that (a) F is completely continuous (i.e. F (X) is compact subset of E for any bounded X E ) and (b) f 1 (0) is bounded (and hence compact). It is well known that the homotopy group π k (LS(E)) is isomorphic to the k-th stable homotopy group of S 0. Given an orthogonal representation of a compact Lie group G on E we will consider some generalizations of the above theorem to the case of G-maps. A. Granas - Topology and nonlinear analysis Abstract: TBA R. Guenther and C.C. Buchanan - Linear barotropic equations. Properties of the solutions Abstract:All flows, including water, are basically compressible. To get the Navier Stokes equations, one has to make fantastically strong assumptions, namely that both the density and the viscosity are constant. But in dealing with deep-water flows, the density depends on the pressure and often on the temperature. Thus, in deep oceans, the Navier Stokes equations do not hold, even after taking salinity into account. Given initial values for pressure and velocity, our goal is to solve the Cauchy problem for p t and v t in a barotropic fluid. This leads us to the class of functions where one can expect existence and uniqueness of the Cauchy problem for our system. We begin with a derivation of the barotropic equations, noting that there is no such thing as an incompressible fluid and in fact that assumption flies in the face of all thermodynamics. The compressibility, c > 0, which we take to be a constant, is defined in terms of density. We derive equations for the expressions of p and p t in Fourier transform form. We obtain the divergence of v and solve a heat equation for each component, which returns both an expression for v, and the fundamental solution for the problem. 2

3 A. Ioffe - Metric regularity in variational analysis Abstract: The talk will be basically focused on discussions concerning (metric) regularity of set-valued mappings between metric spaces, metric fixed point theory, mainly in connection with the so called two maps paradigm when we have two set-valued mappings acting in opposite directions, and von Neumann s alternate projection method with emphasis on the case of two non-convex sets and the role of transversality. Some other regularity related problems may also be addressed if time permits. M. Frigon - On a notion of category depending on a functional Abstract: A notion of category depending on a functional is introduced. This notion permits to obtain a better lower bound on the number of critical points of a functional than the classical Lusternik-Schnirelman category. Our results are presented for continuous functionals defined on locally contractible metric spaces. We study the relation between the notion of linking and the relative category depending on a functional. An application to Hamiltonian systems is also presented. M. Koenig - S. Bernstein s idea for bounding the gradient of solutions to the quasilinear Dirichlet s problem Abstract: In my talk I speak about S. Bernsteins idea to estimate u x 0,0,Ω of a solution u(x) to the quasi-linear elliptic differential equation. Especially I focuse on the case n > 2. Up to about 1956 investigations of nonlinear problems deal with n = 2. A large number of methods were developed, but none of them were applicable for n > 2. The maximum-minimum principle had been a powerfull tool to find bounds. In case of n > 2, as I outline, this tool is not available. H.O. Cordes gave estimates in 1956 for the case n > 2. Later O. A. Ladythenskaya and N. N. Ural tseva establish estimates in the Sobolev space W2 2 (Ω). In line of their reasoning they use the idea of S. Bernstein and investigate the transformed function v(x) with u(x) = φ(v(x)). In my talk I will speak about a more simpler proof of theese results in the classical Banach space C 2,α (Ω). A. M. Micheletti - Multiplicity of solutions for coupled elliptic systems on Riemannian manifolds Abstract: Let (M, g) be a smooth compact 3-dimensional Riemannian manifold. Given real numbers a > 0, q > 0, b 2 < a and 2 < p < 6, we consider the following Klein Gordon Maxwell (KGM) systems 3

4 ɛ 2 g u + au = u p 2 u + (b 2 (qv 1) 2 )u g v + (1 + q 2 u 2 )v = qu 2 on M u > 0, v > 0 on M We show that the topology of the manifold (M, g) has an effect on the number of solutions of KGM systems. In particular we consider the Lusternik Schnirelmann category cat(m) of the manifold M in itself. Also, the geometry of the manifold (M, g) influences the number of solutions. We show that the scalar curvature Sg, relative to the metric g is the geometric property which influences the number of solutions. J.Pejsachowicz - Topology and bifurcation Abstract: Jorge Ize and James Alexander can be considered as the co-founders of topological multiparameter bifurcation theory. They both discovered the role played by Whithead s J-homomorphism in linearized bifurcation. The J-homomorphism had made his first appearance in Ize s PhD thesis Bifurcation theory for Fredholm operators, although without any specific mention because at that time Jorge Ize hadn t realized yet that his complementing method was related to this well known invariant from homotopy theory. Shortly after, J.Alexander and J.Yorke used the stable J-homomorphism in their paper on the global Hopf bifurcation of periodic orbits. In the first part of this talk I will look back to Jorge Ize s ideas at the time of his PhD thesis. In the second part I will speak about some further developments of these ideas in my own recent work T. Popelensky - Polytopes and K-theory Abstract: In a series of papers W.Bruns and J.Gubeladze had defined a new version of algebraic K-theory K i (R, P ), i 2 for commutative ring R with unit. The second argument is s polytope P satisfying certain properties. For the standard simplex one obtains Quillen K-theory K i (R). There are some natural questions concerning this generalization. First of all one is interested in calculating K i (R, P ) for different polytopes P. Bruns and Gubeladze found an answer for balanced polygones. Namely they proved that there are only six different variants. All available calculations show that K i (R, P ) is isomorphic to the Quillen algebraic K theory K i (R) or to the sum of several copies of K i (R) 4

5 (in some known cases the statement holds provided R is good enough ). There are some partial results on three dimensional polytopes. We plan to present a review of Bruns-Gubeladze construction and discuss our results in this direction. M.Starostka - Infinite dimensional cohomology theory and the Conley index Abstract: Motivated by low dimensional geometry and topology we define a cohomological Conley (HC) index in a Hilbert space. This is done using Gęba-Granas cohomology of an index pair. We show that HC is isomorphic to the Floer cohomology. Possible applications and further directions are discussed. H. Steinlein - 70 years of asymptotic fixed point theory Abstract: Asymptotic fixed point theory is the part of topological fixed point theory where assumptions on the mapping f are partly replaced by assumptions on iterates of f. We shall give a brief survey on ideas, methods, applications, and the history of this theory. N. Waterstraat - A family index theorem for periodic Hamiltonian systems and bifurcation Abstract: We introduce an index theorem for families of linear periodic Hamiltonian systems, which is reminiscent of the Atiyah-Singer index theorem for selfadjoint elliptic operators. For the special case of one-parameter families, we compare our theorem with a classical result of Salamon and Zehnder. Finally, we use the index theorem to study bifurcation of branches of periodic solutions for families of nonlinear Hamiltonian systems. V. Zvyagin - The degree theory of equivariant Fredholm maps Abstract: The talk is devoted to the degree theory of Fredholm mappings of index zero commuting with the action of certain groups: the cyclic group Z p of prime order p, the unit circle S 1, and the torus T n. We give formulas providing infor- 5

6 mation about the degree of such mappings. We also consider differences between the degree theory for Fredholm mappings of index zero on one hand and finitedimensional mappings and completely continuous vector fields on the other. H. Żołądek - The Hess-Appelrot system Abstract: We, i.e. Henryk Żołądek and Paweł Lubowiecki (my used to be graduate student), study the Hess-Appelrot case of the Euler-Poisson system which describes dynamics of a rigid body about a.fixed point. We prove existence of an invariant torus which supports hyperbolic or parabolic or elliptic periodic or elliptic quasi-periodic dynamics. In the elliptic cases we study the question of normal hyperbolicity of the invariant torus in the case when the torus is close to a critical circle. It turns out that the normal hyperbolicity takes place only in the case of 1 : q resonance. Next we study limit cycles which appear after perturbation of the above situation. We estimate the number of such cycles by analysis of some non-standard Melnikov integrals. 6