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IWOTA 2010 July 12–16 Technische Universit¨atBerlin Book of Abstracts Some boundary value problems for operator-differential equations of mixed type and their applications to inverse problems N.L. Abasheeva First we consider the questions on existence and uniqueness of a so- lution to the boundary value problem B(t)ut = L(t)u + f(t), t ∈ (0,T ), T < ∞, u(T ) − u(0) = u0. Here B(t) are self-adjoint operators and L(t) are uniformly dissipative operators acting in a complex Hilbert space E with the inner product (·, ·) and the norm k · k. The operator B(t) can be noninvertible and has arbitrary arrangement of the spectrum. Then we apply obtained results to the following linear inverse problem: to find a function u(t) and an element ϕ satisfying the equation B(t)ut = Lu + f(t) + B(0)ϕ and the boundary conditions u(0) = u0, u(T ) = u1. Trace formula in perturbation theory V. Adamyan In this talk we discuss some old and recent results on the trace formula of perturbation theory in Hilbert and Krein spaces and the contribution of Heinz Langer to the elaboration of this issue. 3 Quasiconformal mappings associated with Gahov’s equation in inverse boundary value problems A. Akhmetova The question on the quantity of roots of so called Gahov’s equation is of importance for the theory of inverse boundary value problems, because it determines the quantity of decisions of that problems. As known, it coincides with the number N of extreme points of conformal radius. R ≡ R(D, z) = |f 0(ζ)|(1 − |ζ|2), (1) where z = f(ζ), ζ ∈ E = {ζ : |ζ| < 1},D = f(E). Here equlity (1) is understood as the graph of function R(D, z), i.e., a surface in R3 over disk E or domain D. The value N coincides with the number of coverings of origin by the range of gradient of the conformal radius ∇R(D, z) = 2Rz¯. (2) In this connection there arises an intrinsic problem of classification of diffeomorphisms (in particular, quasiconformal mappings) of form (2). The present work is a review of results of the papers [4], [5] and [6], supplemented by new effects on quasiconformal mappings (2) for domains f(rE), 0 < r < 1, and on calculation of Gahov’s radiuses for solution of inverse boundary value problems. The talk is based on a joint work with L. Aksent’ev. References [1] G.G. Tumashev and M.T. Nuzhin, Inverse Boundary Value Prob- lems with Applications, Kazan Univ. Publ., Kazan (1965) (in Russian). [2] F.D. Gakhov, Boundary Value Problems, Nauka, Moskow (1977) (in Russian). (F.D. Gakhov, Boundary Value Problems, Addison-Wesley, New York (1966).) [3] L.A. Aksent’ev, Communication of Outer Inverse Boundary Value Problem with an Inner Radius of Domain, Izv. VUZov, Mathematics, 2(1984), 3-11 (in Russian). [4] F.G. Avkhadiev , K.-J. Wirths, The conformal radius as a function and its gradient image, Israel J. of Math., 145 (2005), 349-374. [5] L.A. Aksent’ev, A.N. Akhmetova, On Mappings Related to the Gradient of the Conformal Radius, Izv. VUZov, Mathematics, 6(2009), 60-64 (in Russian). 4 [6] L.A. Aksent’ev, A.N. Akhmetova, On Mappings Related to the Gradient of the Conformal Radius, Mat. Zametki, 87(1)(2010), 3-12. Reduction of the plasticity model to an operator equation and regularity H.-D. Alber We show that the model equations of plasticity and viscoplasticity, which use internal variables to model history dependent material behav- ior, can be reduced to an evolution equation with a nonlinear monotone evolution operator. Based on standard properties of this evolution equa- tion we prove interior and boundary regularity of the solutions. The most interesting part is the investigation of the boundary regularity, since one cannot apply the standard method to prove regularity of the tangential derivatives and subsequentially extend this result to the normal deriva- tives by solving the equations for this derivative. Accordingly, one can show that the stress field belongs to H1,loc, but at the boundary one only gets H1/2−ε-regularity. Recently this result could be improved in [3] to H1/2+ε. The talk is based on joint work with S. Nesenenko. References [1] H.-D. Alber, S. Nesenenko: Local and global regularity in time de- pendent viscoplasticity. In: D. Reddy (ed.): Proceedings of the IUTAM symposium on theortetical, modelling and computational aspects of in- elastic media, Cape Town January 14 – 18, 2008, 363–372, Springer 2008. 1 1 −δ [2] H.-D. Alber, S. Nesenenko: Local H -regularity and H 3 –regularity up to the boundary in time dependent viscoplasticity. Asymptotic Anal- ysis 63, No. 3 (2009), 151–187. [3] J. Frehse, D. L¨obach: Regularity results for three-dimensional isotrop- ic and kinematic hardening including boundary differentiability. Math. Models Methods Appl. Sci. 19 (2009), 2231-2262. [4] D. Knees: On global spatial regularity in elasto-plasticity with linear hardening. Calc. Var. Partial Differ. Equ. 36, No. 4 (2009), 611–625 5 The dynamics of the multiple tunnel effect F. Ali Mehmeti We consider the Klein-Gordon equation on a star-shaped network composed of n half-axes connected at their origins, adding a potential which is constant but different on each branch. Our main result [3, 4] is an explicit construction of a spectral representation of the corresponding spatial operator using generalized eigenfunctions, exhibiting what we call the multiple tunnel effect. Results in experimental physics [7, 8], theoretical physics [6] and func- tional analysis [1, 5] describe new phenomena created by the dynamics of the (simple) tunnel effect: the delayed reflection and advanced transmis- sion near nodes issuing two branches. It is of major importance for the comprehension of the vibrations of networks to understand these phenom- ena near ramification nodes i.e. nodes with at least 3 branches, motivating our interest for the multiple tunnel effect. Related possible applications are the L∞−time decay, the extension to coupled transmission conditions and, in the case of semi linear equations, global existence and causality (cf. [2] for the case of two branches). References [1] F. Ali Mehmeti, V. R´egnier, Delayed reflection of the energy flow at a potential step for dispersive wave packets. Math. Meth. Appl. Sci. 27 (2004), 1145–1195. [2] F. Ali Mehmeti, V. R´egnier, Global existence and causality for a transmission problem with a repulsive nonlinearity. Nonlinear Anal- ysis, 69/2 (2008), 408-424. [3] F. Ali Mehmeti, R. Haller-Dintelmann, V. R´egnier. Expansions in generalized eigenfunctions of the weighted Laplacian on star-shaped networks. H. Amann, W. Arendt, M. Hieber, F. Neubrander, S. Nicaise, J. von Below (eds): Functional analysis and Evolution Equa- tions. The G¨unter Lumer Volume (2007), 1-16. [4] F. Ali Mehmeti, R. Haller-Dintelmann, V. R´egnier, The Klein- Gordon Equation with Multiple Tunnel Effect on a Star-Shaped Net- work: Expansions in generalized eigenfunctions. arxiv:0906.3230v1 [math.SP] 6 [5] Y. Daikh. Temps de passage de paquets d’ondes de basses fr´equences ou limit´esen bandes de fr´equences par une barri`ere de potentiel. Th`esede doctorat, Valenciennes, France, 2004. [6] J. M. Deutch, F. E. Low, Barrier Penetration and Superluminal Ve- locity. Annals of Physics 228 (1993), 184-202. [7] A. Enders, G. Nimtz, On superluminal barrier traversal. J. Phys. I France, 2 (1992) 1693-1698. [8] A. Haibel, G. Nimtz, Universal relationship of time and frequency in photonic tunnelling. Ann. Physik (Leipzig) 10 (2001), 707-712. The talk is based on a joint work with R. Haller-Dintelmann and V. R´egnier. Linear stochastic systems: a white noise space approach D. Alpay We present a new approach to input-output systems n X yn = hmun−m, n = 0, 1,..., m=0 and state space equations xn+1 = Axn + Bxn, yn = Cxn + Dun, n = 0, 1,... when randomness is allowed both in the input sequence (un) and in the impulse response (hn) or in the matrices A, B, C, D, which determine the state space equations. We use Hida’s white noise space setting, and the Kondratiev spaces of stochastic test functions and stochastic distribu- tions. The key to our approach is that the pointwise product between complex numbers is now replaced by the Wick product ♦ between random variables. Thus we have systems of the form n X yn = hm♦un−m, n = 0, 1,..., m=0 7 which can be shown to be (in an appropriate basis) double convolution systems. The Hermite transform maps the white noise space onto the hz,wi reproducing kernel Hilbert space F with reproducing kernel e `2 with z, w ∈ `2, that is, onto the Fock space. This allows us to transfer most, if not all, problems from classical system theory into problems for analytic functions where now there is a countable number of variables. We will describe some stability theorems. The image of the Kondratiev space of distributions under the Hermite transform is a commutative ring without divisors, and we will also explain links with the theory of linear systems over commutative rings. Finally, we will describe the parallels with double convolution systems associated to a new approach to multiscale systems, developed with M. Mboup. The talk is based on various joint works with H. Attia, D. Levanony, M. Mboup and A. Pinhas. A Schur algorithm for a large class of functions D. Alpay We present a framework within the indefinite Schur algorithm can be set, and which allows to consider generalized Schur functions, generalized Nevanlinna functions, and the like in a unified way. We consider a wide class of kernels which includes all such classical kernels as the kernels associated to Schur functions and Nevanlinna functions, and for which a Schur algorithm and general theory of interpolation can be developed.
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