TY - INPR A1 - Chen, Hua A1 - Yu, Chun T1 - Asymptotic behaviour of the trace for Schrödinger operator on fractal drums T3 - Preprint - (2001) 32 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26157 ER - TY - INPR A1 - Savin, Anton A1 - Sternin, Boris T1 - Index defects in the theory of nonlocal boundary value problems and the η-invariant N2 - The paper deals with elliptic theory on manifolds with boundary represented as a covering space. We compute the index for a class of nonlocal boundary value problems. For a nontrivial covering, the index defect of the Atiyah-Patodi-Singer boundary value problem is computed. We obtain the Poincaré duality in the K-theory of the corresponding manifolds with singularities. T3 - Preprint - (2001) 31 KW - elliptic operator KW - boundary value problem KW - finiteness theorem KW - nonlocal problem KW - covering KW - relative η-invariant KW - index KW - modn-index Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26146 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Sternin, Boris T1 - Some problems of control of semiclassical states for the Schrödinger equation N2 - Contents: Introduction Controlled Quantum Systems The Asymptotic Controllability Problem The Stabilization Problem Unitarily Nonlinear Equations The Quantum Problem The Stabilization Problem for the Schrödinger Equation with a Unitarily Non-linear Control T3 - Preprint - (2001) 30 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26130 ER - TY - INPR A1 - Messina, Francesca T1 - Local solvability for semilinear Fuchsian equations N2 - Contents: 1 Introduction 2 Differential operators on manifolds with conical singularities 3 When H up(s,y) (B) is an algebra 4 Statement of the main result 5 Proof of the theorem T3 - Preprint - (2001) 29 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26124 ER - TY - INPR A1 - Harutyunyan, Anahit V. T1 - Toeplitz operators and division theorems in anisotropic spaces of holomorphic functions in the polydisc N2 - This work is an introduction to anisotropic spaces, which have an ω-weight of analytic functions and are generalizations of Lipshitz classes in the polydisc. We prove that these classes form an algebra and are invariant with respect to monomial multiplication. These operators are bounded in these (Lipshitz and Djrbashian) spaces. As an application, we show a theorem about the division by good-inner functions in the mentioned classes is proved. T3 - Preprint - (2001) 28 KW - Toeplitz operators KW - anisotropic spaces KW - polydisc KW - good-inner function Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26110 ER - TY - INPR A1 - Junker, Wolfgang A1 - Schrohe, Elmar T1 - Adiabatic vacuum states on general spacetime manifolds : definition, construction, and physical properties N2 - Adiabatic vacuum states are a well-known class of physical states for linear quantum fields n Robertson-Walker spacetimes. We extend the definition of adiabatic vacua to general spacetime manifolds by using the notion of the Sobolev wavefront set. This definition is also applicable to interacting field theories. Hadamard states form a special subclass of the adiabatic vacua. We analyze physical properties of adiabatic vacuum representations of the Klein-Gordon field on globally hyperbolic spacetme manifolds (factoriality, quasiequivalence, local definteness, Haag duality) and construct them explicitly, if the manifold has a compact Cauchy surface. T3 - Preprint - (2001) 27 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26100 ER - TY - INPR A1 - Shlapunov, Alexander A1 - Tarkhanov, Nikolai Nikolaevich T1 - Duality by reproducing kernels N2 - Let A be a determined or overdetermined elliptic differential operator on a smooth compact manifold X. Write Ssub(A)(D) for the space of solutions to thesystem Au = 0 in a domain D ⊂ X. Using reproducing kernels related to various Hilbert structures on subspaces of Ssub(A)(D) we show explicit identifications of the dual spaces. To prove the "regularity" of reproducing kernels up to the boundary of D we specify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known example being the ∂-Neumann problem. The duality itself takes place only for those domains D which possess certain convexity properties with respect to A. T3 - Preprint - (2001) 26 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26095 ER - TY - INPR A1 - Paneah, Boris T1 - On a new problem in integral geometry related to boundary problems for partial differential equations N2 - Contents: 1 Introduction 2 Statement of the problem and definitions 3 The main results 4 Proof of theorem 2 4.1 Reduction of problem (2) to functional - integral equations 4.2 The uniqueness of a solution of equation (2) 4.3 The existence of a solution of equation (2) 5 Proof of theorem 1 6 Proof of theorem 3 7 First boundary problem for hyperbolic differential equations 7.1 Statement of the problem 7.2 The formulation of the result and a sketch of the proof T3 - Preprint - (2001) 25 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26089 ER - TY - INPR A1 - Galstian, Anahit T1 - Lp - Lq decay estimates for the equation with exponentially growing coefficient N2 - Contents: 1 Introduction 1 Representation formulas 2 Consideration in the pseudodifferential zone: e up(t) |ξ| ≤ 1 3 Consideration in he hyperbolic zone: e up(t) |ξ| ≥ 1 T3 - Preprint - (2001) 24 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26079 ER - TY - INPR A1 - Witt, Ingo T1 - Asymptotic algebras N2 - The concept of asymptotic type that primarily appears in singular and asymptotic analysis is developed. Especially, asymptotic algebras are introduced. T3 - Preprint - (2001) 23 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26069 ER - TY - INPR A1 - Yagdjian, Karen T1 - Geometric optics for the nonlinear hyperbolic systems of Kirchhoff-type N2 - Contents: 1 Introduction 2 Main result 3 Construction of the asymptotic solutions 3.1 Derivation of the equations for the profiles 3.2 Exsistence of the principal profile 3.3 Determination of Usub(2) and the remaining profiles 4 Stability of the samll global solutions. Justification of One Phase Nonlinear Geometric Optics for the Kirchhoff-type equations 4.1 Stability of the global solutions to the Kirchhoff-type symmetric hyperbolic systems 4.2 The nonlinear system of ordinary differential equations with the parameter 4.3 Some energies estimates 4.4 The dependence of the solution W(t, ξ) on the function s(t) 4.5 The oscillatory integrals of the bilinear forms of the solutions 4.6 Estimates for the basic bilinear form Γsub(s)(t) 4.7 Contraction mapping 4.8 Stability of the global solution 4.9 Justification of One Phase Nonlinear Geometric Optics for the Kirchhoff-type equations T3 - Preprint - (2001) 22 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26059 ER - TY - INPR A1 - Kapanadze, David A1 - Schulze, Bert-Wolfgang T1 - Symbolic calculus for boundary value problems on manifolds with edges N2 - Boundary value problems for (pseudo-) differential operators on a manifold with edges can be characterised by a hierarchy of symbols. The symbol structure is responsible or ellipicity and for the nature of parametrices within an algebra of "edge-degenerate" pseudo-differential operators. The edge symbol component of that hierarchy takes values in boundary value problems on an infinite model cone, with edge variables and covariables as parameters. Edge symbols play a crucial role in this theory, in particular, the contribution with holomorphic operatot-valued Mellin symbols. We establish a calculus in s framework of "twisted homogenity" that refers to strongly continuous groups of isomorphisms on weighted cone Sobolev spaces. We then derive an equivalent representation with a particularly transparent composition behaviour. T3 - Preprint - (2001) 21 KW - pseudo-differential boundary value problems KW - operators on manifolds with singularities Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26046 ER - TY - INPR A1 - Maniccia, L. A1 - Mughetti, M. T1 - Weyl calculus for a class of subelliptic operators N2 - Weyl-Hörmander calculus is used to get a parametrix in OPS¹-m sub(½, ½)(Ω)for a class of subelliptic pseudodifferential operators in OPS up(m)sub(1, 0)(Ω) with real non-negative principal symbol. T3 - Preprint - (2001) 19 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26038 ER - TY - INPR A1 - Yihong, Du A1 - Li, Ma T1 - Some remarks related to De Giorgi's conjecture N2 - For several classes of functions including the special case f(u) = u − u³, we obtain boundedness and symmetry results for solutions of the problem −Δu = f(u) defined on R up(n). Our results complement a number of recent results related to a conjecture of De Giorgi. T3 - Preprint - (2001) 18 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26027 ER - TY - INPR A1 - Ma, Li A1 - Xu, Xingwang T1 - Positive solutions of a logistic equation on unbounded intervals N2 - In this paper, we study the existence of positive solutions of a one-parameter family of logistic equations on R+ or on R. These equations are stationary versions of the Fisher equations and the KPP equations. We also study the blow up region of a sequence of the solutions when the parameter approachs a critical value and the nonexistence of positive solutions beyond the critical value. We use the direct method and the sub and super solution method. T3 - Preprint - (2001) 17 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26015 ER - TY - INPR A1 - Krainer, Thomas A1 - Schulze, Bert-Wolfgang T1 - On the inverse of parabolic systems of partial differential equations of general form in an infinite space-time cylinder [Part 3: Chapter 6+7] N2 - We consider general parabolic systems of equations on the infinite time interval in case of the underlying spatial configuration is a closed manifold. The solvability of equations is studied both with respect to time and spatial variables in exponentially weighted anisotropic Sobolev spaces, and existence and maximal regularity statements for parabolic equations are proved. Moreover, we analyze the long-time behaiour of solutions in terms of complete asymptotic expansions. These results are deduced from a pseudodifferential calculus that we construct explicitly. This algebra of operators is specifically designed to contain both the classical systems of parabolic equations of general form and their inverses, parabolicity being reflected purely on symbolic level. To this end, we assign t = ∞ the meaning of an anisotropic conical point, and prove that this interprtation is consistent with the natural setting in the analysis of parabolic PDE. Hence, major parts of this work consist of the construction of an appropriate anisotropiccone calculus of so-called Volterra operators. In particular, which is the most important aspect, we obtain the complete characterization of the microlocal and the global kernel structure of the inverse of parabolicsystems in an infinite space-time cylinder. Moreover, we obtain perturbation results for parabolic equations from the investigation of the ideal structure of the calculus. T3 - Preprint - (2001) 16 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26000 ER - TY - INPR A1 - Krainer, Thomas A1 - Schulze, Bert-Wolfgang T1 - On the inverse of parabolic systems of partial differential equations of general form in an infinite space-time cylinder [Part 2: Chapter 3-5] N2 - We consider general parabolic systems of equations on the infinite time interval in case of the underlying spatial configuration is a closed manifold. The solvability of equations is studied both with respect to time and spatial variables in exponentially weighted anisotropic Sobolev spaces, and existence and maximal regularity statements for parabolic equations are proved. Moreover, we analyze the long-time behaiour of solutions in terms of complete asymptotic expansions. These results are deduced from a pseudodifferential calculus that we construct explicitly. This algebra of operators is specifically designed to contain both the classical systems of parabolic equations of general form and their inverses, parabolicity being reflected purely on symbolic level. To this end, we assign t = ∞ the meaning of an anisotropic conical point, and prove that this interprtation is consistent with the natural setting in the analysis of parabolic PDE. Hence, major parts of this work consist of the construction of an appropriate anisotropiccone calculus of so-called Volterra operators. In particular, which is the most important aspect, we obtain the complete characterization of the microlocal and the global kernel structure of the inverse of parabolicsystems in an infinite space-time cylinder. Moreover, we obtain perturbation results for parabolic equations from the investigation of the ideal structure of the calculus. T3 - Preprint - (2001) 15 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25992 ER - TY - INPR A1 - Krainer, Thomas A1 - Schulze, Bert-Wolfgang T1 - On the inverse of parabolic systems of partial differential equations of general form in an infinite space-time cylinder [Part 1: Chapter 1+2] N2 - We consider general parabolic systems of equations on the infinite time interval in case of the underlying spatial configuration is a closed manifold. The solvability of equations is studied both with respect to time and spatial variables in exponentially weighted anisotropic Sobolev spaces, and existence and maximal regularity statements for parabolic equations are proved. Moreover, we analyze the long-time behaiour of solutions in terms of complete asymptotic expansions. These results are deduced from a pseudodifferential calculus that we construct explicitly. This algebra of operators is specifically designed to contain both the classical systems of parabolic equations of general form and their inverses, parabolicity being reflected purely on symbolic level. To this end, we assign t = ∞ the meaning of an anisotropic conical point, and prove that this interprtation is consistent with the natural setting in the analysis of parabolic PDE. Hence, major parts of this work consist of the construction of an appropriate anisotropiccone calculus of so-called Volterra operators. In particular, which is the most important aspect, we obtain the complete characterization of the microlocal and the global kernel structure of the inverse of parabolicsystems in an infinite space-time cylinder. Moreover, we obtain perturbation results for parabolic equations from the investigation of the ideal structure of the calculus. T3 - Preprint - (2001) 14 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25987 ER - TY - INPR A1 - Kytmanov, Aleksandr A1 - Myslivets, Simona A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - Elliptic problems for the Dolbeault complex N2 - The inhomogeneous ∂-equations is an inexhaustible source of locally unsolvable equations, subelliptic estimates and other phenomena in partial differential equations. Loosely speaking, for the anaysis on complex manifolds with boundary nonelliptic problems are typical rather than elliptic ones. Using explicit integral representations we assign a Fredholm complex to the Dolbeault complex over an arbitrary bounded domain in C up(n). T3 - Preprint - (2001) 13 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25979 ER - TY - INPR A1 - Coriasco, Sandro A1 - Schrohe, Elmar A1 - Seiler, Jörg T1 - Bounded imaginary powers of differential operators on manifolds with conical singularities N2 - We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted Lp-spaces over B,1 < p < ∞. Under suitable ellipticity assumptions we can define a family of complex powers A up(z), z ∈ C. We also obtain sufficient information on the resolvent of A to show the boundedness of the pure imaginary powers. Examples concern unique solvability and maximal regularity of the solution of the Cauchy problem u' - Δu = f, u(0) = 0, for the Laplacian on conical manifolds. T3 - Preprint - (2001) 12 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25962 ER -