TY - INPR A1 - Krainer, Thomas A1 - Schulze, Bert-Wolfgang T1 - The conormal symbolic structure of corner boundary value problems N2 - Ellipticity of operators on manifolds with conical singularities or parabolicity on space-time cylinders are known to be linked to parameter-dependent operators (conormal symbols) on a corresponding base manifold. We introduce the conormal symbolic structure for the case of corner manifolds, where the base itself is a manifold with edges and boundary. The specific nature of parameter-dependence requires a systematic approach in terms of meromorphic functions with values in edge-boundary value problems. We develop here a corresponding calculus, and we construct inverses of elliptic elements. T3 - Preprint - (2004) 01 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26662 ER - TY - INPR A1 - Shlapunov, Alexander A1 - Tarkhanov, Nikolai Nikolaevich T1 - Mixed problems with a parameter N2 - Let X be a smooth n -dimensional manifold and D be an open connected set in X with smooth boundary ∂D. Perturbing the Cauchy problem for an elliptic system Au = f in D with data on a closed set Γ ⊂ ∂D we obtain a family of mixed problems depending on a small parameter ε > 0. Although the mixed problems are subject to a non-coercive boundary condition on ∂D\Γ in general, each of them is uniquely solvable in an appropriate Hilbert space DT and the corresponding family {uε} of solutions approximates the solution of the Cauchy problem in DT whenever the solution exists. We also prove that the existence of a solution to the Cauchy problem in DT is equivalent to the boundedness of the family {uε}. We thus derive a solvability condition for the Cauchy problem and an effective method of constructing its solution. Examples for Dirac operators in the Euclidean space Rn are considered. In the latter case we obtain a family of mixed boundary problems for the Helmholtz equation. T3 - Preprint - (2004) 02 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26677 ER - TY - INPR A1 - Berman, Gennady A1 - Tarkhanov, Nikolai Nikolaevich T1 - Quantum dynamics in the Fermi-Pasta-Ulam problem N2 - We study the dynamics of four wave interactions in a nonlinear quantum chain of oscillators under the "narrow packet" approximation. We determine the set of times for which the evolution of decay processes is essentially specified by quantum effects. Moreover, we highlight the quantum increment of instability. T3 - Preprint - (2004) 05 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26695 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Differential operators on manifolds with singularities : analysis and topology : Chapter 7: The index problem on manifolds with singularities N2 - Contents: Chapter 7: The Index Problemon Manifolds with Singularities Preface 7.1. The Simplest Index Formulas 7.1.1. General properties of the index 7.1.2. The index of invariant operators on the cylinder 7.1.3. Relative index formulas 7.1.4. The index of general operators on the cylinder 7.1.5. The index of operators of the form 1 + G with a Green operator G 7.1.6. The index of operators of the form 1 + G on manifolds with edges 7.1.7. The index on bundles with smooth base and fiber having conical points 7.2. The Index Problem for Manifolds with Isolated Singularities 7.2.1. Statement of the index splitting problem 7.2.2. The obstruction to the index splitting 7.2.3. Computation of the obstruction in topological terms 7.2.4. Examples. Operators with symmetries 7.3. The Index Problem for Manifolds with Edges 7.3.1. The index excision property 7.3.2. The obstruction to the index splitting 7.4. Bibliographical Remarks T3 - Preprint - (2004) 06 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26700 ER - TY - INPR A1 - Popivanov, Petăr R. T1 - Lorenz transformations and creation of logarithmic singularities to the solutions of some nonstrictly hyperbolic semilinear systems with two space variables T3 - Preprint - (2004) 07 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26710 ER - TY - INPR A1 - Harutjunjan, Gohar A1 - Schulze, Bert-Wolfgang T1 - Boundary problems with meromorphic symbols in cylindrical domains N2 - We show relative index formulas for boundary value problems in cylindrical domains and Sobolev spaces with different weigths at ±∞. The amplitude functions are meromorphic in the axial covariable and take values in the space of boundary value problems on the cross section of the cylinder. T3 - Preprint - (2004) 12 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26735 ER - TY - INPR A1 - Xiaochun, Liu A1 - Schulze, Bert-Wolfgang T1 - Boundary value problems in edge representation N2 - Edge representations of operators on closed manifolds are known to induce large classes of operators that are elliptic on specific manifolds with edges, cf. [9]. We apply this idea to the case of boundary value problems. We establish a correspondence between standard ellipticity and ellipticity with respect to the principal symbolic hierarchy of the edge algebra of boundary value problems, where an embedded submanifold on the boundary plays the role of an edge. We first consider the case that the weight is equal to the smoothness and calculate the dimensions of kernels and cokernels of the associated principal edge symbols. Then we pass to elliptic edge operators for arbitrary weights and construct the additional edge conditions by applying relative index results for conormal symbols. T3 - Preprint - (2004) 14 KW - Boundary value problems KW - edge singularities KW - ellipticity in the edge calculus Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26746 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Differential operators on manifolds with singularities : analysis and topology : Chapter 6: Elliptic theory on manifolds with edges N2 - Contents: Chapter 6: Elliptic Theory on Manifolds with Edges Introduction 6.1. Motivation and Main Constructions 6.1.1. Manifolds with edges 6.1.2. Edge-degenerate differential operators 6.1.3. Symbols 6.1.4. Elliptic problems 6.2. Pseudodifferential Operators 6.2.1. Edge symbols 6.2.2. Pseudodifferential operators 6.2.3. Quantization 6.3. Elliptic Morphisms and the Finiteness Theorem 6.3.1. Matrix Green operators 6.3.2. General morphisms 6.3.3. Ellipticity, Fredholm property, and smoothness Appendix A. Fiber Bundles and Direct Integrals A.1. Local theory A.2. Globalization A.3. Versions of the Definition of the Norm T3 - Preprint - (2004) 15 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26757 ER - TY - INPR A1 - Nazaikinskii, Vladimir E. A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - On the homotopy classification of elliptic operators on manifolds with edges N2 - We obtain a stable homotopy classification of elliptic operators on manifolds with edges. T3 - Preprint - (2004) 16 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26769 ER - TY - INPR A1 - Kytmanov, Aleksandr A1 - Myslivets, Simona A1 - Tarkhanov, Nikolai Nikolaevich T1 - Power sums of roots of a nonlinear system N2 - For a system of meromorphic functions f = (f1, . . . , fn) in Cn, an explicit formula is given for evaluating negative power sums of the roots of the nonlinear system f(z) = 0. T3 - Preprint - (2004) 18 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26788 ER - TY - INPR A1 - Kytmanov, Aleksandr A1 - Myslivets, Simona A1 - Tarkhanov, Nikolai Nikolaevich T1 - Zeta-function of a nonlinear system N2 - Given a system of entire functions in Cn with at most countable set of common zeros, we introduce the concept of zeta-function associated with the system. Under reasonable assumptions on the system, the zeta-function is well defined for all s ∈ Zn with sufficiently large components. Using residue theory we get an integral representation for the zeta-function which allows us to construct an analytic extension of the zeta-function to an infinite cone in Cn. T3 - Preprint - (2004) 19 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26795 ER - TY - INPR A1 - Gil, Juan B. A1 - Krainer, Thomas A1 - Mendoza, Gerardo A. T1 - Geometry and spectra of closed extensions of elliptic cone operators N2 - We study the geometry of the set of closed extensions of index 0 of an elliptic cone operator and its model operator in connection with the spectra of the extensions, and give a necessary and sufficient condition for the existence of rays of minimal growth for such operators. T3 - Preprint - (2004) 21 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26815 ER - TY - INPR A1 - Gil, Juan B. A1 - Krainer, Thomas A1 - Mendoza, Gerardo A. T1 - Resolvents of elliptic cone operators N2 - We prove the existence of sectors of minimal growth for general closed extensions of elliptic cone operators under natural ellipticity conditions. This is achieved by the construction of a suitable parametrix and reduction to the boundary. Special attention is devoted to the clarification of the analytic structure of the resolvent. T3 - Preprint - (2004) 22 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26820 ER - TY - INPR A1 - Dines, Nicoleta A1 - Liu, X. A1 - Schulze, Bert-Wolfgang T1 - Edge quantisation of elliptic operators N2 - The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy σ = (σψ, σ∧), where the second component takes value in operators on the infinite model cone of the local wedges. In general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the elliptcity of the principal edge symbol σ∧ which includes the (in general not explicitly known) number of additional conditions on the edge of trace and potential type. We focus here on these queations and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet-Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich-Dynin formula for edge boundary value problems, and we establish relations of elliptic operators for different weights, via the spectral flow of the underlying conormal symbols. T3 - Preprint - (2004) 24 KW - Boundary value problems KW - edge singularities KW - ellipticity KW - spectral flow Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26838 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Volpato, A. T1 - Green operators in the edge calculus N2 - Green operators on manifolds with edges are known to be an ingredient of parametrices of elliptic (edge-degenerate) operators. They play a similar role as corresponding operators in boundary value problems. Close to edge singularities the Green operators have a very complex asymptotic behaviour. We give a new characterisation of Green edge symbols in terms of kernels with discrete and continuous asymptotics in the axial variable of local model cones. T3 - Preprint - (2004) 25 KW - operators on manifolds with edges KW - weighted spaces with asymptotics KW - Green and Mellin edge operators Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26846 ER - TY - INPR A1 - Jaiani, George A1 - Schulze, Bert-Wolfgang T1 - Some degenerate elliptic systems and applications to cusped plates N2 - The tension-compression vibration of an elastic cusped plate is studied under all the reasonable boundary conditions at the cusped edge, while at the noncusped edge displacements and at the upper and lower faces of the plate stresses are given. T3 - Preprint - (2004) 27 KW - Casped plates KW - vibration KW - degenerate elliptic systems KW - weighted spaces KW - Hardy‘s inequality KW - Korn’s weighted inequality Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26866 ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich T1 - Harmonic integrals on domains with edges N2 - We study the Neumann problem for the de Rham complex in a bounded domain of Rn with singularities on the boundary. The singularities may be general enough, varying from Lipschitz domains to domains with cuspidal edges on the boundary. Following Lopatinskii we reduce the Neumann problem to a singular integral equation of the boundary. The Fredholm solvability of this equation is then equivalent to the Fredholm property of the Neumann problem in suitable function spaces. The boundary integral equation is explicitly written and may be treated in diverse methods. This way we obtain, in particular, asymptotic expansions of harmonic forms near singularities of the boundary. T3 - Preprint - (2004) 20 KW - domains with singularities KW - de Rham complex KW - Neumann problem KW - Hodge theory Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26800 ER - TY - INPR A1 - Egorov, Jurij V. A1 - Kondratiev, V. A. A1 - Schulze, Bert-Wolfgang T1 - On the completeness of root functions of elliptic boundary problems in a domain with conical points on the boundary N2 - Contents: 1 Introduction 2 Definitions 3 Rays of minimal growth 4 Proof of Theorem 2. 5 The growth of the resolvent 6 Proof of Theorem 3. 7 The completeness of root functions 8 Some generalizations T3 - Preprint - (2004) 17 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26773 ER - TY - INPR A1 - Calin, Ovidium A1 - Der-Chen, Chang T1 - The geometry on a step 3 Grushin model N2 - In this article we study the geometry associated with the sub-elliptic operator ½ (X²1 +X²2), where X1 = ∂x and X2 = x²/2 ∂y are vector fields on R². We show that any point can be connected with the origin by at least one geodesic and we provide an approximate formula for the number of the geodesics between the origin and the points situated outside of the y-axis. We show there are in¯nitely many geodesics between the origin and the points on the y-axis. T3 - Preprint - (2004) 08 KW - Grushin operator KW - subRiemannian geometry KW - geodesics KW - Hamilton-Jacobi theory KW - elliptic functions KW - Euler's theta functions Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26724 ER - TY - INPR A1 - Gauthier, Paul M. A1 - Tarkhanov, Nikolai Nikolaevich T1 - A covering property of the Riemann zeta-function N2 - For each compact subset K of the complex plane C which does not surround zero, the Riemann surface Sζ of the Riemann zeta function restricted to the critical half-strip 0 < Rs < 1/2 contains infinitely many schlicht copies of K lying ‘over’ K. If Sζ also contains at least one such copy, for some K which surrounds zero, then the Riemann hypothesis fails. T3 - Preprint - (2004) 03 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26683 ER -