@unpublished{Krainer2001, author = {Krainer, Thomas}, title = {The calculus of Volterra Mellin pseudodifferential operators with operator-valued symbols}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26185}, year = {2001}, abstract = {We introduce the calculus of Mellin pseudodifferential operators parameters based on "twisted" operator-valued Volterra symbols as well aas the abstract Mellin calclus with holomorphic symbols. We establish the properties of the symblic and operational calculi, and we give and make use of explicit oscillatory integral formulas on the symbolic side, e. g., for the Leibniz-product, kernel cut-off, and Mellin quantization. Moreover, we introduce the notion of parabolicity for the calculi of Volterra Mellin operators, and construct Volterra parametrices for parabolic operators within the calculi.}, language = {en} } @unpublished{XiaochunWitt2002, author = {Xiaochun, Liu and Witt, Ingo}, title = {Pseudodifferential calculi on the half-line respecting prescribed asymptotic types}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26255}, year = {2002}, abstract = {Contents: 1. Introduction 2. Preliminaries 3. Basic Elements of the Calculus 4. Further Elements of the Calculus}, language = {en} } @unpublished{Tarkhanov2002, author = {Tarkhanov, Nikolai Nikolaevich}, title = {Anisotropic edge problems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26280}, year = {2002}, abstract = {We investigate elliptic pseudodifferential operators which degenerate in an anisotropic way on a submanifold of arbitrary codimension. To find Fredholm problems for such operators we adjoint to them boundary and coboundary conditions on the submanifold.The algebra obtained this way is a far reaching generalisation of Boutet de Monvel's algebra of boundary value problems with transmission property. We construct left and right regularisers and prove theorems on hypoellipticity and local solvability.}, language = {en} } @unpublished{Witt2002, author = {Witt, Ingo}, title = {A calculus for a class of finitely degenerate pseudodifferential operators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26246}, year = {2002}, abstract = {For a class of degenerate pseudodifferential operators, local parametrices are constructed. This is done in the framework of a pseudodifferential calculus upon adding conditions of trace and potential type, respectively, along the boundary on which the operators degenerate.}, language = {en} } @unpublished{LiuYangLu2002, author = {Liu, Weian and Yang, Yin and Lu, Gang}, title = {Viscosity solutions of fully nonlinear parabolic systems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26215}, year = {2002}, abstract = {In this paper, we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second order degenerate parabolic equations and their Cauchy-Dirichlet problem. We prove the existence, uniqueness and continuity of viscosity solution by combining Perron's method with the technique of coupled solutions. The results here generalize those in [2] and [3].}, language = {en} } @unpublished{SchroheSeiler1999, author = {Schrohe, Elmar and Seiler, J{\"o}rg}, title = {Ellipticity and invertibility in the cone algebra on Lp-Sobolev spaces}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25621}, year = {1999}, abstract = {Given a manifold B with conical singularities, we consider the cone algebra with discrete asymptotics, introduced by Schulze, on a suitable scale of Lp-Sobolev spaces. Ellipticity is proven to be equivalent to the Fredholm property in these spaces, it turns out to be independent of the choice of p. We then show that the cone algebra is closed under inversion: whenever an operator is invertible between the associated Sobolev spaces, its inverse belongs to the calculus. We use these results to analyze the behaviour of these operators on Lp(B).}, language = {en} } @unpublished{Fedosov1999, author = {Fedosov, Boris}, title = {Pseudo-differential operators and deformation quantization}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25651}, year = {1999}, abstract = {Using the Riemannian connection on a compact manifold X, we show that the algebra of classical pseudo-differential operators on X generates a canonical deformation quantization on the cotangent manifold T*X. The corresponding Abelian connection is calculated explicitly in terms of the of the exponential mapping. We prove also that the index theorem for elliptic operators may be obtained as a consequence of the index theorem for deformation quantization.}, language = {en} } @unpublished{Sadykov1999, author = {Sadykov, Timour}, title = {Hypergeometric systems of differential equations and amoebas of rational functions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25665}, year = {1999}, abstract = {We study the approach to the theory of hypergeometric functions in several variables via a generalization of the Horn system of differential equations. A formula for the dimension of its solution space is given. Using this formula we construct an explicit basis in the space of holomorphic solutions to the generalized Horn system under some assumptions on its parameters. These results are applied to the problem of describing the complement of the amoeba of a rational function, which was posed in [12].}, language = {en} } @unpublished{Shlapunov2000, author = {Shlapunov, Alexander}, title = {On Iterations of double layer potentials}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25687}, year = {2000}, abstract = {We prove the existence of Hp(D)-limit of iterations of double layer potentials constructed with the use of Hodge parametrix on a smooth compact manifold X, D being an open connected subset of X. This limit gives us an orthogonal projection from Sobolev space Hp(D) to a closed subspace of Hp(D)-solutions of an elliptic operator P of order p ≥ 1. Using this result we obtain formulae for Sobolev solutions to the equation Pu = f in D whenever these solutions exist. This representation involves the sum of a series whose terms are iterations of double layer potentials. Similar regularization is constructed also for a P-Neumann problem in D.}, language = {en} } @unpublished{Myslivets2000, author = {Myslivets, Simona}, title = {On the boundary behaviour of the logarithmic residue integral}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25733}, year = {2000}, abstract = {A formula of multidimensional logarithmic residue is proved for holomorphic maps with zeroes on the boundary of a bounded domain in Cn.}, language = {en} } @unpublished{CoriascoSchroheSeiler2001, author = {Coriasco, Sandro and Schrohe, Elmar and Seiler, J{\"o}rg}, title = {Bounded imaginary powers of differential operators on manifolds with conical singularities}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25962}, year = {2001}, abstract = {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.}, language = {en} } @unpublished{XiaochunWitt2001, author = {Xiaochun, Liu and Witt, Ingo}, title = {Asymptotic expansions for bounded solutions to semilinear Fuchsian equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25912}, year = {2001}, abstract = {It is shown that bounded solutions to semilinear elliptic Fuchsian equations obey complete asymptoic expansions in terms of powers and logarithms in the distance to the boundary. For that purpose, Schuze's notion of asymptotic type for conormal asymptotics close to a conical point is refined. This in turn allows to perform explicit calculations on asymptotic types - modulo the resolution of the spectral problem for determining the singular exponents in the asmptotic expansions.}, language = {en} } @unpublished{MaXu2001, author = {Ma, Li and Xu, Xingwang}, title = {Positive solutions of a logistic equation on unbounded intervals}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26015}, year = {2001}, abstract = {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.}, language = {en} } @unpublished{Tarkhanov2004, author = {Tarkhanov, Nikolai Nikolaevich}, title = {Harmonic integrals on domains with edges}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26800}, year = {2004}, abstract = {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.}, language = {en} } @unpublished{Krainer2002, author = {Krainer, Thomas}, title = {On the inverse of parabolic boundary value problems for large times}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26310}, year = {2002}, abstract = {We construct algebras of Volterra pseudodifferential operators that contain, in particular, the inverses of the most natural classical systems of parabolic boundary value problems of general form. Parabolicity is determined by the invertibility of the principal symbols, and as a result is equivalent to the invertibility of the operators within the calculus. Existence, uniqueness, regularity, and asymptotics of solutions as t → ∞ are consquences of the mapping properties of the operators in exponentially weighted Sobolev spaces and subspaces with asymptotics. An important aspect of this work is that the microlocal and global kernel structure of the inverse operator (solution operator) of a parabolic boundary value problem for large times is clarified. Moreover, our approach naturally yields qualitative pertubation results for the solvability theory of parabolic boundary value problems. To achieve these results, we assign t = ∞ the meaning of a conical point and treat the operators as totally characteristic pseudodifferential boundary value problems.}, language = {en} } @unpublished{NazaikinskiiSternin2002, author = {Nazaikinskii, Vladimir and Sternin, Boris}, title = {Relative elliptic theory}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26400}, year = {2002}, abstract = {This paper is a survey of relative elliptic theory (i.e. elliptic theory in the category of smooth embeddings), closely related to the Sobolev problem, first studied by Sternin in the 1960s. We consider both analytic aspects to the theory (the structure of the algebra of morphismus, ellipticity, Fredholm property) and topological aspects (index formulas and Riemann-Roch theorems). We also study the algebra of Green operators arising as a subalgebra of the algebra of morphisms.}, language = {en} } @unpublished{Davis2002, author = {Davis, Simon}, title = {On the absence of large-order divergences in superstring theory}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26452}, year = {2002}, abstract = {The genus-dependence of multi-loop superstring ams is estimated at large orders in perturbation theory using the super-Schottky group parameterization of supermoduli space. Restriction of the integration region to a subset of supermoduli space and a single fundamental domain of the super-modular group suggests an exponential dependence on the genus. Upper bounds for these estimates are obtained for arbitrary N-point superstring scattering amplitudes and are shown to be consistent with exact results obtained for special type II string amplitudes for orbifold or Calabi-Yau compactifications. The genus-dependence is then obtained by considering the effect of the remaining contribution to the superstring amplitudes after the coefficients of the formally divergent parts of the integrals vanish as a result of a sum over spin structures. The introduction of supersymmetry therefore leads to the elimination of large-order divergences in string pertubation theory, a result which is based only on the supersymmetric generalization of the polyakov measure and not the gauge group of the string model.}, language = {en} } @unpublished{JunkerSchrohe2001, author = {Junker, Wolfgang and Schrohe, Elmar}, title = {Adiabatic vacuum states on general spacetime manifolds : definition, construction, and physical properties}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26100}, year = {2001}, abstract = {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.}, language = {en} } @unpublished{Yagdjian2001, author = {Yagdjian, Karen}, title = {Geometric optics for the nonlinear hyperbolic systems of Kirchhoff-type}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26059}, year = {2001}, abstract = {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}, language = {en} } @unpublished{YihongLi2001, author = {Yihong, Du and Li, Ma}, title = {Some remarks related to De Giorgi's conjecture}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26027}, year = {2001}, abstract = {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.}, language = {en} }