TY - INPR A1 - Antoniouk, Alexandra Viktorivna A1 - Kiselev, Oleg A1 - Stepanenko, Vitaly A1 - Tarkhanov, Nikolai Nikolaevich T1 - Asymptotic solutions of the Dirichlet problem for the heat equation at a characteristic point N2 - The Dirichlet problem for the heat equation in a bounded domain is characteristic, for there are boundary points at which the boundary touches a characteristic hyperplane t = c, c being a constant. It was I.G. Petrovskii (1934) who first found necessary and sufficient conditions on the boundary which guarantee that the solution is continuous up to the characteristic point, provided that the Dirichlet data are continuous. This paper initiated standing interest in studying general boundary value problems for parabolic equations in bounded domains. We contribute to the study by constructing a formal solution of the Dirichlet problem for the heat equation in a neighbourhood of a characteristic boundary point and showing its asymptotic character. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)25 KW - Heat equation KW - the first boundary value problem KW - characteristic boundary point KW - cusp Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-61987 ER - TY - INPR A1 - Alsaedy, Ammar A1 - Tarkhanov, Nikolai Nikolaevich T1 - The method of Fischer-Riesz equations for elliptic boundary value problems N2 - We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)24 KW - Boundary value problems for first order systems KW - Green formula KW - Fischer-Riesz equations KW - regularisation Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-61792 ER - TY - INPR A1 - Dyachenko, Evgueniya A1 - Tarkhanov, Nikolai Nikolaevich T1 - Degeneration of boundary layer at singular points N2 - We study the Dirichlet problem in a bounded plane domain for the heat equation with small parameter multiplying the derivative in t. The behaviour of solution at characteristic points of the boundary is of special interest. The behaviour is well understood if a characteristic line is tangent to the boundary with contact degree at least 2. We allow the boundary to not only have contact of degree less than 2 with a characteristic line but also a cuspidal singularity at a characteristic point. We construct an asymptotic solution of the problem near the characteristic point to describe how the boundary layer degenerates. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)23 KW - Heat equation KW - Dirichlet problem KW - characteristic points KW - boundary layer Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-60135 ER - TY - INPR A1 - Alsaedy, Ammar A1 - Tarkhanov, Nikolai Nikolaevich T1 - Spectral projection for the dbar-Neumann problem N2 - We show that the spectral kernel function of the dbar-Neumann problem on a non-compact strongly pseudoconvex manifold is smooth up to the boundary. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)12 KW - dbar-Neumann problem KW - strongly pseudoconvex domains KW - spectral kernel function Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-58616 SN - 2193-6943 ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich T1 - A simple numerical approach to the Riemann hypothesis N2 - The Riemann hypothesis is equivalent to the fact the the reciprocal function 1/zeta (s) extends from the interval (1/2,1) to an analytic function in the quarter-strip 1/2 < Re s < 1 and Im s > 0. Function theory allows one to rewrite the condition of analytic continuability in an elegant form amenable to numerical experiments. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 9 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-57645 SN - 2193-6943 ER - TY - INPR A1 - Shlapunov, Alexander A1 - Tarkhanov, Nikolai Nikolaevich T1 - On completeness of root functions of Sturm-Liouville problems with discontinuous boundary operators N2 - We consider a Sturm-Liouville boundary value problem in a bounded domain D of R^n. By this is meant that the differential equation is given by a second order elliptic operator of divergent form in D and the boundary conditions are of Robin type on bD. The first order term of the boundary operator is the oblique derivative whose coefficients bear discontinuities of the first kind. Applying the method of weak perturbation of compact self-adjoint operators and the method of rays of minimal growth, we prove the completeness of root functions related to the boundary value problem in Lebesgue and Sobolev spaces of various types. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)11 KW - Sturm-Liouville problems KW - discontinuous Robin condition KW - root functions KW - Lipschitz domains Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-57759 SN - 2193-6943 ER - TY - INPR A1 - Grudsky, Serguey A1 - Tarkhanov, Nikolai Nikolaevich T1 - Conformal reduction of boundary problems for harmonic functions in a plane domain with strong singularities on the boundary N2 - We consider the Dirichlet, Neumann and Zaremba problems for harmonic functions in a bounded plane domain with nonsmooth boundary. The boundary curve belongs to one of the following three classes: sectorial curves, logarithmic spirals and spirals of power type. To study the problem we apply a familiar method of Vekua-Muskhelishvili which consists in using a conformal mapping of the unit disk onto the domain to pull back the problem to a boundary problem for harmonic functions in the disk. This latter is reduced in turn to a Toeplitz operator equation on the unit circle with symbol bearing discontinuities of second kind. We develop a constructive invertibility theory for Toeplitz operators and thus derive solvability conditions as well as explicit formulas for solutions. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)10 KW - singular integral equations KW - nonsmooth curves KW - boundary value problems Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-57745 ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich A1 - Wallenta, Daniel T1 - The Lefschetz number of sequences of trace class curvature N2 - For a sequence of Hilbert spaces and continuous linear operators the curvature is defined to be the composition of any two consecutive operators. This is modeled on the de Rham resolution of a connection on a module over an algebra. Of particular interest are those sequences for which the curvature is "small" at each step, e.g., belongs to a fixed operator ideal. In this context we elaborate the theory of Fredholm sequences and show how to introduce the Lefschetz number. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 3 KW - Perturbed complexes KW - curvature KW - Lefschetz number Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-56969 ER - TY - INPR A1 - Kiselev, Oleg M. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Scattering of autoresonance trajectories upon a separatrix N2 - We study asymptotic properties of solutions to the primary resonance equation with large amplitude on a long time interval. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 2 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-56880 ER -