Institut für Mathematik
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We define the Dirichlet to Neumann operator for an elliptic complex of first order differential operators on a compact Riemannian manifold with boundary. Under reasonable conditions the Betti numbers of the complex prove to be completely determined by the Dirichlet to Neumann operator on the boundary.
We prove a theorem on separation of boundary null points for generators of continuous semigroups of holomorphic self-mappings of the unit disk in the complex plane. Our construction demonstrates rather strikingly the particular role of the binary operation au broken vertical bar given by 1/ f au broken vertical bar g = 1/f + 1/g on generators.
We show that it is possible to approximate the zeta-function of a curve over a finite field by meromorphic functions which satisfy the same functional equation and moreover satisfy (respectively do not satisfy) an analog of the Riemann hypothesis. In the other direction, it is possible to approximate holomorphic functions by simple manipulations of such a zeta-function. No number theory is required to understand the theorems and their proofs, for it is known that the zeta-functions of curves over finite fields are very explicit meromorphic functions. We study the approximation properties of these meromorphic functions.
On completeness of root functions of Sturm-Liouville problems with discontinuous boundary operators
(2013)
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 partial derivative D. 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 selfadjoint 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. (C) 2013 Elsevier Inc. All rights reserved.
We consider a solution of the nonlinear Klein-Gordon equation perturbed by a parametric driver. The frequency of parametric perturbation varies slowly and passes through a resonant value, which leads to a solution change. We obtain a new connection formula for the asymptotic solution before and after the resonance.
We study the autoresonant solution of Duffing's equation in the presence of dissipation. This solution is proved to be an attracting set. We evaluate the maximal amplitude of the autoresonant solution and the time of transition from autoresonant growth of the amplitude to the mode of fast oscillations. Analytical results are illustrated by numerical simulations.
We discuss the Cauchy problem for the so-called Chaplygin system which often appears in gas, aero- and hydrodynamics. This system can be thought of as a nonlinear analogue of the Cauchy-Riemann system in the plane. We pose Cauchy data on a part of the boundary and apply variational approach to construct a solution to this ill-posed problem. The problem actually gives insight to fundamental questions related to instable problems for nonlinear equations.
Anisotropic edge problems
(2002)
We show a Lefschetz fixed point formula for holomorphic functions in a bounded domain D with smooth boundary in the complex plane. To introduce the Lefschetz number for a holomorphic map of D, we make use of the Bergman kernel of this domain. The Lefschetz number is proved to be the sum of the usual contributions of fixed points of the map in D and contributions of boundary fixed points, these latter being different for attracting and repulsing fixed points