@unpublished{LyTarkhanov2013, author = {Ly, Ibrahim and Tarkhanov, Nikolai Nikolaevich}, title = {Generalised Beltrami equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-67416}, year = {2013}, abstract = {We enlarge the class of Beltrami equations by developping a stability theory for the sheaf of solutions of an overdetermined elliptic system of first order homogeneous partial differential equations with constant coefficients in the Euclidean space.}, language = {en} } @unpublished{ShlapunovTarkhanov2013, author = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich}, title = {Sturm-Liouville problems in domains with non-smooth edges}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-67336}, year = {2013}, abstract = {We consider a (generally, non-coercive) mixed boundary value problem in a bounded domain for a second order elliptic differential operator A. The differential operator is assumed to be of divergent form and the boundary operator B is of Robin type. The boundary is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset of the boundary and control the growth of solutions near this set. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set. Moreover, we prove the completeness of root functions related to L.}, language = {en} } @unpublished{AlsaedyTarkhanov2013, author = {Alsaedy, Ammar and Tarkhanov, Nikolai Nikolaevich}, title = {Normally solvable nonlinear boundary value problems}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-65077}, year = {2013}, abstract = {We study a boundary value problem for an overdetermined elliptic system of nonlinear first order differential equations with linear boundary operators. Such a problem is solvable for a small set of data, and so we pass to its variational formulation which consists in minimising the discrepancy. The Euler-Lagrange equations for the variational problem are far-reaching analogues of the classical Laplace equation. Within the framework of Euler-Lagrange equations we specify an operator on the boundary whose zero set consists precisely of those boundary data for which the initial problem is solvable. The construction of such operator has much in common with that of the familiar Dirichlet to Neumann operator. In the case of linear problems we establish complete results.}, language = {en} } @unpublished{KiselevTarkhanov2013, author = {Kiselev, Oleg and Tarkhanov, Nikolai Nikolaevich}, title = {The capture of a particle into resonance at potential hole with dissipative perturbation}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64725}, year = {2013}, abstract = {We study the capture of a particle into resonance at a potential hole with dissipative perturbation and periodic outside force. The measure of resonance solutions is evaluated. We also derive an asymptotic formula for the parameter range of those solutions which are captured into resonance.}, language = {en} } @unpublished{ShlapunovTarkhanov2007, author = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich}, title = {Formal Poincar{\´e} lemma}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30231}, year = {2007}, abstract = {We show how the multiple application of the formal Cauchy-Kovalevskaya theorem leads to the main result of the formal theory of overdetermined systems of partial differential equations. Namely, any sufficiently regular system Au = f with smooth coefficients on an open set U ⊂ Rn admits a solution in smooth sections of a bundle of formal power series, provided that f satisfies a compatibility condition in U.}, language = {en} } @unpublished{MaergoizTarkhanov2006, author = {Maergoiz, L. and Tarkhanov, Nikolai Nikolaevich}, title = {Optimal recovery from a finite set in Banach spaces of entire functions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30199}, year = {2006}, abstract = {We develop an approach to the problem of optimal recovery of continuous linear functionals in Banach spaces through information on a finite number of given functionals. The results obtained are applied to the problem of the best analytic continuation from a finite set in the complex space Cn, n ≥ 1, for classes of entire functions of exponential type which belong to the space Lp, 1 < p < 1, on the real subspace of Cn. These latter are known as Wiener classes.}, language = {en} } @unpublished{KrupchykTarkhanovTuomela2006, author = {Krupchyk, K. and Tarkhanov, Nikolai Nikolaevich and Tuomela, J.}, title = {Elliptic quasicomplexes in Boutet de Monvel algebra}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30122}, year = {2006}, abstract = {We consider quasicomplexes of Boutet de Monvel operators in Sobolev spaces on a smooth compact manifold with boundary. To each quasicomplex we associate two complexes of symbols. One complex is defined on the cotangent bundle of the manifold and the other on that of the boundary. The quasicomplex is elliptic if these symbol complexes are exact away from the zero sections. We prove that elliptic quasicomplexes are Fredholm. As a consequence of this result we deduce that a compatibility complex for an overdetermined elliptic boundary problem operator is also Fredholm. Moreover, we introduce the Euler characteristic for elliptic quasicomplexes of Boutet de Monvel operators.}, language = {en} } @unpublished{Tarkhanov2006, author = {Tarkhanov, Nikolai Nikolaevich}, title = {Euler characteristic of Fredholm quasicomplexes}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30117}, year = {2006}, abstract = {By quasicomplexes are usually meant perturbations of complexes small in some sense. Of interest are not only perturbations within the category of complexes but also those going beyond this category. A sequence perturbed in this way is no longer a complex, and so it bears no cohomology. We show how to introduce Euler characteristic for small perturbations of Fredholm complexes. The paper is to appear in Funct. Anal. and its Appl., 2006.}, language = {en} } @unpublished{MakhmudovNiyozovTarkhanov2006, author = {Makhmudov, O. and Niyozov, I. and Tarkhanov, Nikolai Nikolaevich}, title = {The cauchy problem of couple-stress elasticity}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30078}, year = {2006}, abstract = {We study the Cauchy problem for the oscillation equation of the couple-stress theory of elasticity in a bounded domain in R3. Both the displacement and stress are given on a part S of the boundary of the domain. This problem is densely solvable while data of compact support in the interior of S fail to belong to the range of the problem. Hence the problem is ill-posed which makes the standard calculi of Fourier integral operators inapplicable. If S is real analytic the Cauchy-Kovalevskaya theorem applies to guarantee the existence of a local solution. We invoke the special structure of the oscillation equation to derive explicit conditions of global solvability and an approximation solution.}, language = {en} } @unpublished{TarkhanovVasilevski2005, author = {Tarkhanov, Nikolai Nikolaevich and Vasilevski, Nikolai}, title = {Microlocal analysis of the Bochner-Martinelli integral}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30012}, year = {2005}, abstract = {In order to characterise the C*-algebra generated by the singular Bochner-Martinelli integral over a smooth closed hypersurfaces in Cn, we compute its principal symbol. We show then that the Szeg{\"o} projection belongs to the strong closure of the algebra generated by the singular Bochner-Martinelli integral.}, language = {en} } @unpublished{KrupchykTarkhanovTuomela2005, author = {Krupchyk, K. and Tarkhanov, Nikolai Nikolaevich and Tuomela, J.}, title = {Generalised elliptic boundary problems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-29994}, year = {2005}, abstract = {For elliptic systems of differential equations on a manifold with boundary, we prove the Fredholm property of a class of boundary problems which do not satisfy the Shapiro-Lopatinskii property. We name these boundary problems generalised elliptic, for they preserve the main properties of elliptic boundary problems. Moreover, they reduce to systems of pseudodifferential operators on the boundary which are generalised elliptic in the sense of Saks (1997).}, language = {en} } @unpublished{AizenbergTarkhanov2005, author = {Aizenberg, Lev A. and Tarkhanov, Nikolai Nikolaevich}, title = {Stable expansions in homogeneous polynomials}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-29925}, year = {2005}, abstract = {An expansion for a class of functions is called stable if the partial sums are bounded uniformly in the class. Stable expansions are of key importance in numerical analysis where functions are given up to certain error. We show that expansions in homogeneous functions are always stable on a small ball around the origin, and evaluate the radius of the largest ball with this property.}, language = {en} } @unpublished{Tarkhanov2005, author = {Tarkhanov, Nikolai Nikolaevich}, title = {Unitary solutions of partial differential equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-29852}, year = {2005}, abstract = {We give an explicit construction of a fundamental solution for an arbitrary non-degenerate partial differential equation with smooth coefficients.}, language = {en} } @unpublished{SchulzeTarkhanov2005, author = {Schulze, Bert-Wolfgang and Tarkhanov, Nikolai Nikolaevich}, title = {Boundary value problems with Toeplitz conditions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-29837}, year = {2005}, abstract = {We describe a new algebra of boundary value problems which contains Lopatinskii elliptic as well as Toeplitz type conditions. These latter are necessary, if an analogue of the Atiyah-Bott obstruction does not vanish. Every elliptic operator is proved to admit up to a stabilisation elliptic conditions of such a kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. The crucial novelty consists of the new type of weighted Sobolev spaces which serve as domains of pseudodifferential operators and which fit well to the nature of operators.}, language = {en} } @unpublished{Tarkhanov2005, author = {Tarkhanov, Nikolai Nikolaevich}, title = {On the root functions of general elliptic boundary value problems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-29822}, year = {2005}, abstract = {We consider a boundary value problem for an elliptic differential operator of order 2m in a domain D ⊂ n. The boundary of D is smooth outside a finite number of conical points, and the Lopatinskii condition is fulfilled on the smooth part of δD. The corresponding spaces are weighted Sobolev spaces H(up s,Υ)(D), and this allows one to define ellipticity of weight Υ for the problem. The resolvent of the problem is assumed to possess rays of minimal growth. The main result says that if there are rays of minimal growth with angles between neighbouring rays not exceeding π(Υ + 2m)/n, then the root functions of the problem are complete in L²(D). In the case of second order elliptic equations the results remain true for all domains with Lipschitz boundary.}, language = {en} } @unpublished{Tarkhanov2005, author = {Tarkhanov, Nikolai Nikolaevich}, title = {Root functions of elliptic boundary problems in domains with conic points of the boundary}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-29812}, year = {2005}, abstract = {We prove the completeness of the system of eigen and associated functions (i.e., root functions) of an elliptic boundary value problem in a domain whose boundary is a smooth surface away from a finite number of points, each of them possesses a neighbourhood where the boundary is a conical surface.}, language = {en} } @unpublished{Tarkhanov2005, author = {Tarkhanov, Nikolai Nikolaevich}, title = {Operator algebras related to the Bochner-Martinelli Integral}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-29789}, year = {2005}, abstract = {We describe a general method of computing the square of the singular integral of Bochner-Martinelli. Any explicit formula for the square applies in a familiar way to describe the C*-algebra generated by this integral.}, 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{KytmanovMyslivetsTarkhanov2004, author = {Kytmanov, Aleksandr and Myslivets, Simona and Tarkhanov, Nikolai Nikolaevich}, title = {Zeta-function of a nonlinear system}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26795}, year = {2004}, abstract = {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.}, language = {en} } @unpublished{KytmanovMyslivetsTarkhanov2004, author = {Kytmanov, Aleksandr and Myslivets, Simona and Tarkhanov, Nikolai Nikolaevich}, title = {Power sums of roots of a nonlinear system}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26788}, year = {2004}, abstract = {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.}, language = {en} }