TY  - JOUR
A1  - Mera, Azal Jaafar Musa
A1  - Tarchanov, Nikolaj Nikolaevič
T1  - The Neumann Problem after Spencer
JF  - Žurnal Sibirskogo Federalʹnogo Universiteta = Journal of Siberian Federal University : Matematika i fizika = Mathematics & physics
N2  - When trying to extend the Hodge theory for elliptic complexes on compact closed manifolds to the case of compact manifolds with boundary one is led to a boundary value problem for the Laplacian of the complex which is usually referred to as Neumann problem. We study the Neumann problem for a larger class of sequences of differential operators on a compact manifold with boundary. These are sequences of small curvature, i.e., bearing the property that the composition of any two neighbouring operators has order less than two.
KW  - elliptic complexes
KW  - manifolds with boundary
KW  - Hodge theory
KW  - Neumann problem
Y1  - 2017
U6  - https://doi.org/10.17516/1997-1397-2017-10-4-474-493
SN  - 1997-1397
SN  - 2313-6022
VL  - 10
SP  - 474
EP  - 493
PB  - Sibirskij Federalʹnyj Universitet
CY  - Krasnojarsk
ER  - 
TY  - INPR
A1  - Schulze, Bert-Wolfgang
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Elliptic complexes of pseudodifferential operators on manifolds with edges
N2  - On a compact closed manifold with edges live pseudodifferential operators which are block matrices of operators with additional edge conditions like boundary conditions in boundary value problems. They include Green, trace and potential operators along the edges, act in a kind of Sobolev spaces and form an algebra with a wealthy symbolic structure. We consider complexes of Fréchet spaces whose differentials are given by operators in this algebra. Since the algebra in question is a microlocalization of the Lie algebra of typical vector fields on a manifold with edges, such complexes are of great geometric interest. In particular, the de Rham and Dolbeault complexes on manifolds with edges fit into this framework. To each complex there correspond two sequences of symbols, one of the two controls the interior ellipticity while the other sequence controls the ellipticity at the edges. The elliptic complexes prove to be Fredholm, i.e., have a finite-dimensional cohomology. Using specific tools in the algebra of pseudodifferential operators we develop a Hodge theory for elliptic complexes and outline a few applications thereof.
T3  - Preprint - (1998) 14 
KW  - manifolds with singularities
KW  - pseudodifferential operators
KW  - elliptic complexes
KW  - Hodge theory
Y1  - 1998
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25257
ER  - 
TY  - INPR
A1  - Fedchenko, Dmitry
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Boundary value problems for elliptic complexes
N2  - The aim of this paper is to bring together two areas which are of great importance for the study of overdetermined boundary value problems. The first area is homological algebra which is the main tool in constructing the formal theory of overdetermined problems. And the second area is the global calculus of pseudodifferential operators which allows one to develop explicit analysis.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 3 
KW  - elliptic complexes
KW  - Fredholm property
KW  - index
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-86705
SN  - 2193-6943
VL  - 5
IS  - 3
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Schulze, Bert-Wolfgang
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A Lefschetz fixed point formula in the relative elliptic theory
N2  - A version of the classical Lefschetz fixed point formula is proved for the cohomology of the cone of a cochain mapping of elliptic complexes. As a particular case we show a Lefschetz formula for the relative de Rham cohomology.
T3  - Preprint - (1998) 01 
KW  - elliptic complexes
KW  - relative cohomology
KW  - Lefschetz number
Y1  - 1998
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25159
ER  - 
TY  - GEN
A1  - Wallenta, Daniel
T1  - A Lefschetz fixed point formula for elliptic quasicomplexes
T2  - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe
N2  - In a recent paper, the Lefschetz number for endomorphisms (modulo trace class operators) of sequences of trace class curvature was introduced. We show that this is a well defined, canonical extension of the classical Lefschetz number and establish the homotopy invariance of this number. Moreover, we apply the results to show that the Lefschetz fixed point formula holds for geometric quasiendomorphisms of elliptic quasicomplexes.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 885 
KW  - elliptic complexes
KW  - Fredholm complexes
KW  - Lefschetz number
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-435471
SN  - 1866-8372
IS  - 885
SP  - 577
EP  - 587
ER  -