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  - JOUR
A1  - Malass, Ihsane
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - The de Rham Cohomology through Hilbert Space Methods
JF  - Journal of Siberian Federal University. Mathematics & physics
N2  - We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler-Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer.
KW  - De Rham complex
KW  - cohomology
KW  - Hodge theory
KW  - Neumann problem
Y1  - 2019
U6  - https://doi.org/10.17516/1997-1397-2019-12-4-455-465
SN  - 1997-1397
SN  - 2313-6022
VL  - 12
IS  - 4
SP  - 455
EP  - 465
PB  - Sibirskij Federalʹnyj Universitet
CY  - Krasnoyarsk
ER  - 
TY  - JOUR
A1  - Malass, Ihsane
A1  - Tarkhanov, Nikolaj Nikolaevič
T1  - A perturbation of the de Rham complex
T1  - Возмущение комплекса де Рама
JF  - Journal of Siberian Federal University : Mathematics & Physics
JF  - Žurnal Sibirskogo Federalʹnogo Universiteta : Matematika i fizika
N2  - We consider a perturbation of the de Rham complex on a compact manifold with boundary. This perturbation goes beyond the framework of complexes, and so cohomology does not apply to it. On the other hand, its curvature is "small", hence there is a natural way to introduce an Euler characteristic and develop a Lefschetz theory for the perturbation. This work is intended as an attempt to develop a cohomology theory for arbitrary sequences of linear mappings.
N2  - Рассмотрим возмущение комплекса де Рама на компактном многообразии с краем. Это возмущение выходит за рамки комплексов, и поэтому когомологии к нему не относятся. С другой стороны, его кривизна "мала", поэтому существует естественный способ ввести характеристику Эйлера и разработать теорию Лефшеца для возмущения. Данная работа предназначена для попытки разработать теорию когомологий для произвольных последовательностей линейных отображений.
KW  - de Rham complex
KW  - cohomology
KW  - Hodge theory
KW  - Neumann problem
KW  - комплекс де Рама
KW  - когомологии
KW  - теория Ходжа
KW  - проблема Неймана
Y1  - 2020
U6  - https://doi.org/10.17516/1997-1397-2020-13-5-519-532
SN  - 1997-1397
SN  - 2313-6022
VL  - 13
IS  - 5
SP  - 519
EP  - 532
PB  - Siberian Federal University
CY  - Krasnojarsk
ER  -