TY  - INPR
A1  - Dyachenko, Evgueniya
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Singular perturbations of elliptic operators
N2  - We develop a new approach to the analysis of pseudodifferential operators with small parameter 'epsilon' in (0,1] on a compact smooth manifold X. The standard approach assumes action of operators in Sobolev spaces whose norms depend on 'epsilon'. Instead we consider the cylinder [0,1] x X over X and study pseudodifferential operators on the cylinder which act, by the very nature, on functions depending on 'epsilon' as well. The action in 'epsilon' reduces to multiplication by functions of this variable and does not include any differentiation. As but one result we mention asymptotic of solutions to singular perturbation problems for small values of 'epsilon'.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 1 
KW  - singular perturbation
KW  - pseudodifferential operator
KW  - ellipticity with parameter
KW  - regularization
KW  - asymptotics
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-69502
SN  - 2193-6943
VL  - 3
IS  - 1
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Shlapunov, Alexander
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Sturm-Liouville problems in domains with non-smooth edges
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)13 
KW  - Second order elliptic equations
KW  - non-coercive boundary conditions
KW  - root functions
KW  - weighted spaces
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-67336
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  -