TY - INPR
A1 - Polkovnikov, Alexander
A1 - Tarkhanov, Nikolai Nikolaevich
T1 - A Riemann-Hilbert problem for the Moisil-Teodorescu system
N2 - In a bounded domain with smooth boundary in R^3 we consider the stationary Maxwell equations
for a function u with values in R^3 subject to a nonhomogeneous condition
(u,v)_x = u_0 on
the boundary, where v is a given vector field and u_0 a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.
T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 6 (2017) 3
KW - Dirac operator
KW - Riemann-Hilbert problem
KW - Fredholm operators
Y1 - 2017
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-397036
VL - 6
IS - 3
ER -
TY - INPR
A1 - Fedchenko, Dmitry
A1 - Tarkhanov, Nikolai Nikolaevich
T1 - A Radó Theorem for the Porous Medium Equation
T2 - Preprints des Instituts für Mathematik der Universität Potsdam
N2 - We prove that each locally Lipschitz continuous function satisfying the porous medium equation away from the set of its zeroes is actually a weak solution of this equation in the whole domain.
T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 6 (2017) 1
KW - quasilinear equation
KW - removable set
KW - porous medium equation
Y1 - 2017
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-102735
VL - 6
IS - 1
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - INPR
A1 - Shlapunov, Alexander
A1 - Tarkhanov, Nikolai Nikolaevich
T1 - Golusin-Krylov Formulas in Complex Analysis
T2 - Preprints des Instituts für Mathematik der Universität Potsdam
N2 - This is a brief survey of a constructive technique of analytic continuation related to an explicit integral formula of Golusin and Krylov (1933). It goes far beyond complex analysis and applies to the Cauchy problem for elliptic partial differential equations as well. As started in the classical papers, the technique is elaborated in generalised Hardy spaces also called Hardy-Smirnov spaces.
T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 6 (2017) 2
KW - analytic continuation
KW - integral formulas
KW - Cauchy problem
Y1 - 2017
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-102774
VL - 6
IS - 2
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - INPR
A1 - Vasiliev, Serguei
A1 - Tarkhanov, Nikolai Nikolaevich
T1 - Construction of series of perfect lattices by layer superposition
N2 - We construct a new series of perfect lattices in n dimensions by the layer superposition method of Delaunay-Barnes.
T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016)11
KW - lattice packing and covering
KW - polyhedra and polytopes
KW - regular figures
KW - division of spaces
Y1 - 2016
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-100591
SN - 2193-6943 (online)
VL - 5
IS - 11
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - INPR
A1 - Shlapunov, Alexander
A1 - Tarkhanov, Nikolai Nikolaevich
T1 - An open mapping theorem for the Navier-Stokes equations
N2 - We consider the Navier-Stokes equations in the layer R^n x [0,T] over R^n with finite T > 0. Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes equations to a nonlinear Fredholm equation of the form (I+K) u = f, where K is a compact continuous operator in anisotropic normed Hölder spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes equations for all t in [0,T]. On using the particular properties of the de Rham complex we conclude that the Fréchet derivative (I+K)' is continuously invertible at each point of the Banach space under consideration and the map I+K is open and injective in the space. In this way the Navier-Stokes equations prove to induce an open one-to-one mapping in the scale of Hölder spaces.
T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016)10
KW - Navier-Stokes equations
KW - weighted Hölder spaces
KW - integral representation method
Y1 - 2016
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-98687
SN - 2193-6943 (online)
VL - 5
IS - 10
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - INPR
A1 - Mera, Azal
A1 - Tarkhanov, Nikolai Nikolaevich
T1 - The Neumann problem after Spencer
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.
T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 6
KW - elliptic complex
KW - manifold with boundary
KW - Hodge theory
KW - Neumann problem
Y1 - 2016
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-90631
SN - 2193-6943 (online)
VL - 5
IS - 6
PB - Universitätsverlag Potsdam
CY - Potsdam
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 (online)
VL - 5
IS - 3
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - INPR
A1 - Alsaedy, Ammar
A1 - Tarkhanov, Nikolai Nikolaevich
T1 - A Hilbert boundary value problem for generalised Cauchy-Riemann equations
N2 - We elaborate a boundary Fourier method for studying an analogue of the Hilbert problem for analytic functions within the framework of generalised Cauchy-Riemann equations. The boundary value problem need not satisfy the Shapiro-Lopatinskij condition and so it fails to be Fredholm in Sobolev spaces. We show a solvability condition of the Hilbert problem, which looks like those for ill-posed
problems, and construct an explicit formula for approximate solutions.
T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 1
KW - Dirac operator
KW - Clifford algebra
KW - Riemann-Hilbert problem
KW - Fredholm operator
Y1 - 2016
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-86109
SN - 2193-6943 (online)
VL - 5
IS - 1
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - INPR
A1 - Mera, Azal
A1 - Shlapunov, Alexander
A1 - Tarkhanov, Nikolai Nikolaevich
T1 - Navier-Stokes equations for elliptic complexes
N2 - We continue our study of invariant forms of the classical equations of mathematical physics,
such as the Maxwell equations or the Lamé system, on manifold with boundary. To this end we interpret them in terms of the de Rham complex at a certain step. On using the structure of the complex we get an insight to predict a degeneracy deeply encoded
in the equations. In the present paper we develop an invariant approach to the classical Navier-Stokes equations.
T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015)12
KW - Navier-Stokes equations
KW - classical solution
Y1 - 2015
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-85592
SN - 2193-6943 (online)
VL - 4
IS - 12
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - INPR
A1 - Makhmudov, K. O.
A1 - Makhmudov, O. I.
A1 - Tarkhanov, Nikolai Nikolaevich
T1 - A nonstandard Cauchy problem for the heat equation
N2 - We consider a Cauchy problem for the heat equation in a cylinder X x (0,T) over a domain X in the n-dimensional space with data on a strip lying on the lateral surface. The strip is of the form
S x (0,T), where S is an open subset of the boundary of X. The problem is ill-posed. Under natural restrictions on the configuration of S we derive an explicit formula for solutions of this problem.
T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015)11
KW - heat equation
KW - Cauchy problem
KW - Carleman formulas
Y1 - 2015
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-83830
SN - 2193-6943 (online)
VL - 4
IS - 11
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -