@unpublished{PolkovnikovTarkhanov2017,
author = {Polkovnikov, Alexander and Tarkhanov, Nikolai Nikolaevich},
title = {A Riemann-Hilbert problem for the Moisil-Teodorescu system},
volume = {6},
number = {3},
url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-397036},
pages = {31},
year = {2017},
abstract = {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.},
language = {en}
}
@unpublished{FedchenkoTarkhanov2017,
author = {Fedchenko, Dmitry and Tarkhanov, Nikolai Nikolaevich},
title = {A Rad{\´o} Theorem for the Porous Medium Equation},
series = {Preprints des Instituts f{\"u}r Mathematik der Universit{\"a}t Potsdam},
volume = {6},
journal = {Preprints des Instituts f{\"u}r Mathematik der Universit{\"a}t Potsdam},
number = {1},
publisher = {Universit{\"a}tsverlag Potsdam},
address = {Potsdam},
url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-102735},
pages = {12},
year = {2017},
abstract = {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.},
language = {en}
}
@unpublished{ShlapunovTarkhanov2017,
author = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich},
title = {Golusin-Krylov Formulas in Complex Analysis},
series = {Preprints des Instituts f{\"u}r Mathematik der Universit{\"a}t Potsdam},
volume = {6},
journal = {Preprints des Instituts f{\"u}r Mathematik der Universit{\"a}t Potsdam},
number = {2},
publisher = {Universit{\"a}tsverlag Potsdam},
address = {Potsdam},
url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-102774},
pages = {25},
year = {2017},
abstract = {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.},
language = {en}
}
@unpublished{VasilievTarkhanov2016,
author = {Vasiliev, Serguei and Tarkhanov, Nikolai Nikolaevich},
title = {Construction of series of perfect lattices by layer superposition},
volume = {5},
number = {11},
publisher = {Universit{\"a}tsverlag Potsdam},
address = {Potsdam},
issn = {2193-6943 (online)},
url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-100591},
pages = {11},
year = {2016},
abstract = {We construct a new series of perfect lattices in n dimensions by the layer superposition method of Delaunay-Barnes.},
language = {en}
}
@unpublished{ShlapunovTarkhanov2016,
author = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich},
title = {An open mapping theorem for the Navier-Stokes equations},
volume = {5},
number = {10},
publisher = {Universit{\"a}tsverlag Potsdam},
address = {Potsdam},
issn = {2193-6943 (online)},
url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-98687},
pages = {80},
year = {2016},
abstract = {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 H{\"o}lder 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 Fr{\´e}chet 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 H{\"o}lder spaces.},
language = {en}
}
@unpublished{MeraTarkhanov2016,
author = {Mera, Azal and Tarkhanov, Nikolai Nikolaevich},
title = {The Neumann problem after Spencer},
volume = {5},
number = {6},
publisher = {Universit{\"a}tsverlag Potsdam},
address = {Potsdam},
issn = {2193-6943 (online)},
url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-90631},
pages = {21},
year = {2016},
abstract = {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.},
language = {en}
}
@unpublished{FedchenkoTarkhanov2016,
author = {Fedchenko, Dmitry and Tarkhanov, Nikolai Nikolaevich},
title = {Boundary value problems for elliptic complexes},
volume = {5},
number = {3},
publisher = {Universit{\"a}tsverlag Potsdam},
address = {Potsdam},
issn = {2193-6943 (online)},
url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-86705},
pages = {12},
year = {2016},
abstract = {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.},
language = {en}
}
@unpublished{AlsaedyTarkhanov2016,
author = {Alsaedy, Ammar and Tarkhanov, Nikolai Nikolaevich},
title = {A Hilbert boundary value problem for generalised Cauchy-Riemann equations},
volume = {5},
number = {1},
publisher = {Universit{\"a}tsverlag Potsdam},
address = {Potsdam},
issn = {2193-6943 (online)},
url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-86109},
pages = {21},
year = {2016},
abstract = {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.},
language = {en}
}
@unpublished{MeraShlapunovTarkhanov2015,
author = {Mera, Azal and Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich},
title = {Navier-Stokes equations for elliptic complexes},
volume = {4},
number = {12},
publisher = {Universit{\"a}tsverlag Potsdam},
address = {Potsdam},
issn = {2193-6943 (online)},
url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-85592},
pages = {27},
year = {2015},
abstract = {We continue our study of invariant forms of the classical equations of mathematical physics, such as the Maxwell equations or the Lam{\´e} 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.},
language = {en}
}
@unpublished{MakhmudovMakhmudovTarkhanov2015,
author = {Makhmudov, K. O. and Makhmudov, O. I. and Tarkhanov, Nikolai Nikolaevich},
title = {A nonstandard Cauchy problem for the heat equation},
volume = {4},
number = {11},
publisher = {Universit{\"a}tsverlag Potsdam},
address = {Potsdam},
issn = {2193-6943 (online)},
url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-83830},
pages = {14},
year = {2015},
abstract = {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.},
language = {en}
}