@unpublished{ShlapunovTarkhanov2016,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},
  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}
}
@article{ShlapunovTarchanov2021,
  author    = {Shlapunov, Alexander and Tarchanov, Nikolaj Nikolaevič},
  title     = {An open mapping theorem for the Navier-Stokes type equations associated with the de Rham complex over R-n},
  series = {Siberian electronic mathematical reports = Sibirskie ėlektronnye matematičeskie izvestija},
  volume    = {18},
  journal   = {Siberian electronic mathematical reports = Sibirskie ėlektronnye matematičeskie izvestija},
  number    = {2},
  publisher = {Institut Matematiki Imeni S. L. Soboleva},
  address   = {Novosibirsk},
  issn      = {1813-3304},
  doi       = {10.33048/semi.2021.18.108},
  pages     = {1433 -- 1466},
  year      = {2021},
  abstract  = {We consider an initial problem for the Navier-Stokes type equations associated with the de Rham complex over R-n x[0, T], n >= 3, with a positive time T. We prove that the problem induces an open injective mappings on the scales of specially constructed function spaces of Bochner-Sobolev type. In particular, the corresponding statement on the intersection of these classes gives an open mapping theorem for smooth solutions to the Navier-Stokes equations.},
  language  = {en}
}
@book{ShlapunovTarkhanov2001,
  author    = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich},
  title     = {Duality by reproducing kernels},
  series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  journal   = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  publisher = {Univ.},
  address   = {Potsdam},
  issn      = {1437-739X},
  pages     = {78 S.},
  year      = {2001},
  language  = {en}
}
@article{ShlapunovTarkhanov2007,
  author    = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich},
  title     = {Formal poincare lemma},
  series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  journal   = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  publisher = {Univ.},
  address   = {Potsdam},
  issn      = {1437-739X},
  pages     = {36 S.},
  year      = {2007},
  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}
}
@misc{ShlapunovTarkhanov2017,
  author    = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich},
  title     = {Golusin-Krylov formulas in complex analysis},
  series = {Complex variables and elliptic equations},
  volume    = {63},
  journal   = {Complex variables and elliptic equations},
  number    = {7-8},
  publisher = {Routledge},
  address   = {Abingdon},
  issn      = {1747-6933},
  doi       = {10.1080/17476933.2017.1395872},
  pages     = {1142 -- 1167},
  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}
}
@book{Shlapunov1999,
  author    = {Shlapunov, Alexander},
  title     = {Iterations of self-adjoint operators and their applications to elliptic systems},
  series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  journal   = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  publisher = {Univ.},
  address   = {Potsdam},
  issn      = {1437-739X},
  pages     = {23 S.},
  year      = {1999},
  language  = {en}
}
@book{ShlapunovTarkhanov2004,
  author    = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich},
  title     = {Mixed problems with a parameter},
  series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  journal   = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  publisher = {Univ.},
  address   = {Potsdam},
  issn      = {1437-739X},
  pages     = {28 S.},
  year      = {2004},
  language  = {en}
}
@article{ShlapunovTarkhanov2005,
  author    = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich},
  title     = {Mixed problems with parameter},
  issn      = {1061-9208},
  year      = {2005},
  abstract  = {Let X be a smooth n-dimensional manifold and D be an open connected set in X with smooth boundary OD. Perturbing the Cauchy problem for an elliptic system Au = f in D with data on a closed set Gamma subset of partial derivativeD, we obtain a family of mixed problems depending on a small parameter epsilon > 0. Although the mixed problems are subjected to a noncoercive boundary condition on partial derivativeDF in general, each of them is uniquely solvable in an appropriate Hilbert space D-T and the corresponding family {u(epsilon)} of solutions approximates the solution of the Cauchy problem in D-T whenever the solution exists. We also prove that the existence of a solution to the Cauchy problem in D-T is equivalent to the boundedness of the family {u(epsilon)}. We thus derive a solvability condition for the Cauchy problem and an effective method of constructing the solution. Examples for Dirac operators in the Euclidean space R-n are treated. In this case, we obtain a family of mixed boundary problems for the Helmholtz equation},
  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},
  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}
}