@article{Ly2020,
  author    = {Ly, Ibrahim},
  title     = {A Cauchy problem for the Cauchy-Riemann operator},
  series = {Afrika Matematika},
  volume    = {32},
  journal   = {Afrika Matematika},
  number    = {1-2},
  publisher = {Springer},
  address   = {Heidelberg},
  issn      = {1012-9405},
  doi       = {10.1007/s13370-020-00810-4},
  pages     = {69 -- 76},
  year      = {2020},
  abstract  = {We study the Cauchy problem for a nonlinear elliptic equation with data on a piece S of the boundary surface partial derivative X. By the Cauchy problem is meant any boundary value problem for an unknown function u in a domain X with the property that the data on S, if combined with the differential equations in X, allows one to determine all derivatives of u on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy-Kovalevskaya theorem. We discuss a variational setting of the Cauchy problem which always possesses a generalized solution.},
  language  = {en}
}
@article{HarutyunyanSchulze2006,
  author    = {Harutyunyan, Gohar and Schulze, Bert-Wolfgang},
  title     = {The Zaremba problem with singular interfaces as a corner boundary value problem},
  series = {Potential analysis : an international journal devoted to the interactions between potential theory, probability theory, geometry and functional analysis},
  volume    = {25},
  journal   = {Potential analysis : an international journal devoted to the interactions between potential theory, probability theory, geometry and functional analysis},
  publisher = {Springer},
  address   = {Dordrecht},
  issn      = {0926-2601},
  doi       = {10.1007/s11118-006-9020-6},
  pages     = {327 -- 369},
  year      = {2006},
  abstract  = {We study mixed boundary value problems for an elliptic operator A on a manifold X with boundary Y, i.e., Au = f in int X, T (+/-) u = g(+/-) on int Y+/-, where Y is subdivided into subsets Y+/- with an interface Z and boundary conditions T+/- on Y+/- that are Shapiro-Lopatinskij elliptic up to Z from the respective sides. We assume that Z subset of Y is a manifold with conical singularity v. As an example we consider the Zaremba problem, where A is the Laplacian and T- Dirichlet, T+ Neumann conditions. The problem is treated as a corner boundary value problem near v which is the new point and the main difficulty in this paper. Outside v the problem belongs to the edge calculus as is shown in Bull. Sci. Math. ( to appear). With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along Z {v} of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.},
  language  = {en}
}
@unpublished{HarutjunjanSchulze2004,
  author    = {Harutjunjan, Gohar and Schulze, Bert-Wolfgang},
  title     = {The Zaremba problem with singular interfaces as a corner boundary value problem},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26855},
  year      = {2004},
  abstract  = {We study mixed boundary value problems for an elliptic operator A on a manifold X with boundary Y , i.e., Au = f in int X, T±u = g± on int Y±, where Y is subdivided into subsets Y± with an interface Z and boundary conditions T± on Y± that are Shapiro-Lopatinskij elliptic up to Z from the respective sides. We assume that Z ⊂ Y is a manifold with conical singularity v. As an example we consider the Zaremba problem, where A is the Laplacian and T- Dirichlet, T+ Neumann conditions. The problem is treated as a corner boundary value problem near v which is the new point and the main difficulty in this paper. Outside v the problem belongs to the edge calculus as is shown in [3]. With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along Z \ {v} of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.},
  language  = {en}
}
@unpublished{HarutjunjanSchulze2005,
  author    = {Harutjunjan, G. and Schulze, Bert-Wolfgang},
  title     = {Conormal symbols of mixed elliptic problems with singular interfaces},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-29885},
  year      = {2005},
  abstract  = {Mixed elliptic problems are characterised by conditions that have a discontinuity on an interface of the boundary of codimension 1. The case of a smooth interface is treated in [3]; the investigation there refers to additional interface conditions and parametrices in standard Sobolev spaces. The present paper studies a necessary structure for the case of interfaces with conical singularities, namely, corner conormal symbols of such operators. These may be interpreted as families of mixed elliptic problems on a manifold with smooth interface. We mainly focus on second order operators and additional interface conditions that are holomorphic in an extra parameter. In particular, for the case of the Zaremba problem we explicitly obtain the number of potential conditions in this context. The inverses of conormal symbols are meromorphic families of pseudo-differential mixed problems referring to a smooth interface. Pointwise they can be computed along the lines [3].},
  language  = {en}
}