TY  - INPR
A1  - Harutjunjan, Gohar
A1  - Schulze, Bert-Wolfgang
T1  - The Zaremba problem with singular interfaces as a corner boundary value problem
N2  - 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.
T3  - Preprint - (2004) 26 
KW  - Zaremba problem
KW  - corner Sobolev spaces with double weights
KW  - pseudodifferential boundary value problems
Y1  - 2004
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26855
ER  - 
TY  - INPR
A1  - Harutjunjan, G.
A1  - Schulze, Bert-Wolfgang
T1  - Conormal symbols of mixed elliptic problems with singular interfaces
N2  - 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].
T3  - Preprint - (2005) 12 
KW  - Corner boundary value problems
KW  - mixed elliptic problems
KW  - interfaces with conical singularities
KW  - Zaremba problem
Y1  - 2005
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29885
ER  - 
TY  - JOUR
A1  - Harutyunyan, Gohar
A1  - Schulze, Bert-Wolfgang
T1  - The Zaremba problem with singular interfaces as a corner boundary value problem
JF  - Potential analysis : an international journal devoted to the interactions between potential theory, probability theory, geometry and functional analysis
N2  - 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.
KW  - Zaremba problem
KW  - corner Sobolev spaces with double weights
KW  - pseudo-differential boundary value problems
Y1  - 2006
U6  - https://doi.org/10.1007/s11118-006-9020-6
SN  - 0926-2601
VL  - 25
SP  - 327
EP  - 369
PB  - Springer
CY  - Dordrecht
ER  - 
TY  - JOUR
A1  - Ly, Ibrahim
T1  - A Cauchy problem for the Cauchy-Riemann operator
JF  - Afrika Matematika
N2  - 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.
KW  - nonlinear PDI
KW  - Cauchy problem
KW  - Zaremba problem
Y1  - 2020
U6  - https://doi.org/10.1007/s13370-020-00810-4
SN  - 1012-9405
SN  - 2190-7668
VL  - 32
IS  - 1-2
SP  - 69
EP  - 76
PB  - Springer
CY  - Heidelberg
ER  -