Browsing by Author "Begehr, Heinrich"
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Article Citation Count: 8AV Bitsadze's observation on bianalytic functions and the Schwarz problem(Taylor & Francis Ltd, 2019) Aksoy, Ümit; Begehr, Heinrich; Celebi, A. Okay; MathematicsAccording to an observation of A.V. Bitsadze from 1948 the Dirichlet problem for bianalytic functions is ill-posed. A natural boundary condition for the polyanalytic operator, however, is the Schwarz condition. An integral representation for the solutions in the unit disc to the inhomogeneous polyanalytic equation satisfying Schwarz boundary conditions is known. This representation is extended here to any simply connected plane domain having a harmonic Green function. Some other boundary value problems are investigated with some Dirichlet and Neumann conditions illuminating that just the Schwarz problem is a natural boundary condition for the Bitsadze operator.Article Citation Count: 7AV Bitsadze's observation on bianalytic functions and the Schwarz problem revisited(Taylor & Francis Ltd, 2021) Aksoy, Ümit; Begehr, Heinrich; Celebi, A. Okay; MathematicsThe extension of the Schwarz representation formula to simply connected domains with harmonic Green function and its polyanalytic generalization is not valid in general. They do hold only for certain domains.Article Citation Count: 6Schwarz problem for higher-order complex partial differential equations in the upper half plane(Wiley-v C H verlag Gmbh, 2019) Aksoy, Ümit; Begehr, Heinrich; Celebi, A. Okay; MathematicsLinear and nonlinear elliptic complex partial differential equations of higher-order are considered under Schwarz conditions in the upper-half plane, Firstly, using the integral representations for the solutions of the inhomogeneous polyanalytic equation with Schvvarz conditions, a class of integral operators is introduced together with some of their properties. Then, these operators are used to transform the problem for linear equations into singular integral equations. In the case of nonlinear equations such a transformation yields a system of integro-differential equations. Existence of the solutions of the relevant boundary value problems for linear and nonlinear equations are discussed via Fredholm theory and fixed point theorems, respectively.