### The Neumann problem for the 2-D Helmholtz equation in a domain, bounded by closed and open curves.

Krutitskii, P.A. (1998)

International Journal of Mathematics and Mathematical Sciences

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Krutitskii, P.A. (1998)

International Journal of Mathematics and Mathematical Sciences

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Krutitskii, P.A. (1999)

Mathematical Problems in Engineering

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Krutitskii, P.A. (2001)

Mathematical Problems in Engineering

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Dagmar Medková (1998)

Archivum Mathematicum

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Dirichlet, Neumann and Robin problem for the Laplace equation is investigated on the open set with holes and nonsmooth boundary. The solutions are looked for in the form of a double layer potential and a single layer potential. The measure, the potential of which is a solution of the boundary-value problem, is constructed.

Maz'ya, Vladimir G.

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Basheleishvili, Michael (1995)

Memoirs on Differential Equations and Mathematical Physics

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Dagmar Medková (1998)

Czechoslovak Mathematical Journal

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For fairly general open sets it is shown that we can express a solution of the Neumann problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series. If the open set is simply connected and bounded then the solution of the Dirichlet problem is the double layer potential with a density given by a similar series.

Alberto Cialdea (1992)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The definition of multiple layer potential for the biharmonic equation in ${\mathbb{R}}^{n}$ is given. In order to represent the solution of Dirichlet problem by means of such a potential, a singular integral system, whose symbol determinant identically vanishes, is considered. The concept of bilateral reduction is introduced and employed for investigating such a system.