A numerical solution of the dirichlet problem on some special doubly connected regions

Dont, Miroslav; Dontová, Eva

doi:10.1023/a:1022296024669

Cited by 2 publications

(1 citation statement)

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“…By this way Sloss (see [23)) and Dontova (see [3]) solved the Dirichlet and the Neumann problem (see [4]) in doubly and multiply connected domains.…”

Section: 'Nllo Z-( Z-h(()mentioning

confidence: 99%

A Reflection Principle and its Applications

Witte

1999

Complex Methods for Partial Differential Equations

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The property of jumps of some functions on the boundary of domains is very interesting in function and potential theory. These jumps can be investigated by a reflection principle, which is proved on weaker assumptions than the Plemelj formulas. Especially one gets rid of Holder continuity and Ljaponov conditions of the boundary. With the help of this reflection principle, a generalization of the Poisson formula is proved, if there exists a suitable reflection function. Because a suitable reflection function generally is easier to obtain than a Green's function, it is natural to ask, for which domains the Dirichlet problem can be solved by the reflection principle. If the domain is simply connected, then this is unfortunately only the case, if the boundary is a circle or straight line. Because nobody has arisen this question till now, the result is new. If the domain is multiply connected, in many cases the Dirichlet problem can be solved by reducing a system of n singular integral equations to a system of n -1 equations with the help of the reflection principle. Also an orthogonal decomposition of the space L2(0) can be obtained by the reflection principle on very weak conditions to the boundary .

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“…By this way Sloss (see [23)) and Dontova (see [3]) solved the Dirichlet and the Neumann problem (see [4]) in doubly and multiply connected domains.…”

Section: 'Nllo Z-( Z-h(()mentioning

confidence: 99%