Abstract
Integral equations which contain as kernels fundamental solutions of differential operators defined on Riemann surfaces and spaces can be used advantageously for the treatment of boundary value problems when the domains have slits or cracks or overlapping parts. The present paper shows that a close relationship exists between this type of fundamental solutions of the plane potential operator and well-known series representations of harmonic functions on sectors. It turns out that, if a fundamental solution on a Riemann surface with an “optimal” number of sheets is chosen, several kernels in integral equations of the second kind for the potential problem vanish on the flanks of a sector. As a consequence, a potential problem on a domain with a corner can be formulated as an integral equation not taken over the whole boundary but only over a part of the boundary which does not contain the neighbourhood of the corner. The results apply also to slits since a slit is a corner with an aperture angle equal to 2π.
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Communicated by S. N. Atluri, January 22, 1993
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Heise, U. Integral equations for the plane potential problem on domains with a corner or a slit. Computational Mechanics 12, 1–18 (1993). https://doi.org/10.1007/BF00370481
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DOI: https://doi.org/10.1007/BF00370481