The Dirichlet problem in the plane with semianalytic raw data, quasianalyticity and o-minimal structures

Details The Dirichlet problem in the plane with semianalytic raw data, quasianalyticity and o-minimal structures The Dirichlet problem in the plane with semianalytic raw data, quasianalyticity and o-minimal structures

2007-05-14

arxiv.org/abs/0705.1926v2

Mathematics

Logic

37 pages

Scientific paper

We investigate the Dirichlet solution for a semianalytic continuous function on the boundary of a semianalytic bounded domain in the plane. We show that the germ of the Dirichlet solution at a boundary point with angle greater than 0 lies in a certain quasianalytic class used by Ilyashenko in his work on Hilbert's 16th problem. With this result we can prove that the Dirichlet solution is definable in an o-minimal structure if the angle at a singular boundary point of the domain is an irrational multiple of $\pi$.

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