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

2007/05/14 by Tobias Kaiser, Kaiser, Tobias
Mathematics · #03C64 #30D05 #30D60 #30E15 #32B20 #35C20 #35J25 #37E35 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Logic (math.LO) #Mathematical Dynamics and Fractals #advanced mathematical theories #math.LO #msc:03C64 #msc:30D05 #msc:30D60 #msc:30E15 #msc:32B20 #msc:35C20 #msc:35J25 #msc:37E35

paper · pdf · doi:10.48550/arxiv.0705.1926

37 pages

openalex publication_date 2007/05/14 · arxiv created 2008/07/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 π.

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