The Dirichlet problem in the plane with semianalytic raw data, quasi analyticity, and o-minimal structure

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 zero lies in a certain quasi-analytic class used by Ilyashenko [21]–[23] 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 angles at the singular boundary points of the domain are irrational multiples of $\pi$