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 $\pi$
@article{1237295910,
author = {Kaiser, Tobias},
title = {The Dirichlet problem in the plane with semianalytic raw data, quasi analyticity, and o-minimal structure},
journal = {Duke Math. J.},
volume = {146},
number = {1},
year = {2009},
pages = { 285-314},
language = {en},
url = {http://dml.mathdoc.fr/item/1237295910}
}
Kaiser, Tobias. The Dirichlet problem in the plane with semianalytic raw data, quasi analyticity, and o-minimal structure. Duke Math. J., Tome 146 (2009) no. 1, pp. 285-314. http://gdmltest.u-ga.fr/item/1237295910/