Duke Mathematical Journal

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

Tobias Kaiser

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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 π

Article information

Source
Duke Math. J., Volume 147, Number 2 (2009), 285-314.

Dates
First available in Project Euclid: 17 March 2009

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1237295910

Digital Object Identifier
doi:10.1215/00127094-2009-012

Mathematical Reviews number (MathSciNet)
MR2495077

Zentralblatt MATH identifier
1168.03028

Subjects
Primary: 03C64: Model theory of ordered structures; o-minimality 32B20: Semi-analytic sets and subanalytic sets [See also 14P15] 35J25: Boundary value problems for second-order elliptic equations 37E35: Flows on surfaces
Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX] 30D60: Quasi-analytic and other classes of functions 30E15: Asymptotic representations in the complex domain 35C20: Asymptotic expansions

Citation

Kaiser, Tobias. The Dirichlet problem in the plane with semianalytic raw data, quasi analyticity, and o-minimal structure. Duke Math. J. 147 (2009), no. 2, 285--314. doi:10.1215/00127094-2009-012. https://projecteuclid.org/euclid.dmj/1237295910


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