Duke Mathematical Journal

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

Tobias Kaiser

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 17 March 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

Export citation


  • P. Ahern and X. Gong, A complete classification for pairs of real analytic curves in the complex plane with tangential intersection, J. Dyn. Control Syst. 11 (2005), 1--71.
  • D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer Monogr. Math., Springer, London, 2001.
  • G. V. Badalyan, Quasipower Series and Quasianalytic Classes of Functions, Transl. Math. Monogr. 216, Amer. Math. Soc., Providence, 2002.
  • E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5--42.
  • J. Bochnak, M. Coste, and M.-F. Roy, Real Algebraic Geometry, Ergeb. Math. Grenzgeb. (3) 36, Springer, Berlin, 1998.
  • F. Cano, J.-M. Lion, and R. Moussu, Frontière d'une hypersurface pfaffienne, Ann. Sci. École Norm. Sup. (4) 28 (1995), 591--646.
  • G. Comte, J.-M. Lion, and J.-P. Rolin, Nature log-analytique du volume des sous-analytiques, Illinois J. Math. 44 (2000), 884--888.
  • J. Denef and L. Van Den Dries, $p$-adic and real subanalytic sets, Ann. of Math. (2) 128 (1988), 79--138.
  • L. Van Den Dries, A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bull. Amer. Math. Soc. 15 (1986), 189--193.
  • —, Tame Topology and O-Minimal Structures, London Math. Soc. Lecture Notes Ser. 248, Cambridge Univ. Press, Cambridge, 1998.
  • L. Van Den Dries, A. Macintyre, and D. Marker, The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. (2) 140 (1994), 183--205.
  • L. Van Den Dries and C. Miller, On the real exponential field with restricted analytic functions, Israel J. Math. 85 (1994), 19--56.
  • —, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497--540.
  • L. Van Den Dries and P. Speissegger, The real field with convergent generalized power series, Trans. Amer. Math. Soc. 350 (1998), 4377--4421.
  • —, The field of reals with multisummable series and the exponential function, Proc. London Math. Soc. (3) 81 (2000), 513--565.
  • H. Dulac, Sur les cycles limites, Bull. Soc. Math. France 51 (1923), 45--188.
  • J. Ecalle, Théorie iterative: Introduction à la théorie des invariants holomorphes, J. Math. Pures App. 51 (1975), 183--258.
  • A. Gabrielov, Multiplicities of Pfaffian intersections and the Łojasiewicz inequality, Selecta Math. (N.S.) 1 (1995), 113--127.
  • A. Gabrielov, N. Vorobjov, and T. Zell, Betti numbers of semialgebraic and sub-Pfaffian sets, J. London Math. Soc. (2) 69 (2004), 27--43.
  • L. L. Helms, Introduction to Potential Theory, Pure Appl. Math. 21, Wiley-Interscience, New York, 1969.
  • Y. S. Ilyashenko, Limit cycles of polynomial vector fields with nondegenerate singular points on the real plane, Funct. Anal. Appl. 18 (1984), 199--209.
  • —, Finiteness Theorems for Limit Cycles, Trans. Math. Monogr. 94, Amer. Math. Soc., Providence, 1991.
  • —, Centennial history of Hilbert's 16th problem, Bull. Amer. Math. Soc. (N.S.) 39 (2002), 301--354.
  • T. Kaiser, Definability results for the Poisson equation. Adv. Geom. 6 (2006), 627--644.
  • —, Dirichlet regularity of subanalytic domains, Trans. Amer. Math. Soc. 360 (2008), 6573--6594.
  • —, The Riemann mapping theorem for semianalytic domains and o-minimality, to appear in Proc. London Math. Soc.
  • T. Kaiser, J.-P. Rolin, and P. Speissegger, Transition maps at non-resonant hyperbolic singularities are o-minimal, to appear in J. Reine Angew. Math., preprint,\arxivmath/0612745v3[math.DS]
  • M. Karpinski and A. Macintyre, A generalization of Wilkie's theorem of the complement, and an application to Pfaffian closure, Selecta Math. (N.S.) 5 (1999), 507--516.
  • A. G. Khovanskii [Hovanskiĭ], On a class of systems of transcendental equations, Soviet Math. Dokl. 22 (1980), 762--765.
  • J.-M. Lion and J.-P. Rolin, Homologie des ensembles semi-pfaffians, Ann. Inst. Fourier (Grenoble) 46 (1996), 723--741.
  • —, Intégration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques, Ann. Inst. Fourier 48 (1998), 755--767.
  • S. łOjasiewicz, Ensembles semi-analytiques, lecture notes, Inst. Hautes Études Sci., Bures-sur-Yvette, France, 1965, no. A66.765.
  • C. Pommerenke, Boundary Behaviour of Conformal Maps, Grundlehren Math. Wiss. 299, Springer, Berlin, 1992.
  • J.-P. Rolin, P. Speissegger, and A. J. Wilkie, Quasianalytic Denjoy-Carleman classes and o-minimality, J. Amer. Math. Soc. 16 (2003), 751--777.
  • M. Shiota, Geometry of Subanalytic and Semialgebraic Sets, Progr. Math. 150, Birkhäuser, Boston, 1997.
  • P. Speissegger, The Pfaffian closure of an o-minimal structure, J. Reine Angew. Math. 508 (1999), 189--211.
  • S. M. Voronin, Analytic classification of germs of conformal mappings $(\mathbf C, 0) \to (\mathbf C, 0)$ with linear part the identity, Funct. Anal. Appl. 15 (1981), 1--17.
  • —, Analytic classification of pairs of involutions and its applications, Funct. Anal. Appl. 16 (1982), 94--100.
  • W. Wasow, Asymptotic development of the solution of Dirichlet's problem at analytic corners, Duke Math. J. 24 (1957), 47--56.
  • A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), 1051--1094.
  • —, A theorem of the complement and some new o-minimal structures, Selecta Math. (N.S.) 5 (1999), 397--421.