Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Dirichlet regularity of subanalytic domains


Author: Tobias Kaiser
Journal: Trans. Amer. Math. Soc. 360 (2008), 6573-6594
MSC (2000): Primary 31B25, 32B20; Secondary 03C64
DOI: https://doi.org/10.1090/S0002-9947-08-04609-6
Published electronically: July 22, 2008
MathSciNet review: 2434300
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\Omega}$ be a bounded and subanalytic domain in $ {{\mathbb{R}}^n}$, $ {n\, \geq \, 2}$. We show that the set of boundary points of $ {\Omega}$ which are regular with respect to the Dirichlet problem is again subanalytic. Moreover, we give sharp upper bounds for the dimension of the set of irregular boundary points. This enables us to decide whether the domain has a classical Green function. In dimensions 2 and 3, this is the case, given some mild and necessary conditions on the topology of the domain.


References [Enhancements On Off] (What's this?)

  • 1. D.H. Armitage, S.J. Gardiner: Classical Potential Theory. Springer Monographs in Mathematics, Springer, London, Berlin, Heidelberg, 2001. MR 1801253 (2001m:31001)
  • 2. E. Bierstone, P.D. Milman: Semianalytic and subanalytic sets. Inst. Hautes Études Sci. Publ. Math. No. 67 (1988), 5-42. MR 972342 (89k:32011)
  • 3. M. Brelot: Familles de Perron et problème de Dirichlet. Acta Litt. Sci. Szeged 9 (1939), 133-153. MR 0000734 (1:121d)
  • 4. J. Bochnak, M. Coste, M.-F. Roy: Real Algebraic Geometry. Springer, Berlin, Heidelberg, New York, 1998. MR 1659509 (2000a:14067)
  • 5. G. Comte: Équisingularité réelle, nombres de Lelong et images polaires, Ann. Sci. École Norm. Sup. (4) 33 (2000), 757-788. MR 1832990 (2002d:32040)
  • 6. J. Denef, L. van den Dries: $ p$-adic and real subanalytic sets. Ann. of Math. (2) 128 (1988), no.1, 79-138. MR 951508 (89k:03034)
  • 7. L. van den Dries: Tame Topology and o-Minimal Structures. London Mathematical Society Lecture Notes 248, Cambridge University Press, Cambridge, New York, 1998. MR 1633348 (99j:03001)
  • 8. W. Gröbner, N. Hofreiter: Integraltafel. Erster Teil: Unbestimmte Integrale. Springer, Wien, New York, 1965. MR 0028904 (10:516f)
  • 9. L.L. Helms: Introduction to potential theory. Wiley-Interscience, New York, London, Sydney, 1969. MR 0261018 (41:5638)
  • 10. H. Hironaka: Subanalytic sets. Number theory, algebraic geometry and commutative algebra, in honour of Yasuo Akizuki, 453-493, Kinokuniya, Tokyo, 1973. MR 0377101 (51:13275)
  • 11. T. Kaiser: Capacity in subanalytic geometry. Illinois Journal of Mathematics, Volume 49, Issue 3 (2005), 719-736. MR 2210256 (2007a:32006)
  • 12. K. Kurdyka: On a subanalytic stratification satisfying a Whitney property with exponent 1. Real Algebraic Geometry, Proceedings Rennes, 1991. M. Coste, ed., Lecture Notes in Mathematics 1524, Springer, Berlin, Heidelberg, New York, 1992. MR 1226263 (94f:32016)
  • 13. K. Kurdyka, G. Raby: Densité des ensembles sous-analytiques. Annales Institut Fourier 39, (1989), 753-771. MR 1030848 (90k:32026)
  • 14. K. Kurdyka, J. Xiao: John functions, quadratic integral forms and o-minimal structures. Illinois Journal of Mathematics 46 (2002), no. 4, 1089-1109. MR 1988252 (2005a:32006)
  • 15. T.L. Loi: Verdier and strict Thom stratification in o-minimal structures. Illinois Journal of Mathematics 42 (1998), no. 2, 347-356. MR 1612771 (99c:32058)
  • 16. S. Łojasiewicz: Ensembles Semianalytiques. Institut des Hautes Etudes Scientifiques. Bures sur Yvette (Seine-et-Oise), France, 1965.
  • 17. S. Łojasiewicz: Sur la géométrie semi- et sous-analytique. Annales Institut Fourier 43, 5 (1993), 1575-1595. MR 1275210 (96c:32007)
  • 18. A. Parusinski: Lipschitz stratification of subanalytic sets. Ann. Sci. École Norm. Sup. (4)27 (1994), no. 6, 661-696. MR 1307677 (96g:32017)
  • 19. E.J. McShane: Extension of range of functions. Bulletin American Mathematical Society 40, (1934), 837-842.
  • 20. O. Perron: Eine neue Behandlung der ersten Randwertaufgabe für $ \Delta u = 0$. Mathematische Zeitschrift 18, (1923), 42-54. MR 1544619
  • 21. J. Wermer: Potential Theory. Lecture Notes in Mathematics 408, Springer, Berlin, Heidelberg, New York, 1981. MR 634962 (82k:31001)
  • 22. N. Wiener: Note on a paper by O. Perron. Journal of Mathematics and Physics of the Massachusettes Institute of Technology 4, (1925), 31-32.
  • 23. N. Wiener: The Dirichlet problem. Journal of Mathematics and Physics of the Massachusettes Institute of Technology 3, (1924), 127-146.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 31B25, 32B20, 03C64

Retrieve articles in all journals with MSC (2000): 31B25, 32B20, 03C64


Additional Information

Tobias Kaiser
Affiliation: Naturwissenschaftliche Fakultät-Mathematik, University of Regensburg, Universitätsstr. 31, 93040 Regensburg, Germany
Email: tobias.kaiser@mathematik.uni-regensburg.de

DOI: https://doi.org/10.1090/S0002-9947-08-04609-6
Received by editor(s): March 23, 2006
Received by editor(s) in revised form: February 5, 2007
Published electronically: July 22, 2008
Additional Notes: This research was supported by DFG-Projekt KN202/5-1
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society