Dirichlet regularity of subanalytic domains

Kaiser, Tobias

doi:10.1090/S0002-9947-08-04609-6

Dirichlet regularity of subanalytic domains
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by Tobias Kaiser PDF

Trans. Amer. Math. Soc. 360 (2008), 6573-6594 Request permission

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

David H. Armitage and Stephen J. Gardiner, Classical potential theory, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2001. MR 1801253, DOI 10.1007/978-1-4471-0233-5
Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5–42. MR 972342
M. Brelot, Familles de Perron et problème de Dirichlet, Acta Litt. Sci. Szeged 9 (1939), 133–153 (French). MR 734
Jacek Bochnak, Michel Coste, and Marie-Françoise Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36, Springer-Verlag, Berlin, 1998. Translated from the 1987 French original; Revised by the authors. MR 1659509, DOI 10.1007/978-3-662-03718-8
Georges Comte, Équisingularité réelle: nombres de Lelong et images polaires, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 6, 757–788 (French, with English and French summaries). MR 1832990, DOI 10.1016/S0012-9593(00)01052-1
J. Denef and L. van den Dries, $p$-adic and real subanalytic sets, Ann. of Math. (2) 128 (1988), no. 1, 79–138. MR 951508, DOI 10.2307/1971463
Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348, DOI 10.1017/CBO9780511525919
Wolfgang Gröbner and Nikolaus Hofreiter, Integraltafel. Erster Teil. Unbestimmte Integrale, Springer-Verlag, Vienna, 1949 (German). MR 0028904
L. L. Helms, Introduction to potential theory, Pure and Applied Mathematics, Vol. XXII, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1969. MR 0261018
Heisuke Hironaka, Subanalytic sets, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 453–493. MR 0377101
Tobias Kaiser, Capacity in subanalytic geometry, Illinois J. Math. 49 (2005), no. 3, 719–736. MR 2210256
Krzysztof Kurdyka, On a subanalytic stratification satisfying a Whitney property with exponent $1$, Real algebraic geometry (Rennes, 1991) Lecture Notes in Math., vol. 1524, Springer, Berlin, 1992, pp. 316–322. MR 1226263, DOI 10.1007/BFb0084630
K. Kurdyka and G. Raby, Densité des ensembles sous-analytiques, Ann. Inst. Fourier (Grenoble) 39 (1989), no. 3, 753–771 (French, with English summary). MR 1030848
K. Kurdyka and J. Xiao, John functions, quadratic integral forms and o-minimal structures, Illinois J. Math. 46 (2002), no. 4, 1089–1109. MR 1988252
Ta Lê Loi, Verdier and strict Thom stratifications in o-minimal structures, Illinois J. Math. 42 (1998), no. 2, 347–356. MR 1612771
S. Łojasiewicz: Ensembles Semianalytiques. Institut des Hautes Etudes Scientifiques. Bures sur Yvette (Seine-et-Oise), France, 1965.
Stanislas Łojasiewicz, Sur la géométrie semi- et sous-analytique, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 5, 1575–1595 (French, with English and French summaries). MR 1275210
Adam Parusiński, Lipschitz stratification of subanalytic sets, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 6, 661–696. MR 1307677
E.J. McShane: Extension of range of functions. Bulletin American Mathematical Society 40, (1934), 837-842.
Oskar Perron, Eine neue Behandlung der ersten Randwertaufgabe für $\Delta u=0$, Math. Z. 18 (1923), no. 1, 42–54 (German). MR 1544619, DOI 10.1007/BF01192395
John Wermer, Potential theory, 2nd ed., Lecture Notes in Mathematics, vol. 408, Springer, Berlin, 1981. MR 634962
N. Wiener: Note on a paper by O. Perron. Journal of Mathematics and Physics of the Massachusettes Institute of Technology 4, (1925), 31-32.
N. Wiener: The Dirichlet problem. Journal of Mathematics and Physics of the Massachusettes Institute of Technology 3, (1924), 127-146.