A multidirectional Dirichlet problem



B.E.J. Dahlberg’s theorems on the mutual absolute continuity of harmonic and surface measures, and on the unique solvability of the Dirichlet problem for Laplace’s equation with data taken in Lp spaces p > 2 − δ are extended to compact polyhedral domains of ℝn. Consequently, for q < 2 + δ, Dahlberg’s reverse Hölder inequality for the density of harmonic measure is established for compact polyhedra that additionally satisfy the Harnack chain condition. It is proved that a compact polyhedral domain satisfies the Harnack chain condition if its boundary is a topological manifold. The double suspension of the Mazur manifold is invoked to indicate that perhaps such a polyhedron need not itself be a manifold with boundary; see the footnote in Section 9. A theorem on approximating compact polyhedra by Lipschitz domains in a certain weak sense is proved, along with other geometric lemmas.

Math Subject Classifications

35J25 31B25 

Key Words and Phrases

Complexes Curtis-Zeeman manifold Lipschitz Mazur manifold NTA PL polyhedron reverse Hölder two-brick 


  1. [1]
    Bing, R.H.Some Aspects of the Topology of 3-ManifoldsRelated to the Poincaré Conjecture, Lectures on modern mathematics, Vol. II, John Wiley & Sons, New York, (1964), 93–128. MR 30 #2474.Google Scholar
  2. [2]
    Bing, R.H. The geometric topology of 3-manifolds, American Mathematical Society Colloquium Publications, Vol. 40,AMS, (1983).Google Scholar
  3. [3]
    Cannon, J.W. Shrinking cell-like decompositions of manifolds, Codimension three,Ann. Math. (2),110(1), 83–112, (1979). MR 80j:57013.CrossRefMathSciNetGoogle Scholar
  4. [4]
    Coifman, R.R. and Weiss, G. Extensions of hardy spaces and their use in analysis,Bull. Am. Math. Soc.,83(4), 569–645, (1977).MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Curtis, M.L. and Zeeman, E.C. On the polyhedral Schoenflies theorem,Proc. Am. Math. Soc.,11, 888–889, (1960). MR 22 #9974.CrossRefMathSciNetGoogle Scholar
  6. [6]
    Dahlberg, B.E.J. Estimates of harmonic measure,Arch. Rational Mech. Anal.,65(3), 275–288, (1977). MR 57 #6470.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    Dahlberg, B.E.J. On the Poisson integral for Lipschitz and C1-domains,Studia Math.,66(1), 13–24, (1979). MR 81g:31007MATHMathSciNetGoogle Scholar
  8. [8]
    Dahlberg, B.E.J. and Kenig, C.E. Hardy spaces and the Neumann problem inL p for Laplace’s equation in Lipschitz domains,Ann. Math. (2),125(3), 437–465, (1987). MR 88d:35044.CrossRefMathSciNetGoogle Scholar
  9. [9]
    Dugundji,Topology, Allyn and Bacon, (1966).Google Scholar
  10. [10]
    Fabes, E.B., Jodeit, M., and Riviere, N.M. Potential techniques for boundary value problems on C1 domains,Acta Math.,141 165–186, (1978).MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Fefferman, C. and Stein, E.M.h p spaces of several variables,Acta Math.,129 137–193, (1972).MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Glaser, L.C.Geometrical Combinatorial Topology, Vol. I, Van Nostrand Reinhold, (1970).Google Scholar
  13. [13]
    Helms, L.L.Introduction to Potential Theory, Wiley-Interscience, (1969).Google Scholar
  14. [14]
    Hurewicz, W. and Wallman, H.Dimension Theory, Princeton University Press, (1941).Google Scholar
  15. [15]
    Hunt, R.A. and Wheeden, R.L. On the boundary values of harmonic functions,Trans. Am. Math. Soc.,132, 307–322, (1968).MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Hunt, R.A. and Wheeden, R.L. Positive harmonic functions on Lipschitz domains,Trans. Am. Math. Soc.,147 507–527, (1970). MR 43 #547.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Jerison, D.S. and Kenig, C.E. Boundary behavior of harmonic functions in nontangentially accessible domains,Adv. in Math.,46, 80–147, (1982).MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Jerison, D.S. and Kenig, C.E. Boundary value problems on lipschitz domains, MAA Studies in Mathematics,Stud. Part. Diff. Eq.,23, 1–68, (1982).MathSciNetGoogle Scholar
  19. [19]
    Jones, P.W. A geometric localization theorem,Adv. in Math.,46(1), 71–79, (1982). MR 84d:31005a.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Kenig, C.E. Harmonic analysis techniques for second order elliptic boundary value problems,AMS, (1994).Google Scholar
  21. [21]
    Kirby and Lickorish, personal communication, (2001).Google Scholar
  22. [22]
    Král, J. and Wendland, W. Some examples concerning applicability of the Fredholm-Radon method in potential theory,Apl. Mat.,31(4), 293–308, (1986). MR 88e:31011MATHMathSciNetGoogle Scholar
  23. [23]
    Mazur, B. A note on some contractible 4-manifolds,Ann. Math.,73(1), 221–228, (1961).CrossRefMathSciNetGoogle Scholar
  24. [24]
    Mitchinson, D.Celebrating Moore, University of California Press, (1998).Google Scholar
  25. [25]
    Moise, E.E.Geometric Topology in Dimensions 2 and 3, Springer-Verlag, (1977).Google Scholar
  26. [26]
    Nečas, J. Sur les domaines du type N,Czechoslovak Math. J.,12, 274–287, (1962).MathSciNetGoogle Scholar
  27. [27]
    Pipher, J. and Verchota, G.C. Dilation invariant estimates and the boundary Garding inequality for higher order elliptic operators,Ann. Math.,142, 1–38, (1995).MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    Rourke, C.P. and Sanderson,B. J. Introduction to Piecewise-Linear Topology, Springer-Verlag, (1972).Google Scholar
  29. [29]
    Rudin, M.E. An unshellable triangulation of a tetrahedron,Bull. Am. Math. Soc.,64, 90–91, (1958). MR 20 #3535.MATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    Rushing, T.B.Topological Embeddings, Academic Press, (1983).Google Scholar
  31. [31]
    Semmes, S. Good metric spaces without good parameterizations,Rev. Mat. Iberoamericana,12(1), 187–275, (1996). MR 97e:57025.MATHMathSciNetGoogle Scholar
  32. [32]
    Siebenmann, L. and Sullivan, D. On complexes that are Lipschitz manifolds, Geometric topology, (Proc. Georgia Topology Conf., Athens, Ga., 1977), Academic Press, New York, 503–525, (1979). MR 80h:57027.Google Scholar
  33. [33]
    Stein, E.M.Singular Integrals and Differentiability Properties of Functions, Princeton University Press, (1970).Google Scholar
  34. [34]
    Thurston, W.P.Three-Dimensional Geometry and Topology, Princeton University Pres, (1997).Google Scholar
  35. [35]
    Verchota, G.C. Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains,J. Funct. Anal.,59(3), 572–611, (1984).MATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    Verchota, G.C. The use of Rellich identities on certain nongraph boundaries, Harmonic analysis and boundary value problems, (Fayetteville, AR, 2000),Am. Math. Soc., Providence, RI, 127–138, (2001). MR 1 840 431Google Scholar
  37. [37]
    Verchota, G.C. and Vogel, A.L. The multidirectional Neumann problem in ℝ4, in progress, (2002).Google Scholar
  38. [38]
    Yemelichev, V.A., Kovalev, M.M., and Kravstov, M.K.Polytopes, Graphs, and Optimization, Cambridge University Press, (1984).Google Scholar
  39. [39]
    Ziegler, G.M. Shelling polyhedral 3-balls and 4-polytopes,Disc. Comput. Geom.,19(2), 159–174, (1998). MR 99b:52022MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2003

Authors and Affiliations

  1. 1.Department of MathematicsSyracuse UniversitySyracuse

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