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An example on null sets of parabolic measures


Author: Jang-Mei Wu
Journal: Proc. Amer. Math. Soc. 107 (1989), 949-961
MSC: Primary 35K20; Secondary 31A25
DOI: https://doi.org/10.1090/S0002-9939-1989-0986653-3
MathSciNet review: 986653
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Abstract: A set $ E$ on the real line of Hausdorff dimension 1 is constructed, such that the graph of $ E$ on the boundary of any $ {C^2}$ domain $ \left\{ {t > \tau (x)} \right\}$, or on the boundary of any $ {\text{Lip}}\tfrac{1}{2}$ domain $ \left\{ {x > \chi (t)} \right\}$, is null with respect to the parabolic measure associated with any parabolic operator $ L = a(x,t){\partial ^2}/\partial {x^2} - \partial /\partial t$ on $ {{\mathbf{R}}^2};a$ is Hölder continuous and $ 0 < {\Lambda _1} \leq a \leq {\Lambda _2} < \infty $ for some constants $ {\Lambda _1}$ and $ {\Lambda _2}$.


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

  • [1] E. Fabes and C. Kenig, Examples of singular parabolic measures and singular transition probability density, Duke Math. J. 48 (1981), 845-856. MR 782580 (86j:35081)
  • [2] R. Kaufman and J.-M. Wu, Parabolic potential theory, J. Diff. Eqs. 43 (1982), 204-234. MR 647063 (83d:31006)
  • [3] -, Parabolic measure on domains of class $ {\text{Lip}}\tfrac{1}{2}$, Compositio Math. 65 (1988), 201-207. MR 932644 (89g:31001)
  • [4] -, An example of highly singular parabolic measure, Ann. Probability 16 (1988), 1821-1831. MR 958218 (90b:35113)
  • [5] -, Dirichlet problem of heat equation for $ {C^2}$ domains, J. Diff. Eqs., (to appear).
  • [6] J. L. Lewis and J. Silver, Parabolic measure and the Dirichlet problem for the heat equation in two dimensions, preprint. MR 982831 (90e:35079)
  • [7] I. G. Petrowsky, Zur ersten Randwertaufgaben der Warmeleitungsgleichung, Compositio Math. 1 (1935), 383-419. MR 1556900
  • [8] N. A. Watson, Green functions, potentials and the Dirichlet problem for the heat equation, Proc. London Math. Soc. 33 (3) (1976), 251-298. MR 0425145 (54:13102)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0986653-3
Article copyright: © Copyright 1989 American Mathematical Society

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