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Transactions of the American Mathematical Society

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Temperatures in several variables: Kernel functions, representations, and parabolic boundary values

Author: John T. Kemper
Journal: Trans. Amer. Math. Soc. 167 (1972), 243-262
MSC: Primary 35K05
MathSciNet review: 0294903
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Abstract: This work develops the notion of a kernel function for the heat equation in certain regions of $ n + 1$-dimensional Euclidean space and applies that notion to the study of the boundary behavior of nonnegative temperatures. The regions in question are bounded between spacelike hyperplanes and satisfy a parabolic Lipschitz condition at points on the lateral boundary.

Kernel functions (normalized, nonnegative temperatures which vanish on the parabolic boundary except at a single point) are shown to exist uniquely. A representation theorem for nonnegative temperatures is obtained and used to establish the existence of finite parabolic limits at the boundary (except for a set of heat-related measure zero).

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  • [1] A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions. II, Proc. Cambridge Philos. Soc. 42 (1946), 1-10. MR 7, 281. MR 0014414 (7:281e)
  • [2] J. R. Hattemer, Boundary behavior of temperatures. I, Studia Math. 25 (1964/65), 111-155. MR 31 #6064. MR 0181838 (31:6064)
  • [3] R. A. Hunt and R. L. Wheeden, On the boundary values of harmonic functions, Trans. Amer. Math. Soc. 132 (1968), 307-322. MR 37 #1634. MR 0226044 (37:1634)
  • [4] -, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc. 147 (1970), 507-528. MR 0274787 (43:547)
  • [5] B. F. Jones, Jr. and C. C. Tu, Non-tangential limits for a solution of the heat equation in a two-dimensional $ {\text{Lip}}_\alpha $ region, Duke Math. J. 37 (1970), 243-254. MR 41 #4026. MR 0259388 (41:4026)
  • [6] J. T. Kemper, Kernel functions and parabolic limits for the heat equation, Thesis, Rice University, Houston, Texas, 1970. MR 0264246 (41:8842)
  • [7] -, Kernel functions and parabolic limits for the heat equation, Bull. Amer. Math. Soc. 76 (1970), 1319-1320. MR 41 #8842. MR 0264246 (41:8842)
  • [8] I. G. Petrowski, Zur Ersten Randwertaufgaben der Warmeleitungsgleichung, Compositio Math. 1 (1935), 383-419. MR 1556900

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Keywords: Heat equation, kernel function, parabolic limit
Article copyright: © Copyright 1972 American Mathematical Society

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