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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
DOI: https://doi.org/10.1090/S0002-9947-1972-0294903-6
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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0294903-6
Keywords: Heat equation, kernel function, parabolic limit
Article copyright: © Copyright 1972 American Mathematical Society

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