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 -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