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

Kemper, John T.

doi:10.1090/S0002-9947-1972-0294903-6

Temperatures in several variables: Kernel functions, representations, and parabolic boundary values
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Trans. Amer. Math. Soc. 167 (1972), 243-262 Request permission

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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