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
HTML articles powered by AMS MathViewer

by John T. Kemper

Trans. Amer. Math. Soc. 167 (1972), 243-262

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

PDF | 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).

References

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 14414, DOI 10.1017/s0305004100022660
J. R. Hattemer, Boundary behavior of temperatures. I, Studia Math. 25 (1964/65), 111–155. MR 181838, DOI 10.4064/sm-25-1-111-155
Richard A. Hunt and Richard L. Wheeden, On the boundary values of harmonic functions, Trans. Amer. Math. Soc. 132 (1968), 307–322. MR 226044, DOI 10.1090/S0002-9947-1968-0226044-7
Richard A. Hunt and Richard L. Wheeden, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc. 147 (1970), 507–527. MR 274787, DOI 10.1090/S0002-9947-1970-0274787-0
B. Frank Jones Jr. and C. C. Tu, Non-tangential limits for a solution of the heat equation in a two-dimensional $\textrm {Lip}_{\alpha }$ region, Duke Math. J. 37 (1970), 243–254. MR 259388
John T. Kemper, Kernel functions and parabolic limits for the heat equation, Bull. Amer. Math. Soc. 76 (1970), 1319–1320. MR 264246, DOI 10.1090/S0002-9904-1970-12658-1
John T. Kemper, Kernel functions and parabolic limits for the heat equation, Bull. Amer. Math. Soc. 76 (1970), 1319–1320. MR 264246, DOI 10.1090/S0002-9904-1970-12658-1
I. Petrowsky, Zur ersten Randwertaufgabe der Wärmeleitungsgleichung, Compositio Math. 1 (1935), 383–419 (German). MR 1556900