Temperatures in several variables: Kernel functions, representations, and parabolic boundary values
Author:
John T. Kemper
Journal:
Trans. Amer. Math. Soc. 167 (1972), 243262
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 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 heatrelated measure zero).
 [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 0014414
(7,281e)
 [2]
J.
R. Hattemer, Boundary behavior of temperatures. I, Studia
Math. 25 (1964/1965), 111–155. MR 0181838
(31 #6064)
 [3]
Richard
A. Hunt and Richard
L. Wheeden, On the boundary values of harmonic
functions, Trans. Amer. Math. Soc. 132 (1968), 307–322. MR 0226044
(37 #1634), http://dx.doi.org/10.1090/S00029947196802260447
 [4]
Richard
A. Hunt and Richard
L. Wheeden, Positive harmonic functions on
Lipschitz domains, Trans. Amer. Math. Soc.
147 (1970),
507–527. MR 0274787
(43 #547), http://dx.doi.org/10.1090/S00029947197002747870
 [5]
B.
Frank Jones Jr. and C.
C. Tu, Nontangential limits for a solution of the heat equation in
a twodimensional 𝐿𝑖𝑝_{𝛼} region, Duke
Math. J. 37 (1970), 243–254. MR 0259388
(41 #4026)
 [6]
John
T. Kemper, Kernel functions and parabolic limits
for the heat equation, Bull. Amer. Math.
Soc. 76 (1970),
1319–1320. MR 0264246
(41 #8842), http://dx.doi.org/10.1090/S000299041970126581
 [7]
John
T. Kemper, Kernel functions and parabolic limits
for the heat equation, Bull. Amer. Math.
Soc. 76 (1970),
1319–1320. MR 0264246
(41 #8842), http://dx.doi.org/10.1090/S000299041970126581
 [8]
I.
Petrowsky, Zur ersten Randwertaufgabe der
Wärmeleitungsgleichung, Compositio Math. 1
(1935), 383–419 (German). MR
1556900
 [1]
 A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions. II, Proc. Cambridge Philos. Soc. 42 (1946), 110. MR 7, 281. MR 0014414 (7:281e)
 [2]
 J. R. Hattemer, Boundary behavior of temperatures. I, Studia Math. 25 (1964/65), 111155. 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), 307322. MR 37 #1634. MR 0226044 (37:1634)
 [4]
 , Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc. 147 (1970), 507528. MR 0274787 (43:547)
 [5]
 B. F. Jones, Jr. and C. C. Tu, Nontangential limits for a solution of the heat equation in a twodimensional region, Duke Math. J. 37 (1970), 243254. 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), 13191320. MR 41 #8842. MR 0264246 (41:8842)
 [8]
 I. G. Petrowski, Zur Ersten Randwertaufgaben der Warmeleitungsgleichung, Compositio Math. 1 (1935), 383419. MR 1556900
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197202949036
PII:
S 00029947(1972)02949036
Keywords:
Heat equation,
kernel function,
parabolic limit
Article copyright:
© Copyright 1972
American Mathematical Society
