Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-24T03:35:14.422Z Has data issue: false hasContentIssue false

A Hausdorff measure classification of polar sets for the heat equation

Published online by Cambridge University Press:  24 October 2008

S. J. Taylor
Affiliation:
University of Virginia, U.S.A.
N. A. Watson
Affiliation:
University of Canterbury, New Zealand

Extract

Our main purpose is to give criteria for determining which subsets of Rn+1 are polar relative to the heat equation

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Besicovitch, A. S.. On the existence of subsets of finite measure of sets of infinite measure. Indag. Math. 14 (1952), 339344.Google Scholar
[2]Carleson, L.. Selected Problems on Exceptional Sets (Van Nostrand, 1967).Google Scholar
[3]Dont, M.. On the continuity of heat potentials. Časopis Pěst. Mat. 106 (1981), 156167.CrossRefGoogle Scholar
[4]Doob, J. L.. Classical Potential Theory and its Probabilistic Counterpart (Springer-Verlag, 1984).Google Scholar
[5]Hawkes, J.. Measures of Hausdorff type and stable processes. Mathematika 25 (1978), 202212.CrossRefGoogle Scholar
[6]Hayman, W. K. and Kennedy, P. B.. Subharmonic Functions, vol. 1 (Academic Press, 1976).Google Scholar
[7]Kahane, J.-P.. Points multiples des processus de Lévy symétriques stables restraints à un ensemble de valeurs du temps. Seminar on Harmonic Analysis 1981–82, pp. 74105. Publ. Math. Orsay 83, 2, Univ. Paris XI, Orsay, 1983.Google Scholar
[8]Kaufman, R. and Wu, J.-M.. Parabolic potential theory. J. Differential Equations 43 (1982), 204234.Google Scholar
[9]Kbál, J.. Hölder-continuous heat potentials. Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 51 (1971), 1719.Google Scholar
[10]Petrowsky, I.. Zur ersten Randwertaufgabe der Wärmeleitungsgleichung. Compositio Math. 1 (1934/1935), 383419.Google Scholar
[11]Rogers, C. A.. Hausdorff Measures (Cambridge University Press, 1970).Google Scholar
[12]Rogers, C. A. and Taylor, S. J.. Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8 (1961), 131.Google Scholar
[13]Taylor, S. J.. On the connexion between Hausdorff measures and generalized capacity. Proc. Cambridge Philos. Soc. 57 (1961), 524531.Google Scholar
[14]Watson, N. A.. Green functions, potentials, and the Dirichlet problem for the heat equation. Proc. London Math. Soc. 33 (1976), 251298;CrossRefGoogle Scholar
Corrigendum, Proc. London Math. Soc., 37 (1978), 3234.Google Scholar
[15]Watson, N. A.. Thermal capacity. Proc. London Math. Soc. 37 (1978), 342362.Google Scholar