Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-25T01:53:55.304Z Has data issue: false hasContentIssue false
  • English
  • Français

CALORIC MEASURE FOR ARBITRARY OPEN SETS

Published online by Cambridge University Press:  04 March 2013

NEIL A. WATSON
NEIL A. WATSON*
Affiliation:
Department of Mathematics and Statistics, University of Canterbury, Private Bag, Christchurch, New Zealand email n.watson@math.canterbury.ac.nz
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a systematic treatment of caloric measure for arbitrary open sets. The caloric measure is defined only on the essential boundary of the set. Our main result gives criteria for the resolutivity of essential boundary functions, and their integral representation in terms of caloric measure. We also characterize the caloric measure null sets in terms of the boundary singularities of nonnegative supertemperatures.

Keywords

hypotemperaturehypertemperaturecaloric measureDirichlet problem
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 3 , June 2012 , pp. 391 - 407
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc.

References

Armitage, D. H. and Gardiner, S. J., Classical Potential Theory (Springer, London, 2001).CrossRefGoogle Scholar
Bousch, T. and Heurteaux, Y., ‘Caloric measure on domains bounded by Weierstrass-type graphs’, Ann. Acad. Sci. Fenn. Math. 25 (2000), 501–522.Google Scholar
Doob, J. L., Classical Potential Theory and its Probabilistic Counterpart, Grundlehren der mathematischen Wissenschaften, 262 (Springer, New York, 1984).CrossRefGoogle Scholar
Fabes, E. and Salsa, S., ‘Estimates of caloric measure and the initial-Dirichlet problem for the heat equation in Lipschitz cylinders’, Trans. Amer. Math. Soc. 279 (1983), 635–650.CrossRefGoogle Scholar
Fabes, E. B., Garofalo, N. and Salsa, S., ‘Comparison theorems for temperatures in noncylindrical domains’, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. 77 (1984), 1–12.Google Scholar
Heurteaux, Y., ‘Mesure harmonique et l’equation de la chaleur’, Ark. Mat. 34 (1996), 119–139.CrossRefGoogle Scholar
Hofmann, S., Lewis, J. L. and Nyström, K., ‘Caloric measure in parabolic flat domains’, Duke Math. J. 122 (2004), 281–346.CrossRefGoogle Scholar
Kaufman, R. and Wu, J.-M., ‘Singularity of parabolic measures’, Compositio Math. 40 (1980), 243–250.Google Scholar
Kaufman, R. and Wu, J.-M., ‘Parabolic potential theory’, J. Differential Equations 43 (1982), 204–234.CrossRefGoogle Scholar
Kaufman, R. and Wu, J.-M., ‘Parabolic measure on domains of class Lip$\frac{1}{2} $’, Compositio Math. 66 (1988), 201–207.Google Scholar
Kaufman, R. and Wu, J.-M., ‘Dirichlet problem of heat equation for ${C}^{2} $ domains’, J. Differential Equations 80 (1989), 14–31.CrossRefGoogle Scholar
Kemper, J. T., ‘Temperatures in several variables: Kernel functions, representations, and parabolic boundary values’, Trans. Amer. Math. Soc. 167 (1972), 243–262.CrossRefGoogle Scholar
Lewis, J. L. and Silver, J., ‘Parabolic measure and the Dirichlet problem for the heat equation in two dimensions’, Indiana Univ. Math. J. 37 (1988), 801–839.CrossRefGoogle Scholar
Nyström, K., ‘The Dirichlet problem for second order parabolic operators’, Indiana Univ. Math. J. 46 (1997), 183–245.CrossRefGoogle Scholar
Suzuki, N., ‘On the essential boundary and supports of harmonic measures for the heat equation’, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), 381–385.CrossRefGoogle Scholar
Watson, N. A., ‘Green functions, potentials, and the Dirichlet problem for the heat equation’, Proc. London Math. Soc. 33 (1976), 251–298.CrossRefGoogle Scholar
Watson, N. A., ‘A unifying definition of a subtemperature’, New Zealand J. Math. 38 (2008), 197–223.Google Scholar
Wu, J.-M. G., ‘On parabolic measures and subparabolic functions’, Trans. Amer. Math. Soc. 251 (1979), 171–185; 259 (1980), 636 (erratum).CrossRefGoogle Scholar
Wu, J.-M., ‘On heat capacity and parabolic measure’, Math. Proc. Cambridge Phil. Soc. 102 (1987), 163–172.CrossRefGoogle Scholar
Wu, J.-M., ‘An example on null sets of parabolic measures’, Proc. Amer. Math. Soc. 107 (1989), 949–961.CrossRefGoogle Scholar