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UNIQUENESS OF EXTENDABLE TEMPERATURES

Part of: Higher-dimensional theory Parabolic equations and systems

Published online by Cambridge University Press:  02 October 2020

NEIL A. WATSON Open the ORCID record for NEIL A. WATSON [Opens in a new window]
NEIL A. WATSON*
Affiliation:
School of Mathematics and Statistics, University of Canterbury, Private Bag, Christchurch, New Zealand
*
e-mail: neil.watson@canterbury.ac.nz
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Abstract

Let E and D be open subsets of $\mathbb {R}^{n+1}$ such that $\overline {D}$ is a compact subset of E, and let v be a supertemperature on E. A temperature u on D is called extendable by v if there is a supertemperature w on E such that $w=u$ on D and $w=v$ on $E\backslash \overline D$ . From earlier work of N. A. Watson, [‘Extendable temperatures’, Bull. Aust. Math. Soc.100 (2019), 297–303], either there is a unique temperature extendable by v, or there are infinitely many; a necessary condition for uniqueness is that the generalised solution of the Dirichlet problem on D corresponding to the restriction of v to $\partial _eD$ is equal to the greatest thermic minorant of v on D. In this paper we first give a condition for nonuniqueness and an example to show that this necessary condition is not sufficient. We then give a uniqueness theorem involving the thermal and cothermal fine topologies and deduce a corollary involving only parabolic and coparabolic tusks.

Keywords

temperaturesupertemperaturefine topologycaloric measure

MSC classification

Primary: 35K05: Heat equation
Secondary: 31B05: Harmonic, subharmonic, superharmonic functions 31B20: Boundary value and inverse problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 2 , April 2021 , pp. 311 - 317
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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