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THERMIC MINORANTS AND REDUCTIONS OF SUPERTEMPERATURES

Part of: Higher-dimensional theory Parabolic equations and systems

Published online by Cambridge University Press:  06 March 2015

NEIL A. WATSON
NEIL A. WATSON*
Affiliation:
School of Mathematics and Statistics, University of Canterbury, Private Bag, Christchurch, New Zealand email [email protected]
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Abstract

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Let $u$ be a supertemperature on an open set $E$, and let $v$ be a related temperature on an open subset $D$ of $E$. For example, $v$ could be the greatest thermic minorant of $u$ on $D$, if it exists. Putting $w=u$ on $E\setminus D$ and $w=v$ on $D$, we investigate whether $w$, or its lower semicontinuous smoothing, is a supertemperature on $E$. We also give a representation of the greatest thermic minorant on $E$, if it exists, in terms of PWB solutions on an expanding sequence of open subsets of $E$ with union $E$.  In addition, in the case of a nonnegative supertemperature, we prove inequalities that relate reductions to Dirichlet solutions. We also prove that the value of any reduction at a given time depends only on earlier times.

Keywords

thermic minorantsupertemperaturereductionPWB solution

MSC classification

Primary: 35K05: Heat equation
Secondary: 31B05: Harmonic, subharmonic, superharmonic functions 31B20: Boundary value and inverse problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 99 , Issue 1 , August 2015 , pp. 128 - 144
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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