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Journal of the London Mathematical Society 2004 69(1):97-106; doi:10.1112/S0024610703004964
© 2004 by London Mathematical Society
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© The London Mathematical Society

On Ratio Inequalities for Heat Content

Burgess Davis and Majid Hosseini

Department of Mathematics, Purdue University West Lafayette, IN 47907, USA, bdavis{at}stat.purdue.edu
Department of Mathematics, State University of New York Suite 9, 75 South Manheim Boulevard, New Paltz, NY 12561-2443, USA, hosseinm{at}newpaltz.edu

Received 3 September 2002. Revision received 29 April 2003.

Let U be a domain, convex in x and symmetric about the y-axis, which is contained in a centered and oriented rectangle S. It is proved that Ht(U+)/Ht(U)≤Ht(S+)/Ht(S) where Ht stands for heat content, that is, the remaining heat in the domain at time t if it initially has uniform temperature 1, with Dirichlet boundary conditions, where A+=A{cap}{(x,y):x>0}. It is also shown that the analog of this inequality holds for some other Schrödinger operators.


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