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ON RATIO INEQUALITIES FOR HEAT CONTENT

Published online by Cambridge University Press:  28 January 2004

BURGESS DAVIS  and
MAJID HOSSEINI
BURGESS DAVIS
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, [email protected]
MAJID HOSSEINI
Affiliation:
Department of Mathematics, State University of New York, Suite 9, 75 South Manheim Boulevard, New Paltz, NY 12561-2443, [email protected]
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Abstract

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 $H_t{(U^+)}/H_t{(U)}\,{\leq}\, H_t{(S^+)}/H_t{(S)}$ where $H_t$ 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.

Keywords

35K05 (primary)35J1060J65 (secondary)
Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 69 , Issue 1 , February 2004 , pp. 97 - 106
Copyright
The London Mathematical Society 2004

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