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A COMPARISON ESTIMATE FOR THE HEAT EQUATION WITH AN APPLICATION TO THE HEAT CONTENT OF THE S-ADIC VON KOCH SNOWFLAKE

Published online by Cambridge University Press:  01 July 1998

M. VAN DEN BERG  and
P. B. GILKEY
M. VAN DEN BERG
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW
P. B. GILKEY
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
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Abstract

Let D be an open set in Euclidean space IRm with boundary ∂D, and let ϕ[ratio ]∂D→[0, ∞) be a bounded, measurable function. Let u[ratio ]D∪∂D×[0, ∞)→[0, ∞) be the unique weak solution of the heat equation

formula here

with initial condition

formula here

and with inhomogeneous Dirichlet boundary condition

formula here

Then u(x; t) represents the temperature at a point x∈D at time t if D has initial temperature 0, while the temperature at a point x∈∂D is kept fixed at ϕ(x) for all t>0. We define the total heat content (or energy) in D at time t by

formula here

In this paper we wish to examine the effect of imposing additional cooling on some subset C on both u and ED.

Type
Notes and Papers
Information
Bulletin of the London Mathematical Society , Volume 30 , Issue 4 , July 1998 , pp. 404 - 412
Copyright
© The London Mathematical Society 1998

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