A Comparison Estimate for the Heat Equation with an Application to the Heat Content of the S-Adic Von Koch Snowflake |
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Authors: | van den Berg M; Gilkey P B |
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Institution: | School of Mathematics, University of Bristol University Walk, Bristol BS8 1TW
Department of Mathematics, University of Oregon Eugene, OR 97403, USA |
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Abstract: | Let D be an open set in Euclidean space Rm with boundary D,and let : D 0, ) be a bounded, measurable function. Let u:D![{cup}](http://blms.oxfordjournals.org/math/cup.gif) Dx0, ) 0, ) be the unique weak solution of the heat equation formula] with initial condition formula] and with inhomogeneous Dirichlet boundary condition formula] Then u(x; t) represents the temperature at a point x D at timet if D has initial temperature 0, while the temperature at apoint x![isin](http://blms.oxfordjournals.org/math/isin.gif) D is kept fixed at (x) for all t>0. We define thetotal heat content (or energy) in D at time t by formula] In this paper we wish to examine the effect of imposing additionalcooling on some subset C on both u and ED. 1991 MathematicsSubject Classification 35K05, 60J65, 28A80. |
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