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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
Author(s) -
van den Berg M.,
Gilkey P. B.
Publication year - 1998
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609398004469
Subject(s) - mathematics , heat equation , bounded function , euclidean space , heat flow , boundary (topology) , open set , dirichlet distribution , dirichlet boundary condition , combinatorics , content (measure theory) , function (biology) , mathematical analysis , boundary value problem , thermodynamics , physics , thermal , evolutionary biology , biology
Let D be an open set in Euclidean space R m with boundary ∂ D , and let φ:∂ D →[0, ∞) be a bounded, measurable function. Let u : D ∪∂ D ×[0, ∞)→[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 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] In this paper we wish to examine the effect of imposing additional cooling on some subset C on both u and E D . 1991 Mathematics Subject Classification 35K05, 60J65, 28A80.