A Comparison Estimate for the Heat Equation with an Application to the Heat Content of the S -Adic Von Koch Snowflake

Abstract

Let D be an open set in Euclidean space Rm 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 ED. 1991 Mathematics Subject Classification 35K05, 60J65, 28A80.