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.