  This paper is devoted to the homogenization of the heat conduction equation,
with a homogeneous Dirichlet boundary condition, having a periodically
oscillating thermal conductivity and a vanishing volumetric heat capacity. A
homogenization result is established by using the evolution settings of
multiscale and very weak multiscale convergence. In particular, we investigate
how the relation between the volumetric heat capacity and the microscopic
structure effects the homogenized problem and its associated local problem. It
turns out that the properties of the microscopic geometry of the problem give
rise to certain special effects in the homogenization result.
