We study homogenisation problems for divergence form equations with rapidly
sign-changing coefficients. With a focus on problems with piecewise constant,
scalar coefficients in a ($d$-dimensional) crosswalk type shape, we will
provide a limit procedure in order to understand potentially ill-posed and
non-coercive settings. Depending on the integral mean of the coefficient and
its inverse, the limits can either satisfy the usual homogenisation formula for
stratified media, be entirely degenerate or be a non-local differential
operator of 4th order. In order to mark the drastic change of nature, we
introduce the `inner spectrum’ for conductivities. We show that even though $0$
is contained in the inner spectrum for all strictly positive periods, the limit
inner spectrum can be empty. Furthermore, even though the spectrum was confined
in a bounded set uniformly for all strictly positive periods and not containing
$0$, the limit inner spectrum might have $0$ as an essential spectral point and
accumulate at $\infty$ or even be the whole of $\mathbb{C}$. This is in stark
contrast to the classical situation, where it is possible to derive upper and
lower bounds in terms of the values assumed by the coefficients in the
pre-asymptotics. In passing, we also develop a theory for Sturm–Liouville type
operators with indefinite weights, reduce the question on solvability of the
associated Sturm–Liouville operator to understanding zeros of a certain
explicit polynomial and show that generic real perturbations of piecewise
constant coefficients lead to continuously invertible Sturm–Liouville
expressions.
