  We provide a new constructive method for obtaining explicit remainder
estimates of eigenvalue counting functions of Neumann Laplacians on domains
with fractal boundary. This is done by establishing estimates for first
non-trivial eigenvalues through Rayleigh quotients. A main focus lies on
domains whose boundary can locally be represented as a limit set of an IFS,
with the classic Koch snowflake and certain Rohde snowflakes being prototypical
examples, to which the new method is applied. Central to our approach is the
construction of a novel foliation of the domain near its boundary.
