On bounds for the remainder term of counting functions of the Neumann Laplacian on domains with fractal boundary

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.