  We consider a membrane both weakly and strongly adhering to a geometrically
structured substrate. The interaction potential is assumed to be local, via the
Deryagin approximation, and harmonic. Consequently, we can analytically
describe a variety of different geometries: as well as randomly rough
self-affine surfaces, smooth substrates interrupted by an isolated cylindrical
pit, a single elongated trench or a periodic array of trenches are
investigated. We present more general expressions for the adhesion energy and
membrane configuration in Fourier space and find that, compared to planar
surfaces, the adhesion energy decreases. We also highlight the possibility of
overshoots occurring in the membrane profile and look at its degree of
penetration into surface indentations.
