  We study the pinning of a two-dimensional membrane to a patterned substrate
within elastic theory both in the bending rigidity and in the strain dominated
regimes. We find that both the in-plane strains and the bending rigidity can
lead to depinning. We show from energetic arguments that the system experiences
a first order phase transition between the attached configuration to a
partially detached one when the relevant parameters of the substrate are
varied, and we construct a qualitative phase diagram. Our results are confirmed
through analytical solutions for some simple geometries of the substrate's
profile.
