  In this work we start from the assumption that normal solid to supersolid
(NS-SS) phase transition is continuous, and develop a phenomenological Landau
theory of the transition in which superfluidity is coupled to the elasticity of
the crystalline $^4$He lattice. We find that the elasticity does not affect the
universal properties of the superfluid transition, so that in an unstressed
crystal the well-known $\lambda$-anomaly in the heat capacity of the superfluid
transition should also appear at the NS-SS transition. We also find that the
onset of supersolidity leads to anomalies in the elastic constants near the
transition; conversely, inhomogeneous strains in the lattice can induce local
variations of the superfluid transition temperature, leading to a broadened
transition.
