  We study the elastic properties of incommensurate solid 4He in the limit of
zero temperature. Specifically, we calculate the pressure dependence of the
five elastic constants (C11, C12, C13, C33, and C44), longitudinal and
transversal speeds of sound, and the T = 0 Debye temperature of incommensurate
and commensurate hcp 4He using the diffusion Monte Carlo method. Our results
show that under compression the commensurate crystal is globally stiffer than
the incommensurate, however at pressures close to melting (i.e. P ~ 25 bars)
some of the elastic constants accounting for strain deformations of the hcp
basal plane (C12 and C13) are slightly larger in the incommensurate solid.
Also, we find that upon the introduction of tiny concentrations of point
defects the shear modulus of 4He (C44) undergoes a small reduction.
