  We study analytically and numerically the process of indentation of
cylindrical rigid indenter with concave face in form of a power-law function.
In the well-known case of a parabolic concave indenter, the contact starts at
sharp edges of the indenter and spreads inwards with increasing indentation
depth. For all profiles with the exponent larger than 2, the contact area first
spreads from the boarder inwards, but then a contact is established in the
center of the indenter. Finally, the outer ring spreads inwards and the central
contact area outwards until the complete contact is achieved. The critical
indentation depth for the full contact is calculated ones proceeding from the
full contact and looking for the condition of vanishing pressure and also
proceeding from incomplete contact (in this case numerically, using Boundary
Element Method). The results of both approaches coincide.
