Sharp regularity for singular obstacle problems

We obtain sharp local C-1,C-alpha regularity of solutions for singular obstacle problems, Euler-Lagrange equation of which is given by Delta p(u) = gamma (u - phi)(gamma-1) in {u >phi}, for 0 < gamma < 1 and p >= 2. At the free boundary partial derivative{u > phi}, we prove optimal C1,tau regularity of solutions, with tau given explicitly in terms of p,gamma and smoothness of phi, which is new even in the linear setting.