Sharp regularity for singular obstacle problems

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