Sharp regularity for singular obstacle problems

Abstract

We obtain sharp local $C^{1,α}$ regularity of solutions for singular obstacle problems, Euler-Lagrange equation of which is given by $$ Δ_p u=γ(u-φ)^{γ-1}\,\text{ in }\,\{u>φ\}, $$ for $0<γ<1$ and $p\ge2$. At the free boundary $\partial\{u>φ\}$, we prove optimal $C^{1,τ}$ regularity of solutions, with $τ$ given explicitly in terms of $p$, $γ$ and smoothness of $φ$, which is new even in the linear setting.