Sharp regularity for singular obstacle problems

Abstract

We obtain sharp local C1,α regularity of solutions for singular obstacle problems, Euler-Lagrange equation of which is given by Δpu=γ(u-φ)γ-1in{u>φ},for 0 < γ< 1 and p≥ 2. At the free boundary ∂{ u> φ} , we prove optimal C1,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.