On the regularity of the free boundary in the p-Laplacian obstacle problem

Abstract

We study the regularity of the free boundary in the obstacle for the p-Laplacian, min⁡{−Δpu,u−φ}=0 in Ω⊂Rn. Here, Δpu=div(|∇u|p−2∇u), and p∈(1,2)∪(2,∞).Near those free boundary points where ∇φ≠0, the operator Δp is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when ∇φ=0 then Δp is singular or degenerate, and nothing was known about the regularity of the free boundary at those points.Here we study the regularity of the free boundary where ∇φ=0. On the one hand, for every p≠2 we construct explicit global 2-homogeneous solutions to the p-Laplacian obstacle problem whose free boundaries have a corner at the origin. In particular, we show that the free boundary is in general not C1 at points where ∇φ=0. On the other hand, under the “concavity” assumption |∇φ|2−pΔpφ<0, we show the free boundary is countably (n−1)-rectifiable and we prove a nondegeneracy property for u at all free boundary points.