ON THE REGULARITY OF THE FREE BOUNDARY IN THE p-LAPLACIAN OBSTACLE PROBLEM

We study the regularity of the free boundary in the obstacle for the p-Laplacian, min { −∆pu, u − φ } = 0 in Ω ⊂ R. Here, ∆pu = div ( |∇u|p−2∇u ) , and p ∈ (1, 2) ∪ (2,∞). Near those free boundary points where ∇φ 6= 0, the operator ∆p is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when ∇φ = 0 then ∆p 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 6= 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 C at points where ∇φ = 0. On the other hand, under the “concavity” assumption |∇φ|∆pφ < 0, we show the free boundary is countably (n− 1)-rectifiable and we prove a nondegeneracy property for u at all free boundary points.