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

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.