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 ⁡ { − Δ p u , u − φ } = 0 in Ω ⊂ R n . Here, Δ p u = div ( | ∇ u | p − 2 ∇ u ) , and p ∈ ( 1 , 2 ) ∪ ( 2 , ∞ ) . Near those free boundary points where ∇ φ ≠ 0 , the operator Δ p is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when ∇ φ = 0 then Δ 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 ≠ 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 1 at points where ∇ φ = 0 . On the other hand, under the “concavity” assumption | ∇ φ | 2 − p Δ p φ < 0 , we show the free boundary is countably ( n − 1 ) -rectifiable and we prove a nondegeneracy property for u at all free boundary points. [ABSTRACT FROM AUTHOR]