Regularity and Free Boundary Regularity for the p Laplacian

In this paper we study regularity and free boundary regularity, below the continuous threshold, for the p Laplace equation in Lipschitz and C1 domains. To formulate our results we let Ω ⊂ Rn be a bounded Lipschitz domain with constant M. Given p, 1 < p < ∞, w ∈ ∂Ω, 0 < r < r0, suppose that u is a positive p harmonic function in Ω ∩B(w, 4r), that u is continuous in Ω ̄ ∩ B̄(w, 4r) and u = 0 on ∆(w, 4r). We first prove, Theorem 1, that ∇u(y) → ∇u(x), for almost every x ∈ ∆(w, 4r), as y → x non tangentially in Ω. Moreover, ‖ log |∇u|‖BMO(∆(w,r)) ≤ c(p, n,M). If, in addition, Ω is C1 regular then we prove, Theorem 2, that log |∇u | ∈ VMO(∆(w, r)). Finally we prove, Theorem 3, that there exists M ̂ , independent of u, such that if M ≤ M ̂ and if log |∇u | ∈ VMO(∆(w, r)