REGULARITY AND FREE BOUNDARY REGULARITY FOR THE p LAPLACIAN IN LIPSCHITZ AND C1 DOMAINS

In this paper we study regularity and free boundary regularity, below the continuous threshold, for the p Laplace equation in Lipschitz and C domains. To formulate our results we let Ω ⊂ R 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 Ω̄ ∩ 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 C regular then we prove, Theorem 2, that log |∇u| ∈ V MO(∆(w, r)). Finally we prove, Theorem 3, that there exists M̂ , independent of u, such that if M ≤ M̂ and if log |∇u| ∈ V MO(∆(w, r)) then the outer unit normal to ∂Ω, n, is in V MO(∆(w, r/2)).