Estimates for solutions to equations of p-Laplace type in Ahlfors regular NTA-domains

Abstract

In this paper we study, for given p, 1<p<∞, the boundary behavior of non-negative solutions to equations of p-Laplace type of the formΔA,pu:=∇⋅((A(x)∇u⋅∇u)p/2−1A(x)∇u)=0. Concerning the matrix A we assume that A is symmetric, positive definite and that its entries are C1,γ-smooth functions. Let Ω⊂ℜn be an Ahlfors regular NTA-domain with constants C,M,r0, and let σ be the surface measure on ∂Ω. Given p,1<p<∞,w∈∂Ω, 0<r<r0, suppose that u is a positive solution to ΔA,pu=0 in Ω∩B(w,4r), that u is continuous in Ω¯∩B(w,4r) and that u=0 on ∂Ω∩B(w,4r). Among other things we prove that ∇u(x)→∇u(y), for σ-a.e. y∈∂Ω∩B(w,4r), as x→y non-tangentially in Ω, and that ‖log|∇u|‖BMO(∂Ω∩B(w,r))⩽c(p,n,M,C,α,αˆ,β,γ) where (α,αˆ,β,γ) are the constants giving the structural restrictions on our class of operators. Furthermore, assuming in addition that Ω⊂ℜn is uniformly δ-approximable by Lipschitz graphs, we prove Hölder continuity of the ratio of two solutions which vanish on a portion of the boundary. Our main contribution is the extension of a toolbox developed by the second author and John Lewis in the context of p-harmonic functions, to non-negative solutions to the equation ΔA,pu=0 in Lipschitz domains.