Regularity and well-posedness of some quasi-linear elliptic and parabolic problems with nonlinear general Wentzell boundary conditions on nonsmooth domains

Abstract

We show that on a bounded domain Ω⊂RN with Lipschitz continuous boundary ∂Ω, if 2N/(N+2)<p≤N−1, then weak solutions of the quasi-linear elliptic equation −Δpu+|u|p−2u+α1(x,u)=f in Ω with the general Wentzell boundary conditions −Δp,Γu+|∇u|p−2∂νu+|u|p−2u+α2(x,u)=g weakly on ∂Ω, are globally Hölder continuous on Ω¯. Here, Δp and Δp,Γ denote the p-Laplace operator, and the p-Laplace–Beltrami operator on the boundary, respectively. The functions α1(x,⋅) (for x∈Ω) and α2(x,⋅) (for x∈∂Ω) are continuous on R and satisfy a certain growth condition. We also obtain that a realization of the operator Δpu−α1(x,u) in C(Ω¯) with the boundary conditions [Δpu−α1(x,u)]|∂Ω−Δp,Γu+|∇u|p−2∂νu+α2(x,u)=0 on ∂Ω generates a strongly continuous semigroup of nonlinear operators on C(Ω¯).