Existence of functions in weighted Sobolev spaces

Abstract

The aim of this paper is to determine when there exists a quasicontinuous Sobolev function $u \in W^{1,p}({\bf R}^n;\mu)$ whose trace $u|_{{\mathbf R}^{n-1}}$ is the characteristic function of a bounded set $E \subset {\bf R}^{n-1}$, where $d\mu(x) = |x_n|^\alpha dx$ with $-1< \alpha < p-1$. As application we discuss the existence of harmonic measures for weighted $p$-Laplacians in the unit ball.