## Nagoya Mathematical Journal

### 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.

#### Article information

Source
Nagoya Math. J., Volume 164 (2001), 75-88.

Dates
First available in Project Euclid: 27 April 2005

https://projecteuclid.org/euclid.nmj/1114631655

Mathematical Reviews number (MathSciNet)
MR1869095

Zentralblatt MATH identifier
1023.46037

#### Citation

Futamura, Toshihide; Mizuta, Yoshihiro. Existence of functions in weighted Sobolev spaces. Nagoya Math. J. 164 (2001), 75--88. https://projecteuclid.org/euclid.nmj/1114631655

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