Illinois Journal of Mathematics

Traces of monotone functions in weighted Sobolev spaces

Juan J. Manfredi and Enrique Villamor

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Consider monotone functions $u\colon \mathbb{B}^n\to \mathbb{R}$ in the weighted Sobolev space $W^{1,p}(\mathbb{B}^n;w)$, where $n-1<p\leq n$ and $w$ is a weight in the class $A_q$ for some $1\leq q<{{p}/({n-1})}$ which has a certain symmetry property with respect to $\partial \mathbb{B}^n$. We prove that $u$ has nontangential limits at all points of $\partial \mathbb{B}^n$ except possibly those on a set $E$ of weighted $(p,w)$-capacity zero. The proof is based on a new weighted oscillation estimate (Theorem 1) that may be of independent interest. In the special case $w(x)=|1-|x||^\alpha$, the weighted $(p,w)$-capacity of a ball can be easily estimated to conclude that the Hausdorff dimension of the set $E$ is smaller than or equal to $\alpha+n-p$, where $0\leq \alpha<{(p-(n-1))}/{(n-1)}$.

Article information

Illinois J. Math., Volume 45, Number 2 (2001), 403-422.

First available in Project Euclid: 13 November 2009

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Zentralblatt MATH identifier

Primary: 31B25: Boundary behavior
Secondary: 30C65: Quasiconformal mappings in $R^n$ , other generalizations 35J25: Boundary value problems for second-order elliptic equations 35J67: Boundary values of solutions to elliptic equations 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems


Manfredi, Juan J.; Villamor, Enrique. Traces of monotone functions in weighted Sobolev spaces. Illinois J. Math. 45 (2001), no. 2, 403--422. doi:10.1215/ijm/1258138347.

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