Traces of monotone functions in weighted Sobolev spaces (Q5954288)

Let \(\Omega\) be an open set in \(\mathbb{R}^n\). A continuous function \(u:\Omega \mathbb{R}\) is monotone if \(\max_{\overline D}u =\max_{\partial D}u\), \(\min_{\overline D}u= \min_{\partial D}u\) for each domain \(D\subset\subset\Omega\). The use of monotone functions in the Dirichlet problem goes back to H. Lebesgue in 1907. In [\textit{J. Manfredi}, \textit{E. Villamore}, J. Geom. Anal. 6, No. 3, 433-444 (1996; Zbl 0896.31002)] the authors considered traces of monotone functions in the Sobolev space \(W^{1,p}\). Now they consider the weighted Sobolev space \(W^{1,p}(B;w)\) in the unit ball \(B\) where the weight \(w\) is an \(A_q\) weight for some \(1\leq q<p/(n-1)\), see [\textit{J. Heinonen}, \textit{T. Kilpeläinen}, and \textit{O. Martio}, Nonlinear potential theory of degenerate elliptic equations (1993; Zbl 0780.31001)]. If the weight satisfies some symmetry conditions and if \(n-1<p\leq n\), then it is shown that a monotone function \(u\in W^{1,p}(B;w)\) has nontangential limits at all points of \(\partial B\) except on a set of weighted \((p,w)\)-capacity zero. For this new weighted oscillation estimates are used. In the case \(w(x)= (1-|x|)^\alpha\) the weighted \((p,w)\)-capacity can be estimated in terms of the Hausdorff dimension. The results can be used to study boundary behavior of solutions of \(\nabla A(x, \nabla u)=0\) where \(|A(x,h)|\approx|h|^{p-1}\).