### Asymptotic paths for subsolutions of quasilinear elliptic equations.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Back to Simple Search
# Advanced Search

In this paper a quite complete picture is given of the absolute continuity on the boundary of a quasiconformal map B → D, where B is the unit 3-ball and D is a Jordan domain in R with boundary 2-rectifiable in the sense of geometric measure theory. Moreover, examples are constructed, for each n ≥ 3, showing that quasiconformal maps from the unit n-ball onto Jordan domains with boundary (n - 1)-rectifiable need not have absolutely continuous boundary values.

In this paper we study quasiconformal homeomorphisms of the unit ball B = B = {x ∈ R: |x| < 1} of R onto John domains. We recall that John domains were introduced by F. John in his study of rigidity of local quasi-isometries [J]; the term John domain was coined by O. Martio and J. Sarvas seventeen years later [MS]. From the various equivalent characterizations we shall adapt the following definition based on diameter carrots, cf. [V4], [V5], [NV].

We use a recent theorem of Semmes to resolve some questions about the boundary absolute continuity of quasiconformal maps in space.

We give an example of a ${\mathcal{C}}^{3-\u03f5}$-smooth quasiregular mapping in 3-space with nonempty branch set. Moreover, we show that the branch set of an arbitrary quasiregular mapping in-space has Hausdorff dimension quantitatively bounded away from . By using the second result, we establish a new, qualitatively sharp relation between smoothness and branching.

It is shown that the $(1,p)$-fine topology defined via a Wiener criterion is the coarsest topology making all supersolutions to the $p$-Laplace equation $$\mathrm{div}\phantom{\rule{0.166667em}{0ex}}\left(\right|\nabla u{|}^{p-2}\nabla u)=0$$ continuous. Fine limits of quasiregular and BLD mappings are also studied.

We consider quasiconformal mappings in the upper half space ${\mathbb{R}}_{+}^{n+1}$ of ${\mathbb{R}}^{n+1}$, $n\ge 2$, whose almost everywhere defined trace in ${\mathbb{R}}^{n}$ has distributional differential in ${L}^{n}\left({\mathbb{R}}^{n}\right)$. We give both geometric and analytic characterizations for this possibility, resembling the situation in the classical Hardy space ${H}^{1}$. More generally, we consider certain positive functions defined on ${\mathbb{R}}_{+}^{n+1}$, called conformal densities. These densities mimic the averaged derivatives of quasiconformal mappings, and we prove analogous trace theorems for them....

**Page 1**