Mappings of Finite Distortion : Compactness of the Branch Set

We show that an entire branched cover of finite distortion cannot have a compact branch set if its distortion satisfies a certain asymptotic growth condition. We furthermore show that this bound is strict by constructing an entire, continuous, open and discrete mapping of finite distortion which is piecewise smooth, has a branch set homeomorphic to an (n - 2)-dimensional torus and distortion arbitrarily close to the asymptotic bound. Peer reviewed

Minimizers for the Thin One‐Phase Free Boundary Problem

We consider the “thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in urn:x-wiley:00103640:media:cpa22011:cpa22011-math-0001 plus the area of the positivity set of that function in urn:x-wiley:00103640:media:cpa22011:cpa22011-math-0002. We establish full regularity of the free boundary for dimensions urn:x-wiley:00103640:media:cpa22011:cpa22011-math-0003, prove almost everywhere regularity of the free boundary in arbitrary dimension, and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight. While our results are typical for…

On proper branched coverings and a question of Vuorinen

We study global injectivity of proper branched coverings from the open Euclidean n$n$-ball onto an open subset of the Euclidean n$n$-space in the case where the branch set is compact. In particular, we show that such mappings are homeomorphisms when n=3$n=3$ or when the branch set is empty. This gives a positive answer to the corresponding cases of a question of Vuorinen. Peer reviewed

Korn inequality on irregular domains

Abstract In this paper, we study the weighted Korn inequality on some irregular domains, e.g., s-John domains and domains satisfying quasihyperbolic boundary conditions. Examples regarding sharpness of the Korn inequality on these domains are presented. Moreover, we show that Korn inequalities imply certain Poincare inequality.

Generalized dimension estimates for images of porous sets in metric spaces

On BLD-mappings with small distortion

We show that every $$L$$ -BLD-mapping in a domain of $$\mathbb {R}^{n}$$ is a local homeomorphism if $$L < \sqrt{2}$$ or $$K_I(f) < 2$$ . These bounds are sharp as shown by a winding map.

Regularity and modulus of continuity of space-filling curves

We study critical regularity assumptions on space-filling curves that possess certain modulus of continuity. The bounds we obtain are essentially sharp, as demonstrated by an example. peerReviewed

Boundary blow-up under Sobolev mappings

We prove that for mappings $W^{1,n}(B^n, \R^n),$ continuous up to the boundary, with modulus of continuity satisfying certain divergence condition, the image of the boundary of the unit ball has zero $n$-Hausdorff measure. For H\"older continuous mappings we also prove an essentially sharp generalized Hausdorff dimension estimate.

Solvability of the divergence equation implies John via Poincaré inequality

Abstract Let Ω ⊂ R 2 be a bounded simply connected domain. We show that, for a fixed (every) p ∈ ( 1 , ∞ ) , the divergence equation div v = f is solvable in W 0 1 , p ( Ω ) 2 for every f ∈ L 0 p ( Ω ) , if and only if Ω is a John domain, if and only if the weighted Poincare inequality ∫ Ω | u ( x ) − u Ω | q d x ≤ C ∫ Ω | ∇ u ( x ) | q  dist  ( x , ∂ Ω ) q d x holds for some (every) q ∈ [ 1 , ∞ ) . This gives a positive answer to a question raised by Russ (2013) in the case of bounded simply connected domains. In higher dimensions similar results are proved under some additional assumptions on the domain in question.