#### Vol. 7, No. 8, 2014

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Boundary blow-up under Sobolev mappings

### Aapo Kauranen and Pekka Koskela

Vol. 7 (2014), No. 8, 1839–1850
##### Abstract

We prove that for mappings in ${W}^{1,n}\left({\mathsc{ℬ}}^{n},{ℝ}^{m}\right)$, continuous up to the boundary and with modulus of continuity satisfying a certain divergence condition, the image of the boundary of the unit ball has zero $n$-Hausdorff measure. For Hölder continuous mappings we also prove an essentially sharp generalised Hausdorff dimension estimate.

##### Keywords
Sobolev mapping, Hausdorff measure, modulus of continuity
##### Mathematical Subject Classification 2010
Primary: 26B10, 26B35, 46E35
##### Milestones
Received: 30 July 2013
Revised: 27 August 2014
Accepted: 22 October 2014
Published: 5 February 2015
##### Authors
 Aapo Kauranen Department of Mathematics and Statistics University of Jyväskylä P.O. Box 35 (MaD) FI-40014 Jyväskylä Finland Pekka Koskela Department of Mathematics and Statistics University of Jyväskylä P.O. Box 35 (MaD) FI-40014 Jyväskylä Finland