Boundary blow-up under Sobolev mappings

Kauranen, Aapo; Koskela, Pekka

doi:10.2140/apde.2014.7.1839

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1948-206X (online)

ISSN 2157-5045 (print)

Author index

To appear

Other MSP journals

Boundary blow-up under Sobolev mappings

Aapo Kauranen and Pekka Koskela

Vol. 7 (2014), No. 8, 1839–1850

DOI: 10.2140/apde.2014.7.1839

Abstract

We prove that for mappings in $W^{1, n} (ℬ^{n}, ℝ^{m})$ , 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

Vol. 7, No. 8, 2014