Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 8 (2014), 1839-1850.

Boundary blow-up under Sobolev mappings

Aapo Kauranen and Pekka Koskela

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We prove that for mappings in W1,n(n,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.

Article information

Anal. PDE, Volume 7, Number 8 (2014), 1839-1850.

Received: 30 July 2013
Revised: 27 August 2014
Accepted: 22 October 2014
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26B10: Implicit function theorems, Jacobians, transformations with several variables 26B35: Special properties of functions of several variables, Hölder conditions, etc. 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Sobolev mapping Hausdorff measure modulus of continuity


Kauranen, Aapo; Koskela, Pekka. Boundary blow-up under Sobolev mappings. Anal. PDE 7 (2014), no. 8, 1839--1850. doi:10.2140/apde.2014.7.1839.

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