Boundary blow-up under Sobolev mappings

Abstract

We prove that for mappings in $W^{1,n}\left({\mathcal{ℬ}}^n,{ℝ}^m\right)$, 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.