Boundary blow up under Sobolev mappings

TL;DR

This paper investigates the boundary behavior of Sobolev mappings, showing under certain conditions that the boundary image has zero Hausdorff measure and providing sharp dimension estimates for Hölder continuous cases.

Contribution

It establishes new boundary measure and dimension results for Sobolev and Hölder continuous mappings, extending understanding of boundary blow-up phenomena.

Findings

Boundary image has zero n-Hausdorff measure under divergence conditions.

Hölder continuous mappings have sharp generalized Hausdorff dimension estimates.

Results extend boundary behavior analysis for Sobolev mappings.

Abstract

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.