Boundary blow-up under Sobolev mappings

Abstract

We prove that for mappings in W 1,n (B n , Ê m ), continuous up to the boundary, 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 generalized Hausdorff dimension estimate. 0 2010 Mathematics Subject Classification: 46E35, 26B35, 26B10

Generalized dimension estimates for images of porous sets in metric spaces

“…In Q-Ahlfors regular metric measure spaces we obtain essentially sharp dimension estimates for images of porous sets under Hajłasz-type Sobolev maps with L Q -gradient. The estimates generalize the previous results of Jones and Makarov [7] and Kauranen and Koskela [8]. …”

“…Theorem 1.1 recovers the result from [8]: ∂B(0, 1) is porous and the set of the continuous maps f : B n (0, 1) → R m in W 1,n (B n (0, 1), R m ) have L n -Hajłasz gradients (see [4, Theorem 6.10]). Theorem 1.1 is a corollary to a more abstract statement proved in Section 3.…”

“…The purpose of this paper is to show that the methods used in [8] can be applied in the setting of Q-Ahlfors regular metric measure spaces and that the obtained generalized Hausdorff dimension estimates are essentially sharp for each Q > 1 and each modulus of continuity satisfying (3). Our main result is the following.…”

“…Modifying the map h of Section 4 in [8] shows that Theorem 1.2 is true even if a = 1 n , but there the construction is different and done only in Euclidean spaces.…”

“…That is, the integral diverges at zero. Recently Kauranen and Koskela [8] generalized this result to the setting of the Sobolev space W 1,n (B(0, 1)) with modulus of continuity ψ satisfying (2)ˆ0 log ψ(t) log t n dt t = ∞.…”