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Let Q be the open unit square in ℝ2. We prove that in a natural complete metric space of BV homeomorphisms f : Q → Q with f|∂Q = Id, residually many homeomorphisms (in the sense of Baire categories) map a null set onto a set of full measure, and vice versa. Moreover, we observe that for 1 ⩽ p < 2, the family of W1,p homemomorphisms satisfying the above property is of the first category.

Let be an open set in ℝn and suppose that is a Sobolev homeomorphism. We study the regularity of f–1 under the Lp-integrability assumption on the distortion function Kf. First, if is the unit ball and p > n – 1, then the optimal local modulus of continuity of f–1 is attained by a radially symmetric mapping. We show that this is not the case when p ⩽ n – 1 and n ⩾ 3, and answer a question raised by S. Hencl and P. Koskela. Second, we obtain the optimal integrability results for ∣Df–1∣ in terms of the Lp-integrability assumptions of Kf.

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Residually many BV homeomorphisms map a null set onto a set of full measure

Mappings of Lp-integrable distortion: regularity of the inverse

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2 results

Residually many BV homeomorphisms map a null set onto a set of full measure

Mappings of Lp-integrable distortion: regularity of the inverse