Bi-Sobolev homeomorphism with zero minors almost everywhere

Robert Černý

doi:10.1515/acv-2013-0011

Published by De Gruyter October 11, 2013

Bi-Sobolev homeomorphism with zero minors almost everywhere

Robert Černý

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/acv-2013-0011

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Abstract

Suppose that N,α,β∈ℕ are satisfying the conditions 2≤α≤N-1 and 2≤β≤N-1. Let us fix p∈[1,min{NN-α+1,N-β+1}) and q∈[1,min{NN-β+1,N-α+1}). We construct a homeomorphism f:[0,1]N↦[0,1]N such that f∈W1,p([0,1]N,ℝN), f is the identity on the boundary, all minors of Df of the α-th order are zero almost everywhere, f-1∈W1,q([0,1]N,ℝN) and all minors of Df-1 of the β-th order are zero almost everywhere. A simplified version of our construction gives a homeomorphism f:[0,1]N↦[0,1]N such that f∈W1,p([0,1]N,ℝN), f is the identity on the boundary and all minors of Df of the α-th order are zero almost everywhere under a less restrictive assumption p∈[1,NN-α+1).