Bi-Sobolev homeomorphism with zero minors almost everywhere

Robert Černý

doi:10.1515/acv-2013-0011

Article

Bi-Sobolev homeomorphism with zero minors almost everywhere

Robert Černý

Published/Copyright: October 11, 2013

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From the journal Advances in Calculus of Variations Volume 8 Issue 1

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).

Keywords: Sobolev mapping; bi-Sobolev mapping; Jacobian; minor; Luzin condition

MSC: 46E35

Funding source: Czech Ministry of Education

Award Identifier / Grant number: ERC CZ LL1203

The author would like to thank Stanislav Hencl for drawing his attention to the problem and for many fruitful discussions.

Received: 2013-4-28

Revised: 2013-7-19

Accepted: 2013-9-3

Published Online: 2013-10-11

Published in Print: 2015-1-1

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Articles in the same Issue

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

Keywords for this article

Sobolev mapping; bi-Sobolev mapping; Jacobian; minor; Luzin condition