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Licensed Unlicensed Requires Authentication Published by De Gruyter October 11, 2013

Bi-Sobolev homeomorphism with zero minors almost everywhere

Robert Černý

Abstract

Suppose that N,α,β are satisfying the conditions 2αN-1 and 2βN-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 fW1,p([0,1]N,N), f is the identity on the boundary, all minors of Df of the α-th order are zero almost everywhere, f-1W1,q([0,1]N,N) 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 fW1,p([0,1]N,N), 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).

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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