Bi-Sobolev homeomorphism with zero minors almost everywhere

作者：Cerny Robert^*

来源：Advances in Calculus of Variations, 2015, 8(1): 1-30.

DOI：10.1515/acv-2013-0011

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

Suppose that N, alpha, beta is an element of N are satisfying the conditions 2 <= alpha <= N - 1 and 2 <= beta <= N - 1. Let us fix p is an element of [1, min{N/N-alpha+1, N - beta + 1}) and q is an element of [1, min{N/N-beta+1, N - alpha + 1}). We construct a homeomorphism f : [0, 1](N) bar right arrow [0, 1](N) such that f is an element of W-1,W-p ([0, 1](N), R-N), f is the identity on the boundary, all minors of Df of the alpha-th order are zero almost everywhere, f(-1) is an element of W-1,W-q ([0, 1](N), R-N) and all minors of Df(-1) of the beta-th order are zero almost everywhere. A simplified version of our construction gives a homeomorphism f : [0, 1](N) bar right arrow [0, 1](N) such that f is an element of W-1,W-p ([0, 1](N), R-N), f is the identity on the boundary and all minors of Df of the alpha-th order are zero almost everywhere under a less restrictive assumption p is an element of [1, N/N-alpha+1).

• 出版日期2015-1