Bi-Sobolev homeomorphisms f with Df and \(Df^{-1}\) of low rank using laminates

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• 作者：Marcos Oliva
• 关键词：Mathematics Subject Classification46E35 ; 26B25 ; 26B35
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：55
• 期：6
• 全文大小：708 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835
• 卷排序：55

文摘

Let \(\Omega \subset \mathbb {R}^{n}\) be a bounded open set. Given \(1\le m_1,m_2\le n-2\), we construct a homeomorphism \(f :\Omega \rightarrow \Omega \) that is Hölder continuous, f is the identity on \(\partial \Omega \), the derivative Df has rank \(m_1\) a.e. in \(\Omega \), the derivative \(D f^{-1}\) of the inverse has rank \(m_2\) a.e. in \(\Omega \), \(Df\in W^{1,p}\) and \(Df^{-1}\in W^{1,q}\) for \(p<\min \{m_1+1,n-m_2\}\), \(q<\min \{m_2+1,n-m_1\}\). The proof is based on convex integration and laminates. We also show that the integrability of the function and the inverse is sharp.