Monotone Sobolev Mappings of Planar Domains and Surfaces

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• 作者：Tadeusz Iwaniec ; Jani Onninen
• 刊名：Archive for Rational Mechanics and Analysis
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：219
• 期：1
• 页码：159-181
• 全文大小：550 KB
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• 作者单位：Tadeusz Iwaniec (1) (2)
  Jani Onninen (3)
  
  1. Department of Mathematics, Syracuse University, Syracuse, NY, 13244, USA
  2. Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
  3. Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), 40014, Jyväskylä, Finland
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mechanics
  Electromagnetism, Optics and Lasers
  Mathematical and Computational Physics
  Complexity
  Fluids
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0673

文摘

An approximation theorem of Youngs (Duke Math J 15, 87–94, 1948) asserts that a continuous map between compact oriented topological 2-manifolds (surfaces) is monotone if and only if it is a uniform limit of homeomorphisms. Analogous approximation of Sobolev mappings is at the very heart of Geometric Function Theory (GFT) and Nonlinear Elasticity (NE). In both theories the mappings in question arise naturally as weak limits of energy-minimizing sequences of homeomorphisms. As a result of this, the energy-minimal mappings turn out to be monotone. In the present paper we show that, conversely, monotone mappings in the Sobolev space \({\,{\fancyscript{W}}^{1,p}\,, \,1 < p < \infty\,}\), are none other than \({\,{\fancyscript{W}}^{1,p}\,}\)-weak (also strong) limits of homeomorphisms. In fact, these are limits of diffeomorphisms. By way of illustration, we establish the existence of traction free energy-minimal deformations for  p -harmonic type energy integrals. Communicated by V. ŠverákT. Iwaniec was supported by the NSF grant DMS-1301558 and the Academy of Finland project 1128331. J. Onninen was supported by the NSF grant DMS-1301570.