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Monotone Sobolev Mappings of Planar Domains and Surfaces

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, Syracuse University, Syracuse, NY, 13244, USA

    Tadeusz Iwaniec

  2. Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland

    Tadeusz Iwaniec

  3. Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), 40014, Jyväskylä, Finland

    Jani Onninen

Authors
  1. Tadeusz Iwaniec
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  2. Jani Onninen
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Correspondence to Jani Onninen.

Additional information

Communicated by V. Šverák

T. 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.

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Iwaniec, T., Onninen, J. Monotone Sobolev Mappings of Planar Domains and Surfaces. Arch Rational Mech Anal 219, 159–181 (2016). https://doi.org/10.1007/s00205-015-0894-6

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