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Deformations of Bi-conformal Energy and a New Characterization of Quasiconformality

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Abstract

The concept of hyperelastic deformations of bi-conformal energy is developed as an extension of quasiconformality. These deformations are homeomorphisms \(\, h :{\mathbb {X}} \xrightarrow []{{}_{\!\!\text{ onto }\,\,\!\!}}{\mathbb {Y}}\,\) between domains \(\,{\mathbb {X}}, {\mathbb {Y}} \subset {\mathbb {R}}^n\,\) of the Sobolev class \(\,{\mathscr {W}}^{1,n}_{\text{ loc }} ({\mathbb {X}}, {\mathbb {Y}})\,\) whose inverse \(\, f {\mathop {=\!\!=}\limits ^{\text{ def }}} h^{-1} :{\mathbb {Y}} \xrightarrow []{{}_{\!\!\text{ onto }\,\,\!\!}}{\mathbb {X}}\,\) also belongs to \(\,{\mathscr {W}}^{1,n}_{\text{ loc }}({\mathbb {Y}}, {\mathbb {X}})\,\). Thus the paper opens new topics in Geometric Function Theory (GFT) with connections to mathematical models of Nonlinear Elasticity (NE). In seeking differences and similarities with quasiconformal mappings we examine closely the modulus of continuity of deformations of bi-conformal energy. This leads us to a new characterization of quasiconformality. Specifically, it is observed that quasiconformal mappings behave locally at every point like radial stretchings; if a quasiconformal map \(\,h\,\) admits a function \(\,\phi \,\) as its optimal modulus of continuity at a point \(\,x_\circ \, \), then \(\,f = h^{-1}\,\) admits the inverse function \(\, \psi = \phi ^{-1}\,\) as its modulus of continuity at \(\, y_\circ = h(x_\circ ) \,\). That is to say, a poor (possibly harmful) continuity of \(\,h\,\) at a given point \(\,x_\circ \,\) is always compensated by a better continuity of \(\,f\,\) at \(\,y_\circ \,\), and vice versa. Such a gain/loss property, seemingly overlooked by many authors, is actually characteristic of quasiconformal mappings. It turns out that the elastic deformations of bi-conformal energy are very different in this respect. Unexpectedly, such a map may have the same optimal modulus of continuity as its inverse deformation. In line with Hooke’s Law, when trying to restore the original shape of the body (by the inverse transformation), the modulus of continuity may neither be improved nor become worse. However, examples to confirm this phenomenon are far from being obvious; indeed, elaborate computations are on the way. We eventually hope that our examples will gain an interest in the materials science, particularly in mathematical models of hyperelasticity.

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Notes

  1. Hereafter the notation \(\,{\mathbf {A}} \preccurlyeq {\mathbf {B}}\) stands for the inequality \(\,{\mathbf {A}} \leqslant c\, {\mathbf {B}}\) in which \(\, c>0\,\), called implied or hidden constant, plays no role. The implied constant may vary from line to line and is easily identified from the context, or explicitely specified if necessary.

  2.  In the above estimates the implied constants depend only on \(\,n\,\).

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Authors and Affiliations

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

    Tadeusz Iwaniec & Jani Onninen

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

    Jani Onninen & Zheng Zhu

Authors
  1. Tadeusz Iwaniec
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  2. Jani Onninen
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  3. Zheng Zhu
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Correspondence to Tadeusz Iwaniec.

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Communicated by D. Kinderlehrer

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T. Iwaniec was supported by the NSF Grant DMS-1802107. J. Onninen was supported by the NSF Grant DMS-1700274. This research was done while Z. Zhu was visiting Mathematics Department at Syracuse University. He wishes to thank SU for the hospitality.

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Iwaniec, T., Onninen, J. & Zhu, Z. Deformations of Bi-conformal Energy and a New Characterization of Quasiconformality. Arch Rational Mech Anal 236, 1709–1737 (2020). https://doi.org/10.1007/s00205-020-01502-w

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