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Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity

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Abstract

We show that a sufficient condition for the weak limit of a sequence of \(W^1_q\)-homeomorphisms with finite distortion to be almost everywhere injective for \(q \ge n-1\), can be stated by means of composition operators. Applying this result, we study nonlinear elasticity problems with respect to these new classes of mappings. Furthermore, we impose loose growth conditions on the stored-energy function for the class of \(W^1_n\)-homeomorphisms with finite distortion and integrable inner as well as outer distortion coefficients.

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Notes

  1. Müller et al. [44] improved the assumption \(q\ge \frac{p}{p-1}\) to \(q\ge \frac{n}{n-1}\).

  2. Considering that \(D\varphi =0\) a.e. if \(J (x, \varphi ) = 0\), it is assumed that \(K_O (x, \varphi ) = 1\) if \(J (x, \varphi ) = 0\).

  3. Considering that \(Adj D\varphi =0\) a.e. if \(J (x, \varphi ) = 0\), it is assumed that \(K_I (x, \varphi ) = 1\) if \(J (x, \varphi ) = 0\).

  4. In this paper we use a notation \(\Vert F \mid L_p(\varOmega )\Vert \) for the norm of \(F(\cdot )\) in \(L_p(\varOmega )\). In some another texts the same norm is denoted by \(\Vert F\Vert _{L_p(\varOmega )}\).

  5. See [61] for another proof of this property under weaker assumptions.

  6. Some authors include condition \(J(x,f) \ge 0\) in Definition 2. We do not use the condition for the Jacobian to be non-negative as it is unnecessary in the context of the theory of composition operators, see details in [61].

  7. Necessity is proved in [68, 69] (see also earlier work [59]), and sufficiency, in Theorem 6 of [61].

  8. The exponent r from Theorem 8 can be expressed as \(r=\frac{n (n-1) s}{ns + 1 - s} \ge n-1\).

  9. \(a \sim b\) means that there exist constants \(C_1\), \(C_2 > 0\), such that \(C_1 a \le b \le C_2 a\).

  10. Let us remind that for \(a_i\), \(b_i \ge 0\), \(\frac{1}{k} + \frac{1}{k'} = 1\), \( \bigl |\sum a_i b_i\bigr | = \big (\sum a_i^k \big )^{1/k} \big (\sum b_i^{k'}\big )^{1/{k'}} \) if and only if \(a_i^k\) and \(b_i^{k'}\) are proportional.

  11. i.e. \(\lim _{r\rightarrow 0} \frac{|A \cap B(x,r)|}{|B(x,r)|} = 1\).

  12. The exponent r from Theorem 8 can be expressed as \(r=\frac{n (n-1) s}{ns + 1 - s} \ge n-1\).

  13. i.e. determinants of the matrix that is formed by taking the elements of the original matrix from the rows whose indexes are in \(({i_1},i_2,\ldots ,{i_l})\) and columns whose indexes are in \(({j_1},j_2,\ldots ,{j_l})\).

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Acknowledgements

We thank the anonymous reviewers for critically reading and comments, which helped improve and clarify this manuscript.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Acad. Koptyug avenue 4, Novosibirsk, Russia, 630090

    Anastasia Molchanova & Sergey Vodopyanov

  2. University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Wien, Austria

    Anastasia Molchanova

  3. Novosibirsk State University, Pirogova Str. 1, Novosibirsk, Russia, 630090

    Sergey Vodopyanov

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  1. Anastasia Molchanova
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Correspondence to Sergey Vodopyanov.

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Communicated by J. Ball.

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This work was partially supported in Sections 1 and 2 by the Ministry of Science and Education of the Russian Federation (Grant 1.3087.2017/4.6) and in Sections 3 and 4 by the Regional Mathematical Center of Novosibirsk State University. The first author was partially supported by Austrian Science Fund (FWF) projects M 2670.

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Molchanova, A., Vodopyanov, S. Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity. Calc. Var. 59, 17 (2020). https://doi.org/10.1007/s00526-019-1671-4

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