Rigidity of pairs of quasiregular mappings whose symmetric part of gradient are close

Lorent, Andrew

doi:10.1016/j.anihpc.2014.08.003

Lorent, Andrew

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 1, pp. 23-65.

Résumé

For $A \in M^{2 \times 2}$ let $S (A) = \sqrt{A^{T} A}$ , i.e. the symmetric part of the polar decomposition of A. We consider the relation between two quasiregular mappings whose symmetric part of gradient are close. Our main result is the following. Suppose $v, u \in W^{1, 2} (B_{1} (0) : R^{2})$ are Q-quasiregular mappings with $\int_{B_{1} (0)} \det {(D u)}^{- p} d z \leq C_{p}$ for some $p \in (0, 1)$ and $\int_{B_{1} (0)} {| D u |}^{2} d z \leq π$ . There exists constant $M > 1$ such that if $\int_{B_{1} (0)} {| S (D v) - S (D u) |}^{2} d z = ϵ$ then

\int_{B_{\frac{1}{2}} (0)} | D v - R D u | d z \leq c C_{p}^{\frac{2}{p}} ϵ^{\frac{p^{2}}{M Q^{5} \log (10 C_{p} Q)}} for some R \in SO (2) .

Taking

u = Id

we obtain a special case of the quantitative rigidity result of Friesecke, James and Müller [13]. Our main result can be considered as a first step in a new line of generalization of Theorem 1 of [13] in which Id is replaced by a mapping of non-trivial degree.

MR Zbl

DOI : 10.1016/j.anihpc.2014.08.003

Classification : 30C65, 26B99
Mots clés : Rigidity, Liouville's Theorem, Symmetric part of gradient, Reshetnyak

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Lorent, Andrew. Rigidity of pairs of quasiregular mappings whose symmetric part of gradient are close. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 1, pp. 23-65. doi : 10.1016/j.anihpc.2014.08.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.08.003/

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