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

Abstract

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 $v,u\in W^{1,2}(B_1(0):{\mathbb{R}}^2)$ are Q-quasiregular mappings with ${\int }_{B_1(0)}\det (Du{)}^{-p}dz\le C_p$ for some $p\in (0,1)$ and ${\int }_{B_1(0)}|Du{|}^{2}dz\le \pi$. There exists constant $M>1$ such that if ${\int }_{B_1(0)}|S(Dv)-S(Du){|}^{2}dz=ϵ$ then

Taking $u=\mathrm{Id}$ 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.