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 $\mathrm{Id}$ is replaced by a mapping of non-trivial degree.