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 $\int_{B_1(0)} \det(Du)^{-p} dz\leq C_p$ for some $p\in (0,1)$ and $\int_{B_1(0)} |Du|^2 dz\leq 1$. There exists constant $M>1$ such that if $$ \int_{B_1(0)} |S(Du)-S(Dv)|^2 dz=\epsilon $$ then $$ \int_{B_{\frac{1}{2}}(0)} |Dv-R Du| dz\leq c C_p^{\frac{1}{p}}\epsilon^{\frac{p^3}{M Q^5\log(10 C_p Q)}}\text{ for some }R\in SO(2). $$ Taking $u=Id$ we obtain a special case of the quantitative rigidity result of Friesecke, James and Muller. Our main result can be considered as a first step in a new line of generalization of F-J-M Theorem in which $Id$ is replaced by a mapping of non-trivial degree.