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

For A∈M2×2A∈M2×2 let S(A)=ATA, 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∈W1,2(B1(0):R2)v,u∈W1,2(B1(0):R2) are Q  -quasiregular mappings with ∫B1(0)det⁡(Du)−pdz≤Cp∫B1(0)det⁡(Du)−pdz≤Cp for some p∈(0,1)p∈(0,1) and ∫B1(0)|Du|2dz≤π∫B1(0)|Du|2dz≤π. There exists constant M>1M>1 such that if ∫B1(0)|S(Dv)−S(Du)|2dz=ϵ∫B1(0)|S(Dv)−S(Du)|2dz=ϵ then∫B12(0)|Dv−RDu|dz≤cCp2pϵp2MQ5log⁡(10CpQ) for some R∈SO(2). Taking u=Idu=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.