Recent studies on Sobolev mappings

A homeomorphism U of the unit disk D⊂R2, U=(u1,u2):D⟶ontoD is a quasiharmonic map, if ui∈Wloc1,1, i=1,2 are solutions to the system (0.1){divB(y)∇u1=0  a.e. in  DdivB(y)∇u2=0  a.e. in  D for a symmetric degenerate elliptic conductivity B=B(y) i.e. (0.2)∣ξ∣2H(y)≤〈B(y)ξ,ξ〉≤H(y)∣ξ∣2  a.e. in  y∈D∀ξ∈R2 where H:D⟶onto[1,∞[ is measurable. A sufficient condition that the Sobolev homeomorphism U∈Wloc1,1 is a quasiharmonic map is that U−1∈Wloc1,1. This condition is not necessary because we construct a quasiharmonic map U such that U−1∈BV∖Wloc1,1 (see Theorem 1.1 ).