Recent studies on Sobolev mappings. (May 2017) - D'Onofrio, Luigi; Sbordone, Carlo; Schiattarella, Roberta
Abstract: A homeomorphism U of the unit disk D ⊂ R 2, U = ( u 1, u 2 ) : D ⟶ onto D is a quasiharmonic map, if u i ∈ W l o c 1, 1, i = 1, 2 are solutions to the system (0.1) { div B ( y ) ∇ u 1 = 0  a.e. in  D div B ( y ) ∇ u 2 = 0  a.e. in  D for a symmetric degenerate elliptic conductivity B = B ( y ) i.e. (0.2) ∣ ξ ∣ 2 H ( y ) ≤ 〈 B ( y ) ξ, ξ 〉 ≤ H ( y ) ∣ ξ ∣ 2  a.e. in  y ∈ D ∀ ξ ∈ R 2 where H : D ⟶ onto [ 1, ∞ [ is measurable. A sufficient condition that the Sobolev homeomorphism U ∈ W l o c 1, 1 is a quasiharmonic map is that U − 1 ∈ W loc 1, 1 . This condition is not necessary because we construct a quasiharmonic map U such that U − 1 ∈ BV ∖ W loc 1, 1 (see Theorem 1.1 ).