Recent studies on Sobolev mappings. (May 2017)

• Record Type: Journal Article
• Authors: D'Onofrio, Luigi; Sbordone, Carlo; Schiattarella, Roberta
• Is Part Of: Nonlinear analysis. Volume 154(2017)
• Page Start: 25
• Page End: 42
• Publication Date: 2017-05
• Subjects: 30C62 -- 31A35; Bi-Sobolev mappings -- Elliptic equations; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.09.013
• Languages: English
• ISSNs: 0362-546X

Abstract

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 ).