SOME REMARKS ON SOBOLEV AND BI-SOBOLEV MAPPINGS

AbstractIn this note, we present the state-of-the-art theory of bi-Sobolev mappings. We recall that f is a Sobolev homeomorphism if f belongs to $$W^{1,1}_\text {loc}\cap \textrm{Hom}(\Omega ; \Omega ')$$ W loc 1 , 1 ∩ Hom ( Ω ; Ω ′ ) , and f is a bi-Sobolev map if and only if f and $$f^{-1}$$ f - 1 are Sobolev homeomorphisms. This concept, introduced in Hencl et al. (J. Math. Anal. Appl. 355, 22–32 2009), plays a central role in Geometric Function Theory. For instance, we just mention here that maps of bi-Sobolev type are strictly related to the notion of mappings of finite distortion; see, among others, the papers (Hencl and Koskela 2014) Pratelli (Nonlinear Anal. 154, 258–268  2017).