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SOME REMARKS ON SOBOLEV AND BI-SOBOLEV MAPPINGS
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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).
Springer Science and Business Media LLC
Title: SOME REMARKS ON SOBOLEV AND BI-SOBOLEV MAPPINGS
Description:
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).
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