Mappings of finite distortion: Monotonicity and continuity

@article{Iwaniec2001MappingsOF,
  title={Mappings of finite distortion: Monotonicity and continuity},
  author={Tadeusz Iwaniec and Pekka Koskela and Jani Onninen},
  journal={Inventiones mathematicae},
  year={2001},
  volume={144},
  pages={507-531}
}
We study mappings f = ( f1, ..., fn) : Ω → Rn in the Sobolev space W loc (Ω,R n), where Ω is a connected, open subset of Rn with n ≥ 2. Thus, for almost every x ∈ Ω, we can speak of the linear transformation D f(x) : Rn → Rn, called differential of f at x. Its norm is defined by |D f(x)| = sup{|D f(x)h| : h ∈ Sn−1}. We shall often identify D f(x) with its matrix, and denote by J(x, f ) = det D f(x) the Jacobian determinant. Thus, using the language of differential forms, we can write 

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