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Differentiability in the Sobolev space \(W^{1,n-1}\). (English) Zbl 1305.26030

A classical result of F. W. Gehring and O. Lehto [Ann. Acad. Sci. Fenn., Ser. A I 272, 9 p. (1959; Zbl 0090.05302)] asserts that a continuous, discrete and open mapping \(f:\Omega\to \mathbb{R}^2\) from \(W^{1,1}_{\text{loc}}\) is almost everywhere differentiable. In dimension \(n\geq 3\), this is not true. The author gives a positive result of this type in \(W^{1,n-1}_{\text{loc}}(\Omega, \mathbb{R}^n)\) \((\Omega\subset\mathbb{R}^n)\) for \(n\geq 3\) in terms of inner and outer distortion.

MSC:

26B10 Implicit function theorems, Jacobians, transformations with several variables
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
28A35 Measures and integrals in product spaces

Citations:

Zbl 0090.05302
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References:

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