Iwaniec–Martin–Šverák conjecture on mappings of integrable inner distortion

Let ΩRn\Omega\subset\mathbb{R}^n be a domain, let fWloc1,n(Ω,Rn)f\in W^{1,n}_{\operatorname{loc}}(\Omega,\mathbb{R}^n), and let KI(,f)K_I(\cdot,f) denote the inner distortion of ff. Assume that KI(,f)Lloc1(Ω)K_I(\cdot,f)\in L^1_{\operatorname{loc}}(\Omega). Iwaniec–Martin–Šverák conjecture. The mapping ff is either constant or both discrete and open. This conjecture would strengthen the results discussed in the paper by implying discreteness and openness under merely integrable inner distortion. Its resolution is not given in the source.

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Primary source

Leonid V. Kovalev and Jani Onninen, “On Invertibility of Sobolev Mappings”, arXiv:0812.2350 (2008).

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