Differentiability in the Sobolev space $$W^{1,n-1}$$ W 1 , n - 1

Abstract

Let Ω ⊂ R n be a domain, n ≥ 2. We show that a continuous, open and discrete mapping f ∈ W 1,n−1 loc (Ω, R n ) with integrable inner distortion is differentiable almost everywhere on Ω. As a corollary we get that the branch set of such a mapping has measure zero.

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“…Кроме того, в работе [4], в частности, было показано, что гомеоморфизмы f класса 1 = − , который оставался неизученным, рассматривается в данной работе. Это продвижение оказалось возможным, прежде всего, благодаря статье [5].…”

Sharpness of the differentiability almost everywhere and capacitary estimates for Sobolev mappings

“…Indeed, it follows from [11, (Ω, R n ) with locally integrable inner distortion function is differentiable almost everywhere. The topological assumptions above can be further relaxed by assuming f to be only continuous, discrete (the set f −1 (y) is a discrete set in Ω for every y ∈ R n ) and open (f (A) is an open set in R n for every open set A in Ω), see [17]. It was asked in [17] whether the local integrability assumption of the inner distortion function above is sharp.…”

“…The topological assumptions above can be further relaxed by assuming f to be only continuous, discrete (the set f −1 (y) is a discrete set in Ω for every y ∈ R n ) and open (f (A) is an open set in R n for every open set A in Ω), see [17]. It was asked in [17] whether the local integrability assumption of the inner distortion function above is sharp. We will now give a positive answer to this question by a novel construction: Theorem 1.1.…”

“…This method does not work if we assume f ∈ W 1,p loc (Ω, R n ) for some p ∈ [1, n). In [17,Lemma 4.4] it was shown by applying Ziemer's duality equation [20,21] that for spherical condensers the K I -inequality is true even for continuous, open and discrete mappings of finite distortion in W 1,n−1 loc…”

“…In our construction we will actually destroy the Lusin's condition on every hyperplane. The reason for this comes from the proof of [17,Lemma 4.4] where the Sobolev regularity of a mapping was only used to show that mapping satisfy the condition (N ) on almost every hyperplane.…”