On differentiability of Sobolev functions with respect to the Sobolev norm

We study connections between the Wp1$W^1_p$‐differentiability and the Lp$L_p$‐differentiability of Sobolev functions. We prove that Wp1$W^1_p$‐differentiability implies the Lp$L_p$‐differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed...

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Veröffentlicht in:Mathematische Nachrichten 2024-10, Vol.297 (10), p.3681-3699
Hauptverfasser: Gol'dshtein, Vladimir, Hashash, Paz, Ukhlov, Alexander
Format: Artikel
Sprache:eng
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Zusammenfassung:We study connections between the Wp1$W^1_p$‐differentiability and the Lp$L_p$‐differentiability of Sobolev functions. We prove that Wp1$W^1_p$‐differentiability implies the Lp$L_p$‐differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed as well. In addition, we consider the Wp1$W^1_p$‐differentiability of Sobolev functions capp$\operatorname{cap}_p$‐almost everywhere.
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.202300545