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2022•arXiv (Cornell University)Open access

On differentiability of Sobolev functions with respect to the Sobolev norm

Vladimir Gol’dshtein, Paz Hashash, Alexander Ukhlov

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Abstract

We study connections between the $W^1_p$-differentiability and the $L_p$-differentiability of Sobolev functions. We prove that, $W^1_p$-differentiability implies the $L_p$-differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed as well. In addition, we consider the $W^1_p$-differentiability of Sobolev functions $\cp_p$-almost everywhere.

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What this paper is about

We study connections between the $W^1_p$-differentiability and the $L_p$-differentiability of Sobolev functions. We prove that, $W^1_p$-differentiability implies the $L_p$-differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed as well. In addition, we consider the $W^1_p$-differentiability of Sobolev functions $\cp_p$-almost everywhere.

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Available abstract

We study connections between the $W^1_p$-differentiability and the $L_p$-differentiability of Sobolev functions. We prove that, $W^1_p$-differentiability implies the $L_p$-differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed as well. In addition, we consider the $W^1_p$-differentiability of Sobolev functions $\cp_p$-almost everywhere.

Key concepts: Differentiable function, Sobolev space, Mathematics, Norm (philosophy), Pure mathematics, Mathematical analysis, Law, Political science

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