On differentiability of Sobolev functions with respect to the Sobolev norm

Vladimir Gol'dshtein; Paz Hashash; Alexander Ukhlov

paper

On differentiability of Sobolev functions with respect to the Sobolev norm

arXiv:2207.08738

Abstract

We study connections between the $W_{p}^{1}$ -differentiability and the $L_{p}$ -differentiability of Sobolev functions. We prove that, $W_{p}^{1}$ -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 $W_{p}^{1}$ -differentiability of Sobolev functions $\cp_p$-almost everywhere.

20 pages