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Regularity of Nemitskii maps on Sobolev spaces

Let $\Omega\subset \mathbb R^N$ be a bounded smooth domain, and $f\colon\Omega\times \mathbb R\to \mathbb R$ be a smooth function (let's say $C^2$). Let $X=W^{1,p}(\Omega)$ with $p>1$ be the Sobole …