Question: Let $f: [0, 1] \to \mathbb R$ be a continuous, nowhere differentiable function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly. Is it true that $\|u_n\|_{W^{1, p}} \to \infty?$
Blowup of Sobolev norms in approximating a nowhere differentiable function
Nate River
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