Let $x\in \mathbb{R}^{n}, x = (x_1,\ldots,x_n)$, and $f(x) = \operatorname{sgn}(x_{1})$. Is $f$ weakly differentiable on $U = B(0,1)$, i.e. unit ball in $\mathbb{R}^{n}$, and what is the weak derivative?
Weak derivative of $\operatorname{sgn}(x_1)$
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distribution-theory
weak-derivatives
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2What exactly do you mean by "weakly differentiable" here? There are several possible definitions in different contexts. Do you want the weak derivative to be a function, a distribution, ...? – 2012-10-29