I'm seeking a function which belongs to $W^{1,p}(\Omega)$ for $p < n$ which is *not differentiable a.e*. There is a standard theorem which shows that if $p > n$ then in fact any function in $W^{1,p}$ is differentiable a.e. I would like an example where a weak derivative exists, lies in $W^{1,p}$ for $p < n$ but fails to be differentiable a.e.



A function in $W^{1,p}(\Omega)$ for $1 < p < n$ which is not differentiable a.e
Pick a number $\alpha$ with $0<\alpha< n/p-1$ and a smooth, nonnegative function $g(r)$ defined for $r>0$ with $g(r)=r^{-\alpha}$ when $r$ is small, $g(r)=0$ when $r$ is large. Then $x\mapsto g(|x|)$ belongs to $W^{1,p}(\mathbb{R}^n)$. Write $$f(x)=\sum\_{i=1}^\infty 2^{-i}g(|x-q\_i|)$$ where $(q\_i)$ is a dense sequence in $\mathbb{R}^n$. The series converges in $W^{1,p}(\mathbb{R}^n)$ and the sum is unbounded in any neighbourhood of any $q\_i$, hence unbounded in any nonempty open set. Differentiability is rather hard to achieve under those circumstances.


(Edit: Need $g\ge0$.)


