Example of a discontinuous and bounded function for the limiting case $W^{1,n}$

Let $\Omega = B(0,1)$ be the open unit disc in $\mathbb{R}^2$. I'm looking for an example of a discontinuous and bounded function in $W^{1,2}(\Omega)$.

I know the example $u(x) = \log \left( \log \left(1 + \frac{1}{|x|}\right)\right)$ of a discontinuous but unbounded function in $W^{1,2}(\Omega)$. I've tried playing with things like $(x,y) \mapsto \frac{x}{(x^2 + y^2)^{1/2}}$ but it didn't get me far. Any insight on how to try and construct such examples and how to expect such functions to behave would be much welcomed!

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1 Answer

One can get an example just by composing the function $u(x,y)$ with the function $f(x) = \sin(x)$. By some variant of a chain rule for Sobolev functions, a composition of function in $u \in W^{1,p}(\Omega)$ with a function $f \in C^1_B(\mathbb{R})$ results in a function in $W^{1,p}(\Omega)$. Choosing for $f(x)$ a bounded function that doesn't have a limit when $x \rightarrow \infty$ and composing it with an unbounded $u$ gives the required example.

Of course, the belonging of $f \circ u$ to $W^{1,2}(\Omega)$ can be easily checked directly.

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