As you can probably guess, I'm currently studying about differential operators and functional analysis. We've studied the following theorem:
A function $f \in L^2 (\Omega) $ lies in $ W^{1,2} ( \Omega) $ if and only if there exists a function $g \in L^2 ( \Omega ) $ such that: $\int_\Omega f \left\{ b_0 (x) \phi(x) - \sum_{i=1}^n \frac{ \partial(b_i (x) \phi(x)}{\partial x_i} \right\} d^n x = \int_\Omega g(x) \phi(x) d^n x $ for every choice of functions $b_i (x) \in C^\infty (\bar{\Omega} ) $, and $\phi \in C_c^\infty (\Omega ) $ .
Can someone help me use this theorem in order to prove that the function $f(x)= \frac{x_1}{|x|} $ is in $W^{1,2}( \{ x \in \mathbb{R} ^n : |x| <1 \} ) $? What should be my $g$ and how can I prove it?
I really need your help !
Thanks !