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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 !

  • 0
    Really, for every choice of $b_i$? What if I multiply each $b_i$ by $2$?2012-07-14
  • 0
    And besides, $x_1/|x|$ does not belong to $W^{1,2}(\{x\in\mathbb R^n:|x|<1\})$ when $n=2$.2012-07-14

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