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the total variation of a function $u\in L^1(\Omega)$, $\Omega\subset \mathbb{R}^n$, can be defined as

$$ \sup \{ \int_\Omega u \; \mathrm{div} g \; dx:\; g \in C_c^1(\Omega,\mathbb{R}^n), \; \lvert g(x) \rvert \leq 1,\; x \in \Omega \} $$

for (weakly) differentiable functions $u$, this supremum equals the $L^1$ norm of the (weak) gradient $\int_\Omega \lvert \nabla u \rvert\; dx$. however, i can't seem to find the rigorous argument to show this. can someone help me?

  • 1
    What do you assume on $\Omega$?2012-03-11
  • 0
    $\Omega$ is an open set, no further assumptions.2012-03-11

3 Answers 3