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I want to find $p$ which maximizes the given functional. $p$ is a function of the form $\mathbb{R}^2 \to \mathbb{R}$. $\Omega$ is a region in the 2-d plane.

$\underset{p}{\sup} \int_\Omega \{ \lambda(\vec{\nabla}\cdot \vec{p}) - \alpha(|p| - C)\}\, dx$

Authors of the paper A Study on Continuous Max-Flow and Min-Cut Approaches has said the following to be an equivalent formulation:

$\underset{|p| \le C}{\sup} \int_\Omega \lambda \vec{\nabla}\cdot \vec{p}\, dx$

The authors further claim that it is a well known result that the above is equal to

$\int_\Omega C|\nabla\lambda|\, dx$

And hence it is the desired answer.

I was wondering if I can get the above answer by some more intuitive approach, or if someone could please explain to me what the authors are trying to say.

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    Which author of which paper is claiming all that?2011-06-04
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    The second equivalence (which the author suggests is well known) is in the paper " A study on continuous max-flow and min-cut approaches" by Boykov et al. The first equivalence is my understanding of the paper and not necessarily what the author is claiming2011-06-04
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    Did you try partial-integration?2011-06-04
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    The first equivalence is true when $\alpha(x) = \infty$ for $x \leq 0$, $0$ otherwise. It's a function that's often used in convex optimization, I don't remember the name for it.2011-06-04
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    @Fabian: The more common term is "integration by parts". I'm German, I used to think it was called "partial integration", but apparently to native English speakers that sounds like you didn't finish the integration :-) (In German it's "partielle Integration".)2011-06-04
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    @joriki: I find both (http://mathworld.wolfram.com/PartialIntegration.html) but it seems integration by parts is much more common...2011-06-04

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