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I've encountered the following, rather elementary, problem:

$K$ is a compact subset of some 2-dimensional oriented manifold with smooth boundary, $f$ is a positive smooth function on $K$ that vanishes on the border $\partial K$. I want to show that the integral of the Laplacian of $f$ is non-positive, where Laplacian is given by $\Delta f = (f_{xx} + f_{yy}) dx \wedge dy$ in local coordinates.

It seems that one can use Stokes to compute: $ \int_K \Delta f = \int_{\partial K} \nabla f \cdot \mathbf{n} \ dl \leq 0 $ where $\mathbf{n}$ is the normal vector and the inequality follows from the fact that since "$f$ decreases in the direction of the boundary", on the boundary we have $\nabla f \cdot \mathbf{n} \leq 0$.

I was wondering if there is some more elegant reason for the inequality in question. Also: is the given argument correct (my analysis is a bit rusty ;) ) ?

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    @joriki: Thank you! I was a bit careless with the formulation, and I got the sign wrong. I'm just about to correct it.2012-03-02

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Since there is likely nothing more to be added in response to the question, I am placing this community wiki answer with the sole purpose of closing the thread (so as not to add to the already excessive number of unanswered questions).