integral of Laplacian of a positive function 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 ;) ) ? 

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).
