Question About Concave Functions It easy to prove that no non-constant positive concave function exists (for example by integrating:
$ u'' \leq 0 \to u' \leq c \to u \leq cx+c_2 $ and since $u>0$ , we obviously get a contradiction.
Can this result be generalized to $ \Bbb R^2 $ and the Laplacian?  Is there an easy way to see this? (i.e.- no non-constant positive real-valued function with non-positive l
Laplacian exists)
Thanks!

Let $u$ strictly concave and twice diferentiable in $\mathbb{R^{2}}$ então $v(x) = u(x,0)$ is strictly concave and twice diferentiable in $\mathbb{R}$. Hence assume negative value.
