Maximum principle at the border

Question

Let $Ω⊂⊂R^{2}$ a bounded regular domain and $u: \overline{Ω} → R$ a function such that $det(∇^{2}u)≤0$ in $Ω$. I'm trying to proof that you have reached your maximum and minimum in $∂Ω$. In my opinion, the Osserman Theorem for convex envelope can be usefull.

Answer 1

Note: I edited this answer due to my counterexample being wrong. I will go over this question again in the coming days.
This looks like a version of the the maximum principle for subharmonic (or super, respectively) functions. It shouldn't work for both minimum and maximum, unless you have $\Delta u=0$.
If you are looking for a proof, you should study sub/superharmonic functions. As for this problem, it might be sufficient to note that $$ det(\nabla^2u)=\lambda_1 \lambda_2 \leq 0 $$ where $\lambda_i$ denotes the eigenvalues of the hessian. This gives you that either one of the 2 eigenvalues is zero or they have different signs. Using the second derivate test will give you an answer in the second case. As for the case $det(\nabla^2u)=0$, you can not get any information at all. I suspect that you can apply a maximum principle here after re-writing the eqation. In the comin days, I will look at this question again, sorry for the mistake earlier.