I've been reading through the Han Lin book on elliptic PDEs and just came across the definition of the upper contact set:
$$\Gamma^+ = \{ y\in\Omega:u(x)\leq u(y) + Du(y)\cdot(x-y) \text{ for any } x\in \Omega \} $$
for some $u\in C^2(\Omega)$. I'm confused as to how this makes any sense, since if $\Gamma^+\neq \emptyset$, we can take any $y_0 \in \Gamma^+$ and we then have for every $x\in \Omega$ that
$$u(x)\leq u(y_0) + Du(y_0)\cdot (x-y_0) = u(y_0) + D_i \ u(y_0)(x^i-y^i_0)$$ $$\implies D_j \ u(x) \leq 0 + D_i \ u(y_0) \ \delta_j^i = D_j \ u(y_0)$$ $$\implies D_{kj} \ u(x) \leq 0$$
Which then implies that the function has nonpositive second derivative everywhere and so is concave and $\Gamma^+ = \Omega$.
My reasoning must be incorrect somewhere here, since this result seems to make no sense, but I cannot figure out which step the mistake is on.