Show that this subharmonic function has a nonnegative normal derivative

Question

Let $d\in\mathbb N$, $x\in\mathbb R^d$, $r>0$, $\Omega:=B_r(x)$ and $w\in C^1(\overline\Omega)\cap C^2(\Omega)$ with

1. $w\le0$ on $\partial\Omega$ and $\partial B_{r/2}(x)$;
2. $\Delta w\ge0$ in $\Omega\setminus\overline B_{r/2}(x)$;
3. $w\le0$ in $\Omega\setminus\overline B_{r/2}(x)$.

Now let $x_0\in\partial\Omega$. Why can we conclude that $\partial_\nu w(x_0)\ge0$?

I guess this is a simple optimization result. I know that if $f:A\to\mathbb R$ function on an open subset $A\subseteq\mathbb R^d$, $U\subseteq A$ is convex and nonempty and $x_\ast\in U$ is a local minimum of $\left.f\right|_U$ such that $f$ is differentiable at $x_\ast$, then $\langle x-x_\ast,\nabla f(x_\ast)\rangle\ge0$ for all $x\in U$.

However, we are not really in this situation. Do we need to argue differently?

Answer 1

It is not true. Take $d=1$, $r=2$, and set $u(x) = -|x|$ in $B_2(0) \setminus B_1(0)$. Then all the conditions are met, as there are no conditions on $u$ in $B_1(0)$. Now choose $u$ on $B_1(0)$ such that $u\in C^2(B_2(0))$, which is possible (take some polynomial).