Let $\Omega$ be a bounded smooth domain and consider $$K=\{ v \in H^1(\Omega) : v \geq 0 \text{ a.e.}\}.$$
The radial cone at a point $v \in K$ is defined as the set
$$R(v) := \{ w \in H^1(\Omega) : \exists C > 0 : v + cw \in K \text{ for all $c < C$}\}$$
Is there a way to explicitly character this set $R(v)$ in terms of the subdomain $\{ x : v(x) = 0 \}$, or some other characterization?