Take $\Omega$ to be a bounded domain.
Define $$K:=\{ u \in H^1(\Omega) : u \geq 0 \text{ a.e. on $\Omega$}\}$$ $$K_0:=\{ u \in H^1_0(\Omega) : u \geq 0 \text{ a.e. on $\Omega$}\}.$$
Let the norm in $H^1_0(\Omega)$ be given by $\lVert u \rVert_{H^1_0}^2 := \int_\Omega |\nabla u|^2$.
Define the orthogonal projection maps $P_K:H^1 \to K$ and $P_{K_0}:H^1_0 \to K_0$.
Is it possible to write $P_K$ in terms of $P_{K_0}$?
Neither operator can be written explicitly in terms of arguments, but I only need to write them in terms of each other.