1
$\begingroup$

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.

  • 0
    If $P_K$ happens to satisfy $P_K(H^1_0)\subset H^1_0$, then $P_{K_0}$ is just the restriction of $P_K$ to $H^1_0$. But I doubt this is true. And if this fails for some function $f\in P_K(H^1_0)$, then there isn't a reason to expect a relation between $P_Kf $ and $P_{K_0}f$ – 2017-01-10
  • 0
    The answer also depends on the norm in $H^1(\Omega)$. – 2017-01-11

0 Answers 0