In the level set method for shape optimization the shape functionals are defined as:
$$J_{dom}(\Omega) = \int_{\Omega} \psi (x) dx, J_{bd}(\Omega) = \int_{\partial \Omega} \psi(x) d \mathcal{H}^{d-1}$$
and $\Omega$ is related via a level set function $\phi$.
What is the function $\psi$ inside level set definitions for shape functionals? Where is it gotten from?
