Which is correct representation for function of variable?
Based on your new explanations, I think the answer is just
$$
E_{u_i}
$$
given that what you're really trying to find a shorthand for is the expression inside the summation sign in (5) of the screenshot. That is:

$$
\min_{U} \left\{ \lambda \sum_{i=1}^N \int_\Omega r_i(\chi, p_i(\chi)) u_i(\chi) d\chi \right\} = \min_{U} \left\{ \lambda \sum_{i=1}^N E_{u_i} \right\}
$$

so that
$$
E_{u_i} = \int_\Omega r_i(\chi, p_i(\chi)) u_i(\chi) d\chi
$$



Original answer below for reference

As per @Fat32's comment, any of:


$E(\sigma, z, p_i)$
$E_{\sigma,p_i}(z)$
$E(z; \sigma,p_i)$


is acceptable.

But you need to tighten up your notation. What is $x$ vs $z$? How does $\Omega$ relate to them? What is $y$?