oc.optimization control - optimize with respect to domain shape

Let $Gamma$ be the set of all closed $C^2$ curves in the plane which enclose unit area and let $Omega$ be the set of all subsets of $mathbb{R}^2$ that are enclosed by some curve in $Gamma$. Now let $f: mathbb{R}^2rightarrowmathbb{R}$ be a real-valued function on the plane. How can we find the set $omegainOmega$ with boundary $gammainGamma$ such that
$
int_omega f
$
is minimized?

A related question (with a more physics-style interpretation): Let $Omega$ and $Gamma$ be as before. Let $u(vec{x})$ be the real-valued function on the plane that solves the PDE
$$
Delta u(vec{x}) = 0 text{for } vec{x}inomega
$$

$$
u(vec{x}) = 1 text{for } vec{x}in partialomega,
$$

where $omega$ is some set in $Omega$. How can we find the $omega$ that minimizes the quantity $int_{gamma}|du/dn|^2$ where $gamma$ is the boundary of $omega$ with unit outward normal $n$ ?

I'm more interested in finding out which branch of mathematics studies questions like this and what concepts/tools are important to approach questions like this. Any references or suggestions to similar problems are greatly appreciated. (A friend suggested I tag this as geometric measure theory, but I don't know how appropriate that is)