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 $f: \mathbb{R}^2\rightarrow\mathbb{R}$ be a real-valued function on the plane. How can we find the set $\omega\in\Omega$ with boundary $\gamma\in\Gamma$ 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 $$ \Delta u(\vec{x}) = 0\ \text{for } \vec{x}\in\omega $$
$$ u(\vec{x}) = 1\ \text{for } \vec{x}\in \partial\omega, $$
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)
