Next year I will be embarking on a dissertation on isoperimetric inequalities; I'm doing some initial research and I am frequently coming across the definition;
Let $\Omega$ be an open set in $R^n$. The perimeter $E$ in $\Omega$ is defined by;
$P$($E$;$\Omega$) = sup$\bigg\{ \int_E div \ \phi \ dx \ : \phi \in C^1(\Omega, R^n)\bigg\}$ where the infinity norm of $\phi$ is less than or equal to 1.
I just dont understand where $\phi$ comes from, how is it chosen? If anyone could give an example of how a certain perimeter of a set within another set can be found that would be amazing.