Analysis & Applied Math
by Aidan Backus (Brown University)
A function u is said to have least gradient if it minimizes its total variation, $\int |Du|$. The coarea formula shows that each level set of a function of least gradient is a minimal hypersurface, but I'll discuss how it's possible to say more: the level sets form a Lipschitz lamination (foliation of a closed set), giving functions of least gradient more regularity than a typical function of bounded variation. As a consequence, we'll see that functions of least gradient can be used to prove a continuous analogue of the max flow/min cut theorem, which generalizes a classic theorem of Strang.
