Some properties of minimizers of a variational problem involving the total variation functional
Florian Krügel · Communications on Pure & Applied Analysis · 2015
The variational problem of minimizing the functional $u \mapsto \int_\Omega |Du| + \frac{1}{p}\int_\Omega |Du|^p - \int_\Omega au$ on a domain $\Omega\subset R^2$ under zero boundary values, which among other things models the laminar flow of a Bingham fluid, shows an interesting phenomenon: its minimizer has a maximum set with positive measure (a 'plateau'). In this work we show properties of the minimizer and its plateau, most notably, connectedness and a lower bound of its measure.In addition we look at the related boundary value problem where $a=0$, $\Omega$ is a convex ring, and two boundary values are given. For this problem we show various results, including quasiconcavity of the minimizer and regularity.