# American Institute of Mathematical Sciences

January  2015, 14(1): 341-360. doi: 10.3934/cpaa.2015.14.341

## Some properties of minimizers of a variational problem involving the total variation functional

 1 Mathematisches Institut der Universität zu Köln, Weyertal 86 -- 90, 50931 Köln, Germany

Received  January 2014 Revised  March 2014 Published  September 2014

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.
Citation: Florian Krügel. Some properties of minimizers of a variational problem involving the total variation functional. Communications on Pure & Applied Analysis, 2015, 14 (1) : 341-360. doi: 10.3934/cpaa.2015.14.341
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