The talk is concerned with minimization problems for the total variation and the area functional, in which either an obstacle constraint is imposed or a lower-order term with a measure datum is present. These problems are naturally set in the space BV of functions of bounded variation, and the focus is on existence of BV minimizers, convex duality, and connections with BV supersolutions to nonlinear PDE. A crucial technical tool is a new Anzellotti type pairing between divergence-measure fields and gradient measures. Most of these results have been obtained in collaboration with Christoph Scheven (Duisburg-Essen).

We establish mass-angular momentum-charge inequalities in higher dimensions, including within the context of minimal supergravity. These in particular give variational characterizations of some well known stationary and static black holes. We also exhibit a special case of the Penrose inequality with angular momentum in the classical 4D setting.