Inserted: 30 jan 2017
Last Updated: 19 mar 2017
We investigate the Dirichlet minimization problem for the total variation and the area functional with a one-sided obstacle. Relying on techniques of convex analysis, we identify certain dual maximization problems for bounded divergence-measure fields, and we establish duality formulas and pointwise relations between (generalized) $\rm BV$ minimizers and dual maximizers. As a particular case, these considerations yield a full characterization of $\rm BV$ minimizers in terms of Euler equations with a measure datum. Notably, our results apply to very general obstacles such as $\rm BV$ obstacles, thin obstacles, and boundary obstacles, and they include information on exceptional sets and up to the boundary. As a side benefit, in some cases we also obtain assertions on the limit behavior of $p$-Laplace type obstacle problems for $p\searrow1$.
On the technical side, the statements and proofs of our results crucially depend on new versions of Anzellotti type pairings which involve general divergence-measure fields and specific representatives of $\rm BV$ functions. In addition, in the proofs we employ several fine results on ($\rm BV$) capacities and one-sided approximation.