Calculus of Variations and Geometric Measure Theory
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C. Scheven - T. Schmidt

On the dual formulation of obstacle problems for the total variation and the area functional

created by schmidt on 30 Jan 2017
modified on 15 Nov 2017


Accepted Paper

Inserted: 30 jan 2017
Last Updated: 15 nov 2017

Journal: Ann. Inst. Henri Poincaré, Anal. Non Linéaire
Year: 2017
Links: Link to the published version


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.


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