JournalsaihpcVol. 35, No. 5pp. 1175–1207

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

  • Christoph Scheven

    Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany
  • Thomas Schmidt

    Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, 20146 Hamburg, Germany
On the dual formulation of obstacle problems for the total variation and the area functional cover
Download PDF

A subscription is required to access this article.

Abstract

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) BV minimizers and dual maximizers. As a particular case, these considerations yield a full characterization of BV minimizers in terms of Euler equations with a measure datum. Notably, our results apply to very general obstacles such as 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 p1p↘1.

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 BV functions. In addition, in the proofs we employ several fine results on (BV) capacities and one-sided approximation.

Cite this article

Christoph Scheven, Thomas Schmidt, On the dual formulation of obstacle problems for the total variation and the area functional. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 5, pp. 1175–1207

DOI 10.1016/J.ANIHPC.2017.10.003