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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 .
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.
@article{AIHPC_2018__35_5_1175_0,
author = {Scheven, Christoph and Schmidt, Thomas},
title = {On the dual formulation of obstacle problems for the total variation and the area functional},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1175--1207},
publisher = {Elsevier},
volume = {35},
number = {5},
year = {2018},
doi = {10.1016/j.anihpc.2017.10.003},
mrnumber = {3813962},
zbl = {1401.35112},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.10.003/}
}
TY - JOUR AU - Scheven, Christoph AU - Schmidt, Thomas TI - On the dual formulation of obstacle problems for the total variation and the area functional JO - Annales de l'I.H.P. Analyse non linéaire PY - 2018 SP - 1175 EP - 1207 VL - 35 IS - 5 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.10.003/ DO - 10.1016/j.anihpc.2017.10.003 LA - en ID - AIHPC_2018__35_5_1175_0 ER -
%0 Journal Article %A Scheven, Christoph %A Schmidt, Thomas %T On the dual formulation of obstacle problems for the total variation and the area functional %J Annales de l'I.H.P. Analyse non linéaire %D 2018 %P 1175-1207 %V 35 %N 5 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.10.003/ %R 10.1016/j.anihpc.2017.10.003 %G en %F AIHPC_2018__35_5_1175_0
Scheven, Christoph; Schmidt, Thomas. On the dual formulation of obstacle problems for the total variation and the area functional. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 5, pp. 1175-1207. doi: 10.1016/j.anihpc.2017.10.003
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