The obstacle problem  for the total variation flow

Bögelein, Verena; Duzaar, Frank; Scheven, Christoph

doi:10.24033/asens.2306

The obstacle problem for the total variation flow
[Problème de l'obstacle lié au flot de variation totale]

Bögelein, Verena ; Duzaar, Frank ; Scheven, Christoph

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 5, pp. 1143-1188.

Résumé
Abstract

Nous démontrons des résultats d'existence pour le problème de l'obstacle lié au flot de variation totale. Pour les obstacles suffisamment réguliers, nous obtenons les solutions via le procédé de minimisation des mouvements. Les résultats pour les obstacles plus généraux sont dérivés par approximation avec des obstacles réguliers dans le sens d'une propriété de stabilité de solutions relative à l'obstacle. Enfin, nous présentons le traitement de la contrepartie parabolique d'un résultat classique concernant les surfaces minimales avec des obstacles minces au moyen de la mesure variationnelle $(n - 1)$ -dimensionnelle introduite par De Giorgi, Colombini et Piccinini.

We prove existence results for the obstacle problem related to the total variation flow. For sufficiently regular obstacles the solutions are obtained via the method of minimizing movements. The results for more general obstacles are derived by approximation with regular obstacles in the sense of a stability property of solutions with respect to the obstacle. Finally, we present the treatment of the evolutionary counterpart of a classical stationary result concerning minimal surfaces with thin obstacles by means of the $(n - 1)$ -dimensional variational measure introduced by De Giorgi, Colombini and Piccinini.

MR Zbl

DOI : 10.24033/asens.2306

Classification : 35K86, 49J40, 49J45.
Keywords: Total variation flow, obstacle problem, minimizing movements, relaxation.
Mot clés : Flot de variation totale, problème de l'obstacle, minimisation des mouvements, relaxation.

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     title = {The obstacle problem  for the total variation flow},
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Bögelein, Verena; Duzaar, Frank; Scheven, Christoph. The obstacle problem  for the total variation flow. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 5, pp. 1143-1188. doi : 10.24033/asens.2306. http://www.numdam.org/articles/10.24033/asens.2306/

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