On the gradient flow of a one-homogeneous functional

Ariela Briani; Antonin Chambolle; Matteo Novaga; Giandomenico Orlandi

hal-00627812, version 2

On the gradient flow of a one-homogeneous functional

Ariela Briani () ¹, Antonin Chambolle () ², Matteo Novaga () ³, Giandomenico Orlandi ⁴

Confluentes Mathematici (CM) 03, 04 (2012) 617-635

Résumé : We consider the gradient flow of a one-homogeneous functional, whose dual involves the derivative of a constrained scalar function. We show in this case that the gradient flow is related to a weak, generalized formulation of the Hele-Shaw flow. The equivalence follows from a variational representation, which is a variant of well-known variational representations for the Hele-Shaw problem. As a consequence we get existence and uniqueness of a weak solution to the Hele-Shaw flow. We also obtain an explicit representation for the Total Variation flow in one dimension and easily deduce basic qualitative properties, concerning in particular the ''staircasing effect''.

1 : Laboratoire de Mathématiques et Physique Théorique (LMPT)
Université François Rabelais - Tours – CNRS : UMR7350
2 : Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP)
Polytechnique - X – CNRS : UMR7641
3 : Dipartimento di Matematica Pura ed Applicata
Università degli studi di Padova
4 : Department of Computer Science / Dipartimento di Informatica [Verona]
University of Verona – Università degli studi di Verona

hal-00627812, version 2
http://hal.archives-ouvertes.fr/hal-00627812
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Contributeur : Ariela Briani <>
Soumis le : Mardi 11 Octobre 2011, 14:37:53
Dernière modification le : Vendredi 1 Mars 2013, 17:38:08

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arXiv : 1109.6765

DOI : 10.1142/S1793744211000461

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