On the gradient flow of a one-homogeneous functional

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 representati...

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Bibliographic Details
Published in:arXiv.org 2011-10
Main Authors: Briani, Ariela, Chambolle, Antonin, Novaga, Matteo, Orlandi, Giandomenico
Format: Article
Language:English
Subjects:
Gradient flow
Representations
Online Access:Get full text
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Summary: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".
ISSN:2331-8422
DOI:10.48550/arxiv.1109.6765