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Journal Articles Confluentes Mathematici Year : 2012

On the gradient flow of a one-homogeneous functional

Abstract

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''.
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Dates and versions

hal-00627812 , version 1 (29-09-2011)
hal-00627812 , version 2 (11-10-2011)

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Ariela Briani, Antonin Chambolle, Matteo Novaga, Giandomenico Orlandi. On the gradient flow of a one-homogeneous functional. Confluentes Mathematici, 2012, 03 (04), pp.617-635. ⟨10.1142/S1793744211000461⟩. ⟨hal-00627812v2⟩
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