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arXiv · 1109.6765

On the gradient flow of a one-homogeneous functional

Ariela Briani·Antonin Chambolle·Matteo Novaga·Giandomenico Orlandi
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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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Ariela Briani, Antonin Chambolle, Matteo Novaga, Giandomenico Orlandi. 2011-10-11. On the gradient flow of a one-homogeneous functional. https://doi.org/10.1142/s1793744211000461

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