On the gradient flow of a one-homogeneous functional

Details On the gradient flow of a one-homogeneous functional On the gradient flow of a one-homogeneous functional

2011-09-30

arxiv.org/abs/1109.6765v2

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

Mathematics

Analysis of PDEs

Scientific paper

10.1142/S1793744211000461

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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