On the gradient flow of a one-homogeneous functional
Analysis of PDEs
2012-03-09 v2
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".
Cite
@article{arxiv.1109.6765,
title = {On the gradient flow of a one-homogeneous functional},
author = {Ariela Briani and Antonin Chambolle and Matteo Novaga and Giandomenico Orlandi},
journal= {arXiv preprint arXiv:1109.6765},
year = {2012}
}