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 a 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 dimension 1, and easily deduce basic qualitative properties, concerning in particular the "staircasing effect".
Ariela Briani 1 ; Antonin Chambolle 1 ; Matteo Novaga 1 ; Giandomenico Orlandi 1
@article{CML_2011__3_4_617_0,
author = {Ariela Briani and Antonin Chambolle and Matteo Novaga and Giandomenico Orlandi},
title = {On the gradient flow of a one-homogeneous functional},
journal = {Confluentes Mathematici},
pages = {617--635},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {3},
number = {4},
year = {2011},
doi = {10.1142/S1793744211000461},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744211000461/}
}
TY - JOUR AU - Ariela Briani AU - Antonin Chambolle AU - Matteo Novaga AU - Giandomenico Orlandi TI - On the gradient flow of a one-homogeneous functional JO - Confluentes Mathematici PY - 2011 SP - 617 EP - 635 VL - 3 IS - 4 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S1793744211000461/ DO - 10.1142/S1793744211000461 LA - en ID - CML_2011__3_4_617_0 ER -
%0 Journal Article %A Ariela Briani %A Antonin Chambolle %A Matteo Novaga %A Giandomenico Orlandi %T On the gradient flow of a one-homogeneous functional %J Confluentes Mathematici %D 2011 %P 617-635 %V 3 %N 4 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S1793744211000461/ %R 10.1142/S1793744211000461 %G en %F CML_2011__3_4_617_0
Ariela Briani; Antonin Chambolle; Matteo Novaga; Giandomenico Orlandi. On the gradient flow of a one-homogeneous functional. Confluentes Mathematici, Tome 3 (2011) no. 4, pp. 617-635. doi : 10.1142/S1793744211000461. https://cml.centre-mersenne.org/articles/10.1142/S1793744211000461/
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