Calculus of Variations and Geometric Measure Theory
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A. Briani - A. Chambolle - M. Novaga - G. Orlandi

On the gradient flow of a one-homogeneous functional

created by novaga on 28 Sep 2011
modified by orlandi on 24 Jul 2012

[BibTeX]

Published Paper

Inserted: 28 sep 2011
Last Updated: 24 jul 2012

Journal: Confluentes Mathematici
Volume: 3
Number: 4
Pages: 1-19
Year: 2011

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.

Keywords: hele-shaw, one-homogeneous functionals


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