L1-Gradient Flow of Convex Functionals
Chambolle, Antonin; Novaga, Matteo (2022), L1-Gradient Flow of Convex Functionals. https://basepub.dauphine.psl.eu/handle/123456789/23084
TypeDocument de travail / Working paper
Series titleCahier de recherche CEREMADE, Université Paris Dauphine-PSL
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Dipartimento di Matematica [Pisa]
Abstract (EN)We are interested in the gradient flow of a general first order convex functional with respect to the L¹-topology. By means of an implicit minimization scheme, we show existence of a global limit solution, which satisfies an energy-dissipation estimate, and solves a non-linear and non-local gradient flow equation, under the assumption of strong convexity of the energy. Under a monotonicity assumption we can also prove uniqueness of the limit solution, even though this remains an open question in full generality. We also consider a geometric evolution corresponding to the L¹-gradient flow of the anisotropic perimeter. When the initial set is convex, we show that the limit solution is monotone for the inclusion, convex and unique until it reaches the Cheeger set of the initial datum. Eventually, we show with some examples that uniqueness cannot be expected in general in the geometric case.
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