# The 1-harmonic flow

## Lorenzo Giacomelli (SBAI Department, Sapienza University of Rome)

created by paolini on 12 Oct 2016

18 oct 2016 -- 14:30   [open in google calendar]

Aula dal Passo, dipartimento di matematica, Roma "Tor Vergata"

SEMINARIO DI ANALISI DI EQUAZIONI DIFFERENZIALI

Abstract.

The 1-harmonic flow is the formal gradient flow -- with respect to the $L^2$-distance -- of the total variation of a manifold-valued unknown function. The problem originates from image processing and has an intrinsic analytical interest as prototype of constrained and vector-valued evolution equations in BV-spaces. For the resulting PDE, I will introduce a notion of solution and I will discuss existence and uniqueness results for two specific manifolds: the hyper-octant of an N-dimensional sphere and a connected sub-arc of a regular Jordan curve. I will also present possible extensions to general manifolds, together with related open questions and conjectures. Based on joint works with Agnese Di Castro, José Mazòn, and Salvador Moll.

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