[CvGmt News] Seminario Felix Otto (21/02/2017)

[Alfonso Sorrentino]

SEMINARIO DI EQUAZIONI DIFFERENZIALI
Dipartimento di Matematica
Universita' degli Studi di Roma "Tor Vergata"


Martedi' 21 Febbraio 2017, ore 14:30 Aula D'Antoni

(NOTARE IL CAMBIO D'AULA)


Prof. Felix Otto (Max Planck Institut fuer Mathematik, Leipzig)


Titolo: The thresholding scheme for mean curvature flow and De Giorgi's
ideas for minimizing movements


Abstract:
We consider the thresholding scheme, a practically relevant time
discretization for mean curvature flow (MCF) introduced by
Bence-Merriman-Osher, and prove a (conditional) convergence result in the
multi-phase case.  The result establishes convergence towards a weak
formulation in the framework of sets of finite perimeter.

The proof is based on the interpretation of the thresholding scheme as a
minimizing movement scheme, which means that the thresholding scheme
preserves the structure of (multi-phase) mean curvature flow as a gradient
flow  w. r. t.  the total interfacial energy. More precisely, the
thresholding scheme is a minimizing movement scheme for an energy
functional that $\Gamma$-converges to the total interfacial energy (joint
work with Selim Esedoglu).

Our proof is similar in spirit to the convergence result by
Luckhaus-Sturzenhecker of the Almgren-Taylor-Wang scheme, a more academic
minimizing movement scheme for MCF.  In particular, ours is a conditional
convergence result, in the sense that we assume that the energy of the
approximation converges to the energy of the limit.

In addition, we appeal to an argument of De Giorgi to show that the  limit
also satisfies Brakke's inequality, a way to encode the gradient flow
structure of MCF.  De Giorgi's abstract set-up of metric slope and
variational interpolation for minimizing movements, as formulated by
Ambrosio-Gigli-Savare, is taylor-made for this limit.
This is joint work with Tim Laux.


Website: http://www.mat.uniroma2.it/~sorrenti/SeminarioED.html



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