## The PIMS Postdoctoral Fellow Seminar: Cintia Pacchiano

- Date: 02/22/2023

Online

Total Variation Flow on metric measure spaces

In this project, we discuss some fine properties and the existence of variational solutions to the Total Variation Flow. Instead of the classical Euclidean setting, we intend to work mostly in the general setting of metric measure spaces.

During the past two decades, a theory of Sobolev functions and BV functions has been developed in this abstract setting. A central motivation for developing such a theory has been the desire to unify the assumptions and methods employed in various specific spaces, such as weighted Euclidean spaces, Riemannian manifolds, Heisenberg groups, graphs, etc.

The Total Variation Flow can be understood as the process of diminishing the total variation using the gradient descent method. This idea can be reformulated using variational solutions, and it gives rise to a definition of parabolic minimizers. The approach’s advantages using a minimization formulation include much better convergence and stability properties. This is essential as the solutions naturally lie only in the space of BV functions.

We give an existence proof for variational solutions u associated to the Total Variation Flow. Here, the functions being considered are defined on a metric measure space \(\left( X, d, \mu \right)\). For such parabolic minimizers that coincide with a time-independent Cauchy Dirichlet datum \(u_0\) on the parabolic boundary of a spacetime-cylinder \(\Omega \times \left(0, T\right)\) with \(\Omega \subset X\) an open set and \(T > 0\), we prove the existence in the weak parabolic function space \(L^{1}_{w}\left(0,T;BV\left(\Omega\right)\right)\). This is a joint project with Vito Buffa and Michael Collins.

**Speaker Biography**:

Cintia Pacchiano is a PIMS Postdoctoral Fellowship at the University of Calgary under the sponsorship of Prof. Cristian Rios. Her research focuses on Mathematical Analysis, more specifically, in existence and regularity theory for a broad class of partial differential equations. She completed her Ph.D. at Aalto University School of Science, in Finland, where was part of the Nonlinear PDE research group under the supervision of Prof. Juha Kinnunen. During her Ph.D. she discovered the analysis on general metric measure spaces. She obtained her Masters degree from the University of Bonn, and a Bachelor in mathematics from Universidad Nacional Autonoma de Mexico (UNAM). Besides research, she enjoy running, taking pictures, painting, and playing the piano.

**Medium**: Read more about our Cintia and their research here.

This event is part of the Emergent Research: The PIMS Postdoctoral Fellow Colloquium Series.

This seminar takes places across multiple time zones: 9:30 AM Pacific/ 10:30 AM Mountain / 11:30 AM Central

Register via Zoom to receive the link for this event and the rest of the series.

See past seminar recordings on MathTube.