The PIMS Postdoctoral Fellow Seminar: Cintia Pacchiano

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 (X,d,μ). For such parabolic minimizers that coincide with a time-independent Cauchy Dirichlet datum u0 on the parabolic boundary of a spacetime-cylinder Ω×(0,T) with Ω⊂X an open set and T>0, we prove the existence in the weak parabolic function space Lw1(0,T;BV(Ω)). This is a joint project with Vito Buffa and Michael Collins.