Public defence in Mathematics, M.Sc. Cintia Pacchiano Camacho
Opponent is Assistant Professor Cristiana De Filippis, University of Parma, Italy
Custos is Professor Juha Kinnunen, Aalto University School of Science, Department of Mathematics and Systems Analysis
Contact details of the doctoral student: [email protected], +358 (50) 4149900
The public defence will be organised on campus.
The doctoral thesis is publicly displayed 10 days before the defence in the publication archive Aaltodoc of Aalto University.
This dissertation studies existence and regularity properties of functions related to the calculus of variations on metric measure spaces that support a weak Poincaré inequality and doubling measure. We concentrate especially on variational solutions to the total variation flow, and quasiminimizers to the (p,q)-Dirichlet integral. The main interest in this work is to extend some classical results of the calculus of variations to metric measure spaces.
Variational methods appeared as an answer to the problem of finding minima of functionals. It is about giving a necessary and sufficient condition for the existence of the minimum, as well as conditions that allow its calculation and algorithms that let us compute it. Variational calculus is intimately linked with the theory of partial differential equations since the conditions for existence of a solution to the minimization problem normally depend on the fact that said solution satisfies a certain differential equation.
This dissertation focuses on various classes of functions related to a Dirichlet type integral. We first define variational solutions to the total variation flow (TVF) in metric measure spaces. We establish their existence and, using energy estimates and the properties of the underlying metric, we give necessary and sufficient conditions for a variational solution to be continuous at a given point. As far as we know, this is the first time that existence and regularity questions are discussed for parabolic problems with linear growth on metric measure spaces. We then take a purely variational approach to a (p,q)-Dirichlet integral. We define its quasiminimizers, and using the concept of upper gradients together with Newtonian spaces, we develop interior regularity and regularity up to the boundary. Lastly, we prove higher integrability together with stability results in the context of general metric measure spaces.
Analysis on metric spaces is nowadays an active and independent field, bringing together researchers from different parts of the mathematical spectrum. It has applications to disciplines as diverse as geometric group theory, nonlinear PDEs, and even theoretical computer science. This can offer us a better understanding of the phenomena and also lead to new results, even in the classical Euclidean case.