Nonlinear variational problems on metric measure spaces

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School of Science | Doctoral thesis (article-based) | Defence date: 2022-09-23
Degree programme
60 + app. 140
Aalto University publication series DOCTORAL THESES, 109/2022
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. The work consists of four articles in which we study the total variation flow and quasiminimizers of a (p,q)-Dirichlet integral. More specifically, we define variational solutions to the total variation flow in metric measure spaces. We establish 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 ata given point. We then take a purely variational approach to a (p,q)-Dirichlet integral, define its quasiminimizers, and using the concept of upper gradients together with Newtonian spaces, we develop interior regularity, as well as regularity up to the boundary, in the context of general metric measure spaces.For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally Hölder continuous and they satisfy Harnack inequality. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for Hölder continuity and aWiener type regularity condition for continuity up to the boundary. Lastly, we prove higher integrability and stability results in metric measure spaces, for quasiminimizers related to the (p,q)-Dirichlet integral. The results and the methods used in the proofs are discussed in detail, and some related open questions are presented.
Supervising professor
Kinnunen, Juha, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland
Thesis advisor
Kinnunen, Juha, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland
partial differential equations, parabolic, nonlinear analysis, total variation flow, Dirichlet integral, regularity theory, calculus of variations, energy estimates, quasiminimizers, metric spaces, doubling measure, Poincaré inequality, upper gradients, Newtonian spaces, Harnack estimate, existence theory, higher integrability, stability, comparison principle
Other note
  • [Publication 1]: Antonella Nastasi and Cintia Pacchiano Camacho. Regularity properties for quasiminimizers of a (p, q)-Dirichlet integral. Calculus of Variations and Partial Differential Equations, 60(6): Paper No. 227, 37, September 2021. DOI 10.1007/s00526-021-02099-y
  • [Publication 2]: Vito Buffa, Michael Collins and Cintia Pacchiano Camacho. Existence of parabolic minimizers to the total variation flow on metric measure spaces. Manuscripta Mathematica, January 2022. DOI 10.1007/s00229-021-01350-2
  • [Publication 3]: Vito Buffa, Juha Kinnunen and Cintia Pacchiano Camacho. Variational solutions to the total variation flow on metric measure spaces. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, 220: Paper No. 112859, February 2022.
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  • [Publication 4]: Antonella Nastasi and Cintia Pacchiano Camacho. Higher integrability and stability of (p, q)-quasiminimizers. Submitted to a journal, March 2022.