Master students in mathematics and mathematical analysis and modelling
Voraussetzungen
Calculus 3 and some basic knowledge in functional analysis
Lernorganisation
Depending on the restrictions of students, we decide which one of Monday and Wednesday is lecture and which is exercise class
Leistungsnachweis
Portfolio exam consisting of oral exam of 20 minutes about the lecture, and presentation of three exercises
Online/Digitale Veranstaltung
Veranstaltung wird online/digital abgehalten.
Hauptunterrichtssprache
englisch
Literaturhinweise
A. Björn and J. Björn, Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, 17. European Mathematical Society(EMS), Zurich, 2011. xii+403 pp.
In the Euclidean space, Sobolev functions and functions of bounded variation (BV functions) are defined as weakly differentiable functions, which are useful for example in the study of various minimization problems. In the past 20 years, a theory of Sobolev and BV functions as well as various other results in analysis has been developed in the abstract setting of metric measure spaces. A metric measure space (X,d,μ) is simply a set X equipped with a distance function d and a measure μ, the typical example being X=Rn equipped with the Lebesgue measure μ=Ln. A metric measure space lacks linear structure so it is not possible to define partial derivatives, but instead one can use so-called upper gradients. Using these, we develop the theory of Sobolev functions and then BV functions, and consider some applications. The main motivation for studying analysis in metric spaces is to develop general theories and robust techniques that do not rely on the specific structure of the space, and thus provide a unified treatment that applies to Euclidean spaces but also certain Riemannian manifolds, Carnot groups, etc. The course is quite self-contained, and knowledge of Sobolev or BV theory in Euclidean spaces is not required, but some basic knowledge of measure theory and functional analysis is assumed. We will mostly follow the monograph "Nonlinear potential theory on metric spaces" by Björn-Björn (2011).
