Learning objectives
Knowledge and ability to understand. At the end of the course, the student will have to possess a robust knowledge of basic elements of abstract measure theory and BV spaces and he will have to be able to understand how these mathematical tools come into play for the resolution of problems of geometric character, using the approach classified under the label geometric measure theory.
Ability to apply knowledge and understanding. Thanks both to examples carried out in the classroom and proofs left as an exercise, the student will learn both to handle the theoretical tools presented in the course with awareness and familiarity, and to apply them to a concrete context of geometric problems mainly related to the existence of minimal surfaces. Furthermore, he must be fully aware of the pivotal examples on which general and abstract constructions are modeled.
Autonomy of judgment. The student must be able to evaluate the consistency and correctness of proofs, constructing and developing logical arguments with a clear identification of assumptions and conclusions. Furthermore, he will have to be able to evaluate the likelihood of general results stemmed from model cases or analogies with he already knows.
Communication skills. The student must be able to communicate in a clear and precise manner, appropriate for a mathematician in an advanced stage of training, mathematical contents related to the program of the course. Lectures and the direct confrontation with the teacher will favor the acquisition of a specific appropriate scientific language.
Learning ability. At the end of the course, the student will be able to autonomously deepen his knowledge in the areas covered, starting from the basic and fundamental knowledge provided by the course itself. He will be able to consult specialist texts in an autonomous way, even outside the topics covered in detail during the lessons, even written in English, in order to effectively address complex problems inherent to these topics, possibly as part of master's degree theses.
Prerequisites
Basic abstract measure theory elements and Lebesgue’s measure.
Course unit content
The course aims to continue the path started in the Advanced Analysis 1 course (as delivered in recent years) with the study of the BV space and, as applications, sets of finite perimeter and isoperimetric inequality. A large introduction will be devoted to general measure theory elements and tools necessary to address the main topics of the course.
Full programme
If necessary: sigma algebras, outer measures, measurability according to Carathéodory, regular, Borel, Radon measures, restrictions of measures, measurable functions, Lusin's and Egoroff’s theorem. L^1 functions, Fatou’s lemma, Lebesgue dominated convergence theorem. Product measures and Fubini-Tonelli theorem. 1 and n-dimensional Lebesgue measure, covering lemmas (Vitali and Besicovitch).
Differentiation of Radon measures, absolute continuity, Radon-Nikodym theorem, Lebesgue decomposition theorem, Lebesgue differentiation theorem, precise representative. (Weak) compactness for measures and for integrable functions.
Hausdorff measures and Hausdorff dimension, Steiner symmetrization, isodiametric inequality. Density points of a set, density estimates.
BV functions, structure theorem for BV, lower semicontinuity of the total variation, approximation of the derivatives in weak topology, compactness in BV, Poincaré’s and Sobolev’s inequality in BV, coarea formula.
Sets of finite perimeter. Isoperimetric inequality. Reduced boundary, blow-up, measure-theoretical normal, structure theorem for sets of finite perimeter, essential boundary. Gauss-Green theorems.
Bibliography
Evans, Gariepy: Measure theory and fine properties of functions (revised edition). Taylor and Francis, 2015.
DiBenedetto: Real analysis. Birkhäuser, 2002.
Giusti: Minimal surfaces and functions of bounded variations. Birkhäuser, 1984.
Teaching methods
The course, compatibly with the emergency situation caused by the Covid pandemic, will be carried on in the classroom with the help of a blackboard. Several simple proofs will be left to the students, in order to increase their level of competence with the topics taught throughout the course.
Assessment methods and criteria
The exam will consist of an oral test in which the knowledge of the results presented in the course, their proofs and the competence in these topics will be tested also by solving simple problems in the context of the theory.
Other information
If time permits, participants will be provided with the course notes.
