Advanced Topics in Real Analysis
A.Y. 2024/2025
Learning objectives
The aim of the course is to provide a modern and robust treatment of measure and integration theory, and of the theory of differentiability of functions, initiated in Mathematical Analysis 4 and Real Analysis. In particular, we will describe the generalization of the fundamental theorems of Calculus to weakly differentiable functions and non-smooth sets in Euclidean spaces, providing a systematic introduction to Geometric Measure Theory (GMT). In this framework, we will study the almost everywhere differentiability of Lipschitz maps, the existence of tangent spaces to rectifiable sets almost everywhere, and the theories of sets of finite perimeter and functions of bounded variation in arbitrary dimension. Moreover, we will discuss the application of GMT techniques to geometric variational problems, and we will provide an introduction to the regularity theory and the analysis of singularities of solutions, with an emphasis on the weak theory of minimal surfaces.
Expected learning outcomes
Knowledge of basic notions and techniques in Geometric Measure Theory in R^n: Lipschitz maps, rectifiable sets, area and coarea formulas and applications, theory of BV functions and sets of finite perimeter, densities and tangent cones, variation formulas for the perimeter and partial regularity of perimeter minimizing sets.
Lesson period: Second semester
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
- Recalls on general measure and integration theory: Borel and Radon measures in Euclidean spaces; Hausdorff measures; theorems of Egorov and Lusin; Riesz representation theorem; weak convergence and weak compactness of Radon measures; the Lebesgue-Besicovitch differentiation theorem; Radon-Nikodym derivative; signed and vector-valued measures and their total variation.
- Covering theorems.
- Lipschitz maps and their properties. Extension theorem. Rademacher's theorem.
- Area formula and its consequences and applications.
- Coarea formula and its consequences and applications.
- Rectifiable sets and their local properties. Existence of geometric and approximate tangent spaces almost everywhere. Area and coarea for rectifiable sets.
- Sets of (locally) finite perimeter and functions of (locally) bounded variation. Gauss-Green measure. Regularization. Compactness under perimeter bounds. Lower semicontinuity of perimeter. Geometric variational problems involving the perimeter and existence of minimizers. The Euclidean isoperimetric problem.
- Reduced boundary and its properties. Rectifiability of reduced boundaries. Federer's theorem and the De Giorgi-Federer divergence theorem.
- First variation formula of perimeter. Mean curvature. Stationary sets. Monotonicity of the mass density ratio. Densities and tangent cones. Density estimates. Second variation of perimeter. Stability.
- An introduction to the regularity and the analysis of singularities of perimeter minimizers and stationary sets. Epsilon-regularity. Simons' theorem on stable stationary cones and the Bernstein problem. Almgren-Federer dimension reduction and Hausdorff dimension estimate of singular sets for perimeter minimizers.
- Covering theorems.
- Lipschitz maps and their properties. Extension theorem. Rademacher's theorem.
- Area formula and its consequences and applications.
- Coarea formula and its consequences and applications.
- Rectifiable sets and their local properties. Existence of geometric and approximate tangent spaces almost everywhere. Area and coarea for rectifiable sets.
- Sets of (locally) finite perimeter and functions of (locally) bounded variation. Gauss-Green measure. Regularization. Compactness under perimeter bounds. Lower semicontinuity of perimeter. Geometric variational problems involving the perimeter and existence of minimizers. The Euclidean isoperimetric problem.
- Reduced boundary and its properties. Rectifiability of reduced boundaries. Federer's theorem and the De Giorgi-Federer divergence theorem.
- First variation formula of perimeter. Mean curvature. Stationary sets. Monotonicity of the mass density ratio. Densities and tangent cones. Density estimates. Second variation of perimeter. Stability.
- An introduction to the regularity and the analysis of singularities of perimeter minimizers and stationary sets. Epsilon-regularity. Simons' theorem on stable stationary cones and the Bernstein problem. Almgren-Federer dimension reduction and Hausdorff dimension estimate of singular sets for perimeter minimizers.
Prerequisites for admission
The contents of Mathematical Analysis 1, 2, 3, 4, and Real Analysis are prerequisites.
Some of the material discussed in Elements of Functional Analysis (particularly some topics in measure theory) will be recalled at the beginning of the course without proof. Hence, having attended Elements of Functional Analysis is recommended, but not mandatory.
Some of the material discussed in Elements of Functional Analysis (particularly some topics in measure theory) will be recalled at the beginning of the course without proof. Hence, having attended Elements of Functional Analysis is recommended, but not mandatory.
Teaching methods
The course is delivered via standard lectures.
Attendance is strongly recommended.
Attendance is strongly recommended.
Teaching Resources
- F. Maggi - Sets of finite perimeter and geometric variational problems. An introduction to Geometric Measure Theory. Cambridge University Press. 2012.
- L.C. Evans and R.F. Gariepy - Measure theory and fine properties of functions. CRC Press. 1992.
- L. Simon - Lectures on Geometric Measure Theory. Australian National University, Centre for Mathematical Analysis. 1983.
- L. Ambrosio, N. Fusco, D. Pallara - Functions of bounded variation and free discontinuity problems. Oxford Science Publications. 2000.
- L.C. Evans and R.F. Gariepy - Measure theory and fine properties of functions. CRC Press. 1992.
- L. Simon - Lectures on Geometric Measure Theory. Australian National University, Centre for Mathematical Analysis. 1983.
- L. Ambrosio, N. Fusco, D. Pallara - Functions of bounded variation and free discontinuity problems. Oxford Science Publications. 2000.
Assessment methods and Criteria
The exam consists of an oral discussion on the topics presented during the lectures. The purpose of the discussion is to verify that the student knows and understands the content of the lectures, can establish relationships among the various topics, and can effectively apply the techniques presented in concrete situations.
Few homework problems may be suggested during the lectures, with the purpose of facilitating the study of the theoretical material with the help of concrete examples. Solving such problems is not mandatory; nonetheless, similar problems may be proposed during the oral examination.
The exam is passed upon successful completion of the oral examination. A final mark in the range 0-30 (with 18 being the minimum passing grade) is given and communicated immediately at the end of the oral examination.
Few homework problems may be suggested during the lectures, with the purpose of facilitating the study of the theoretical material with the help of concrete examples. Solving such problems is not mandatory; nonetheless, similar problems may be proposed during the oral examination.
The exam is passed upon successful completion of the oral examination. A final mark in the range 0-30 (with 18 being the minimum passing grade) is given and communicated immediately at the end of the oral examination.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor:
Stuvard Salvatore
Professor(s)
Reception:
Please, request an appointment via email
Room 1041, Department of Mathematics, Via Cesare Saldini 50, first floor or via Zoom conference call