- Docente: Berardo Ruffini
- Credits: 6
- SSD: MAT/05
- Language: English
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: Second cycle degree programme (LM) in Mathematics (cod. 6730)
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from Sep 17, 2025 to Dec 19, 2025
Learning outcomes
At the end of the course, students will possess the knowledge of the main instruments of advance mathematical analysis: Sobolev spaces, spaces of generalized functions, Fourier transform. These tools will be the main instruments necessary to the quantitative and qualitative study of properties of the solutions to PDEs.
Course contents
The topics covered may be subject to changes depending on the students' preparation, but generally include the following:
- Basic real analysis on lines: upper and lower limits, monotonic and convex functions, functions of bounded variation.
- Review of basic measure theory: concepts of measurability and pathological examples (Cantor sets, non-measurable or non-Borel sets).
- Lebesgue and Hausdorff measures.
- Review of L^p spaces.
- Advanced measure theory: Vitali and Besicovitch covering theorems, Rademacher's theorem, Lebesgue points theorem.
- Review of Fourier transform theory.
- Sobolev spaces
Readings/Bibliography
Rudin: Real and harmonic analysis
Evans-Gariepy: Measure Theory and fine properties of functions
Evans : PDEs
Teaching methods
frontal lectures
Assessment methods
Written exam at the end of the course, mostly based on the theoretical aspects of the course.
Teaching tools
Beside the suggested bibliography, exercises and notes on virtuale (almamater's website for teaching support).
Office hours
See the website of Berardo Ruffini