- 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 change based on the students' background, but will generally include the following:
PART I: Measure Theory-
Review of real analysis tools: Limits inferior and superior, an overview of cardinality.
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Basic measure theory: Measurability of sets and functions, convergence theorems. Pathological examples (Cantor sets) and their use in the study of non-measurable or non-Borelian sets.
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Lebesgue and Hausdorff measures and their comparison.
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Review of L^p spaces.
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Continuous functions and measures: The Riesz Representation Theorem.
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Fine properties of measure theory: Vitali and Besicovitch covering theorems, differentiation of measures, Rademacher's Theorem, Lebesgue's differentiation theorem.
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Definition and motivation.
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Approximation by smooth functions.
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Embedding theorems (Sobolev, Poincaré, and trace inequalities).
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Review of Fourier analysis and fractional 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