EQUAZIONI ALLE DERIVATE PARZIALI
Academic year and teacher
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- Versione italiana
- Academic year
- 2018/2019
- Teacher
- DAVID JORNET CASANOVA
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/05
Training objectives
- To learn the basics of locally convex spaces and some standard operators between them. In particular, operators between spaces of analytic or differentiable functions like multiplication, composition operators and linear partial differential operators. To study suitable spaces for differentiation. Namely, spaces of smooth functions, rapidly decreasing functions, spaces of real-analytic functions, and their duals, and analyse differential operators on them. Malgrange-Ehrenpreis theorem, fundamental solutions and hypoellipticity.
Prerequisites
- Banach spaces and operators between Banach spaces. It’s convenient to have attended the course of “Analisi funzionale”
Course programme
- 1. Convergence of sequences of functions (2 hours)
a. Pointwise and uniform convergence
b. Series of functions
i. Power series in the complex plane
ii. Fourier series
2. Locally convex spaces (10 hours)
a. Topological preliminaries
b. Norms, seminorms, normed spaces and topologies
c. Linear and continuous operators. Examples. Linear partial differential operators with constant coefficients
3. Introduction to the theory of distributions of L. Schwartz (10 hours)
a. Test functions and distributions
b. The space of rapidly decreasing functions
c. Temperate distributions
d. Fourier transform
4. Partial differential operators I (10 hours)
a. Fundamental solutions. Examples
b. Malgrange-Ehrenpreis theorem
5. Partial differential operators II (10 hours)
a. Hypoelliptic operators
b. Elliptic operators. Real-analytic functions Didactic methods
- Lessons will be held at the blackboard.
There will be held lectures and exercises.
In performing the exercises we will also try to engage students. There will be assigned exercises to do at home. Learning assessment procedures
- Series of problems to do as homework individually, for each of the subjects of the course.
The evaluation is based on the correction of the exercises done by the students, and the exercises will be discussed with each student individually. Reference texts
- Elementary Functional Analysis (Barbara MacCluer)
Composition operators and classical function theory (Shapiro, Joel)
Introduction to functional analysis (Meise, Reinhold)
Analisis real y complejo (Walter Rudin)
Functional analysis (Walter Rudin)
Distributions and Operators (Gerd Grubb)
