TOPICS IN ADVANCED ANALYSIS B
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The main pre-requisite are some basic real analysis, Lebesgue measure on R^n, integration theory, theory of functions of one complex variable.
Oral examination. The exam consists of a discussion on the results and techniques presented during the course. The purpose of the exam is to verify: the level of knowledge and deepening of the topics addressed; the full understanding of the solving techniques and of the properties of the solutions; the ability to state theorems and expose proofs in a mathematically rigorous way; the ability to discuss the examples presented in class.
The aim of the course is to present several concepts and techniques from the modern theory of partial differential equations (PDEs). PDEs are a very rich subject and there exists no general theory to study the solvability of all the PDEs. This course will specifically focus on the study of nonlinear dispersive equations, with particular emphasis on the nonlinear Schrödinger equation.
By the end of the course, the student will:
- have acquired and be able to prove some fundamental results of modern analysis, such as the Riesz-Thorin interpolation theorem;
- be able to state and prove theorems on the existence, uniqueness, and stability of solutions to the Cauchy problem for the Schrödinger equation;
- be able to derive representation formulas and discuss the fundamental properties of solutions to the Cauchy problem for the Schrödinger equation;
- state and prove theorems on the local existence and uniqueness of solutions to the Cauchy problem for the nonlinear Schrödinger equation;
- state and prove results on the asymptotic (in time) behavior of solutions to the Cauchy problem for the nonlinear Schrödinger equation.
The course begins with a review of the fundamental properties of Lebesgue spaces, followed by the proof of the Riesz-Thorin interpolation theorem. Subsequently, the Fourier transform will be introduced, serving as a tool to define Sobolev spaces and to provide a representation formula for the solution of the Cauchy problem associated to the linear Schrödinger equation. Dispersive and Strichartz estimates will then be introduced and applied to investigate the local existence of solutions of the non-linear Schrödinger equation. Analogous results will be discussed for the Korteweg-de Vries equation, which is another example of dispersive equation. The course will finish with a discussion on the long-time behavior of solutions, distinguishing between global existence and finite-time blowup.
- Lebesgue spaces (overview)
- Schwartz functions and tempered distributions
- The Fourier transform
-- Hausdorff-Young inequality
- Interpolation theory
-- Riesz-Thorin interpolation theorem
-- Marcinkiewicz interpolation theorem
- Sobolev spaces
- Hardy-Littlewood theorem
- Hardy-Littlewood-Sobolev inequality
- The linear Schrödinger equation
-- Dispersive estimates
-- Strichartz estimates
- The nonlinear Schrödinger equation
-- Local theory in L^2 and H^1
-- Conservation laws
-- Global existence and finite-time blowup
-- Stability
Frontal lecture making use of a blackboard or equivalent tool. 64 hours
Office hours: by appointment.
e-mail: claudio.cacciapuoti@uninsubria.it
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Degree course in: MATHEMATICS