Advanced analysis
- Code: DT0113
- Unit Coordinator: Corrado Lattanzio
- ECTS Credits: 9
- Semester: 1
- Year: 2
- Campus: University of L'Aquila
- Language: English
- Aims:
LEARNING OBJECTIVES.
The course aims at providing advanced mathematical notions used in the field of (applied) mathematical analysis and their applications to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematics was verified at the time of the planning of the master program.
LEARNING OUTCOMES.
At the end of the course, the student should:
1. know the advanced mathematical notions used in the field of (applied) mathematical analysis, as measure theory, Sobolev Spaces, distributions, and their applications to the theory of linear and non-linear partial differential equations;
2. understand and be able to explain thesis and proofs in advanced mathematical analysis;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics. - Content:
Distributions. Locally integrable functions. The space of test function D(Ω). Distributions associated to locally integrable functions. Singular distributions. Examples. Operations on distributions: sum, products times functions, change of variables, restrictions, tensor product. Differentiation and his properties; comparison with classical derivatives. Differentiation of jump functions. Partition of unity. Support of a distribution; compactly supported distributions. Fourier transform and tempered distributions. Convolution between distributions and regularization of distributions. Denseness of D(Ω) in D′(Ω).
Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(Ω) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo–Nirenberg–Sobolev inequality and embedding theorem for p < n. Hölder spaces. Morrey inequality. Embedding theorem for p > n. Sobolev inequalities in the general case. Compact embeddings: Rellich–Kondrachov theorem, Poincaré inequalities. Embedding theorem for p = n. Characterization of the dual space H-1.
Second order parabolic equations. Definition of parabolic operator. Weak solutions for linear parabolic equations. Existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions.
First order nonlinear hyperbolic equations. Scalar conservation laws: derivation, examples. Weak solutions, Rankine-Hugoniot conditions, entropy conditions. L1 stability, uniqueness and comparison for weak entropy solutions. Convergence of the vanishing viscosity and existence of the weak, entropy solution. Riemann problem.
Measures. System of sets, Positive Measures, Outer Measures, Construction of Measures, Signed Measures, Borel and Radon Measures
Integration. Measurable Functions, Simple Functions, Convergence Almost Everywhere, Integral of Measurable Functions, Convergences of Integrals, Fubini-Tonelli Theorems.
Differentiation. The Radon-Nikodym Theorem, Differentiation on Euclidean space, Differentiation of the real line.
Radon measures and continuous functions. Spaces of continuous functions, Riesz Theorem. - Pre-requisites:
Basic notions of functional analysis, functions of complex values, standard properties of classical solutions of semilinear first order equations, heat equation, wave equation, Laplace and Poisson's equations.
- Reading list:
- L. Ambrosio, G. Da Prato, A. Mennucci. Introduction to measure theory and integration. Edizioni della Normale.
- L. Ambrosio, N. Fusco, D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs.
- V.I. Bogachev. Measure theory, Volume I, Springer.
- H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer.
- P. Cannarsa, T. D’Aprile. Introduction to Measure Theory and Functional Analysis. Springer.
- C.M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics, Springer.
- L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS.
- L. Evans, R. Gariepy. Measure Theory and Fine Properties of Functions, CRC Press. Revised Edition.
- G.B. Folland. Real analysis: Modern techniques and their applications. New York Wiley
- G. Gilardi. Analisi 3. McGraw–Hill.
- L. Grafakos, Classical Fourier Analysis. Springer.
- V.S. Vladimirov. Equations of Mathematical Physics. Marcel Dekker, Inc
