# Course Details

#### Name:

**Advanced Analysis / Advanced Analysis **

### Basic information

##### Credits:

*Master Degree in Mathematics:* 12 Ects (b)

*Master Degree in Mathematical Engineering:* 6 Ects (b)

##### Term:

*Module Advanced Analysis 1:* 1° semester

*Module Advanced Anaysis 2:* 2° semester

##### Degree(s):

Compulsory 1^{st} year Master Degree in Mathematics curriculum Generale

##### Language:

English

### Course Objectives

**Module Advanced Analysis 1:** Knowledge of mathematical methods that are widely used by researchers in the area of Applied Mathematics,
as Sobolev Spaces, distributions.
Application of this knowledge to a variety of topics, including the basic equations of mathematical physics
and some current research topics about linear and nonlinear partial differential equations.

**Module Advanced Anaysis 2:** Aim of the course is the knowledge of advanced techniques of mathematical analysis
and in particular the basic techniques of the modern theory of the
partial differential equations.

### Course Content

##### Module Advanced Analysis 1

- Distributions. Locally integrable functions. The space of test function D(U).
Distributions. 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.
- Convolution. Convolution in Lp spaces.
Regularity of the convolution. Regularizing sequences and smoothing by
means of convolutions. Convolution between distributions and regularization of
distributions. Denseness of D(U) in D'(U).
- Sobolev spaces. Definition of weak derivatives and his motivation.
Sobolev spaces Wk,p(U) and their properties. Interior and global approximation
by smooth functions. Extensions. Traces. Embeddings theorems:
Gagliardo-Nirenberg-Sobolev inequality and Embedding theorem for p < n.
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. Characterization of the dual space H-1.
- Second order parabolic equations. Definition of parabolici operator.
Weak solutions for linear parabolici equations. existence of weak solutions:
Galerkin approximation, construction of approximating solutions, energy estimates,
existence and uniqueness of solutions.
Existence of solutions of viscous scalar conservation laws.
- 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.
Definition of hyperbolic system. Quasilinear hyperbolic systems,
symmetric and symmetrizable systems. Existence of solutions: approximations,
a priori estimate, local existence of classical solutions.

##### Module Advanced Anaysis 2

- Abstract Measure theory
- AC and BV functions.
- Second order elliptic equations.
- Variational methods.
- Fourier transforms.

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

##### Module Advanced Anaysis 2

- Aim of the course is to acquire Knowledge and Understanding of Advanced Techniques of 'Mathematical Analysis.
- applying the techniques learned to problems of partial differential equations
- Acquire the ability to understand what methods and techniques can be used in various problems involving the partial differential equations.
- Acquire the ability 'to expose, explain and elaborate concepts and advanced analysis techniques.
- Acquire the ability 'to study and understand theorems and analysis techniques from books and advanced research products.

### Prerequisites and Learning Activities

**Module Advanced Analysis 1:** 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.

**Module Advanced Anaysis 2:** A good knowledge of the basic arguments of a course of Functional Analysis, in particular,
a good knowledge of the theory
of Lebesgue's integral ande the L^p spaces.
The first module of the course, in particular a good knowledge of the theory of distributions and
Sobolev spaces.

### Teaching Methods

**Language**: English

**Module Advanced Analysis 1:** Lectures

**Module Advanced Anaysis 2:** Lectures.

### Assessment Methods and Criteria

**Module Advanced Analysis 1:** Oral exam

**Module Advanced Anaysis 2:** Written exams.

### Textbooks

##### Module Advanced Analysis 1

- G. Gilardi,
**Analisi 3**. McGraw–Hill. * *
- V.S. Vladimirov,
**Equations of Mathematical Physics**. Marcel Dekker, Inc.. * *
- C.M. Dafermos,
**Hyperbolic Conservation Laws in Continuum Physics**. Springer. * *
- L.C. Evans,
**Partial Differential Equations**. AMS. * *
- M.E. Taylor,
**Partial Differential Equations, Nonlinear equations**. Springer. * *
- H. Brezis,
**Sobolev Spaces and Partial Differential Equations**. Springer. * *

##### Module Advanced Anaysis 2

- L. Evand and R. Garipey,
**Measure Theory and Fine Properties of Functions (Revised Edition)**. * *
- P. Cannarsa and T. D'aprile,
**Introduction to Measure Theory and Functional Analysis **. * *
- L.C. Evans,
**Partial differential equations**. * *
- L. Grafakos,
**Classical Fourier Analysis **. * *

### Online Teaching Resources

### Recent teaching material

This list contains only the latest published resources. Resources marked with an asterisk belong to other courses (indicated between brackets)

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### Course page updates

This course page is available (with possible updates) also for the following academic years:

*Course information last updated on: 16 novembre 2016, 13:11*