Programme

Preliminaries
- Weak topologies and weak convergence, weak lower semi-continuity of the
norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.

Sobolev spaces and variational formulation of boundary value problems in
dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle

Sobolev spaces and variational formulation of boundary value problems in
dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Omega)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle

Core Documentation

Functional analysis, H. Bre'zis

Type of delivery of the course

The course plans lectures.

Attendance

Attendance is not required but strongly suggested.

Type of evaluation

Seminar on a research paper.

Programme

Definition and elementary properties of the Sobolev spaces. Extensions theorem. Sobolev inequalities. Trace operator, compactness. Duality. Fourier transform method. Second order elliptic equations: existence of weak solutions. Regularity:interior/boundary. Maximum principles. Arguments of evolutions problems: the wave equation.

Core Documentation

Haim Breziz - Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer.
Lawrence C. Evans - Partial Differential Equations. AMS

Type of delivery of the course

Lectures on theory and exercises will be held.

Type of evaluation

An oral test will also be held to evaluate the knowledge of the course topics.

Programme

Preliminaries
- Weak topologies and weak convergence, weak lower semi-continuity of the
norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.

Sobolev spaces and variational formulation of boundary value problems in
dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle

Sobolev spaces and variational formulation of boundary value problems in
dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Omega)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle

Core Documentation

Functional analysis, H. Bre'zis

Type of delivery of the course

The course plans lectures.

Attendance

Attendance is not required but strongly suggested.

Type of evaluation

Seminar on a research paper.

Programme

Definition and elementary properties of the Sobolev spaces. Extensions theorem. Sobolev inequalities. Trace operator, compactness. Duality. Fourier transform method. Second order elliptic equations: existence of weak solutions. Regularity:interior/boundary. Maximum principles. Arguments of evolutions problems: the wave equation.

Core Documentation

Haim Breziz - Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer.
Lawrence C. Evans - Partial Differential Equations. AMS

Type of delivery of the course

Lectures on theory and exercises will be held.

Type of evaluation

An oral test will also be held to evaluate the knowledge of the course topics.

Programme

Preliminaries
- Weak topologies and weak convergence, weak lower semi-continuity of the
norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.

Sobolev spaces and variational formulation of boundary value problems in
dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle

Sobolev spaces and variational formulation of boundary value problems in
dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Omega)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle

Core Documentation

Functional analysis, H. Bre'zis

Type of delivery of the course

The course plans lectures.

Attendance

Attendance is not required but strongly suggested.

Type of evaluation

Seminar on a research paper.

Programme

Definition and elementary properties of the Sobolev spaces. Extensions theorem. Sobolev inequalities. Trace operator, compactness. Duality. Fourier transform method. Second order elliptic equations: existence of weak solutions. Regularity:interior/boundary. Maximum principles. Arguments of evolutions problems: the wave equation.

Core Documentation

Haim Breziz - Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer.
Lawrence C. Evans - Partial Differential Equations. AMS

Type of delivery of the course

Lectures on theory and exercises will be held.

Type of evaluation

An oral test will also be held to evaluate the knowledge of the course topics.