## Function Spaces and Applications - MATH97102

### Aims

The purpose of this course is to introduce the basic function spaces and to train the student into the basic methodologies needed to undertake the analysis of Partial Differential Equations and to prepare them for the course ‘Advanced topics in Partial Differential Equations’’ where this framework will be applied. The course is designed as a stand-alone course. No background in topology or measure theory is needed as these concepts will be reviewed at the beginning of the course.

The course will span the basic aspects of modern functional spaces: integration theory, Banach spaces, spaces of differentiable functions and of integrable functions, convolution and regularization, compactness and Hilbert spaces. The concepts of Distributions, Fourier transforms and Sobolev spaces will be taught in the follow-up course ‘’Advanced topics in Partial Differential Equations’’ as they are tightly connected to the resolution of elliptic PDE’s and the material taught in the present course is already significant.

### Role

Lecturer

## Advanced Topics in Partial Differential Equations - M5M12

### Aims

This course develops the analysis of boundary value problems for elliptic and parabolic PDE’s using the variational approach. It is a follow-up of ‘Function spaces and applications’ but is open to other students as well provided they have sufficient command of analysis. An introductory Partial Differential Equation course is not needed either, although certainly useful.

The course consists of three parts. The first part (divided in two chapters) develops further tools needed for the study of boundary value problem, namely distributions and Sobolev spaces. The following two parts are devoted to elliptic and parabolic equations on bounded domains. They present the variational approach and spectral theory of elliptic operators as well as their use in the existence theory for parabolic problems.

Extra reading about elliptic equations on the whole space case will be given. The aim of the course is to expose the students some important aspects of Partial Differential Equation theory, aspects that will be most useful to those who will further work with Partial Differential Equations be it on the Theoretical side or on the Numerical one.

### Role

Lecturer

## Function Spaces and Applications - MATH97025

### Aims

The purpose of this course is to introduce the basic function spaces and to train the student into the basic methodologies needed to undertake the analysis of Partial Differential Equations and to prepare them for the course ‘Advanced topics in Partial Differential Equations’’ where this framework will be applied. The course is designed as a stand-alone course. No background in topology or measure theory is needed as these concepts will be reviewed at the beginning of the course.

The course will span the basic aspects of modern functional spaces: integration theory, Banach spaces, spaces of differentiable functions and of integrable functions, convolution and regularization, compactness and Hilbert spaces. The concepts of Distributions, Fourier transforms and Sobolev spaces will be taught in the follow-up course ‘’Advanced topics in Partial Differential Equations’’ as they are tightly connected to the resolution of elliptic PDE’s and the material taught in the present course is already significant.

### Role

Lecturer

## Advanced Topics in Partial Differential Equations - MATH97026

### Aims

This course develops the analysis of boundary value problems for elliptic and parabolic PDE’s using the variational approach. It is a follow-up of ‘Function spaces and applications’ but is open to other students as well provided they have sufficient command of analysis. An introductory Partial Differential Equation course is not needed either, although certainly useful.

The course consists of three parts. The first part (divided in two chapters) develops further tools needed for the study of boundary value problem, namely distributions and Sobolev spaces. The following two parts are devoted to elliptic and parabolic equations on bounded domains. They present the variational approach and spectral theory of elliptic operators as well as their use in the existence theory for parabolic problems.

Extra reading about elliptic equations on the whole space case will be given. The aim of the course is to expose the students some important aspects of Partial Differential Equation theory, aspects that will be most useful to those who will further work with Partial Differential Equations be it on the Theoretical side or on the Numerical one.

### Role

Lecturer