The course concerns the basic arguments in the theory of Partial differential equations of the main families: elliptic, hyperbolic, parabolic, Hamilton-Jacobi.

**Program**

- Elements in the theory of distributions: multiplication, differentiation, tensor product and convolution. Notion of fundamental solutions for a partial differential operator and construction in the main cases: Laplace, Heat and Wave operators. Application to non-homogeneous PDE's; hypoellipticity; harmonic functions and mean property.
- Poisson equation, properties of Laplace operator, eigenvalues and eigenvectors of Laplace operator, Galerkin method for Heat and Wave equations.
- Elliptic equations. Dirichlet problem. Existence, uniqueness and qualitative theory of weak solutions; interior regularity, regularity at the boundary, higher regularity, smoothness. Weak and strong maximum principles, Harnack inequality, eigenvalues and eigenvectors of an elliptic operator, Fredholm alternative.
- Elements on semilinear elliptic equations.
- Elements on unbounded operators: closed, dissipative and maximal dissipative operators. Fundamental theorems of semigroup theory. Hille theorem, infinitesimal generators and their properties. Hille-Yosida, Lumer-Phillips and Stone theorems. Cauchy problem for abstract equations, Duhamel formula. Construction of (semi)groups associated to Heat, Wave, Klein Gordon and Schrödinger equations.
- Semilinear abstract problem, local solution, extension, global solution, continuous dependence on data, conservations laws. Special properties of Heat semigroup. Example of a nonlinear problem for Klein-Gordon equation.
- Viscosity solutions for Hamilton-Jacobi equations: existence, uniqueness, stability.

**Textbooks.**

F. Hirsch, G. Lacombe, Elements in functional analysis, Springer.

H. Brezis, Functional analysis, Sobolev spaces and Partial differential equations, Springer

L. C. Evans, Partial differential equations, American Mathematical Society

T. Cazenave, A. Haraux, Introduction to semilinear evolution equations. Clarendon Press - Oxford.

**Aims**

Purpose of the course is to give to the students a solid background on the theory of the most studied partial differential equations from a functional analytic point of view, by understanding and applying the fundamental tools of Functional analysis to concrete and important problems. By this way, the course is, simultaneously, an activity devoted to learn fundamental arguments in mathematics and a practice in order to apply the notions of Functional analysis. Indeed the course makes wide and deep use of: fundamental principles (Hahn-Banach, Baire, Banach-Steinhaus, open mapping and closed graph theorems); weak differentiation and distributions; main tools in managing basic function spaces like Hilbert, Lebesgue and Sobolev spaces, that is to say Schwartz, Bessel, Holder and Young inequalities, Hilbert bases, convolution, regularization, separability, reflexivity, duality, convexity, weak and strong (relative) compactness, weak topologies, Riesz and Lax-Milgram theorems, fixed point theorems; Sobolev embeddings, extensions, traces; theory of linear (compact, bounded, unbounded) operators; spectral theory; harmonic analysis, viscosity. Due to the fact that the course covers various and different fields in functional analysis, the student may appreciate and learn several methods and techniques. At the end of the course, the student is expected to be able to manage all techniques mentioned above, by applying them to Partial differential equations.

**Prerequisites**

Basic elements of functional analysis and of distribution theory and of Sobolev spaces. These prerequisites are covered by the courses Advanced Analysis, parts A and B, and can be found in the indicated sections of following textbooks.

H. Brezis, Functional analysis, Sobolev spaces and Partial differential equations:

chapters 1, 2, 3, 4, 5, 6, 8, 9.

F. Hirsch, G. Lacombe, Elements in functional analysis:

chapters 1, 3, 4, 5, 6, 7.

**Lectures**

The course consists in lectures of the instructor, in problems suggested to the students, in discussions during the lectures and in other situations, even online.

**Exam**

The exam consists in an oral test (40-60 minutes) with questions devoted to the aim of ascertain if the student did actually understand the contents of the course.