Partial Differential Equations

Credits

Prerequisites

None.

Scientific-disciplinary sector (SSD)

MAT/05 Mathematical Analysis.

Examination method

Oral exam.

Learning
objectives

The course aims to provide the basic elements of the classical theory of the Poisson, heat and wave equations and a fairly detailed introduction to Sobolev functions and to the theory of weak solutions of elliptic equations.

Syllabus

Laplace’s equation: fundamental solution and Newtonian potential. Harmonic functions: mean value theorem, maximum principle, Liouville’s theorem, Harnack’s inequality, Weyl’s lemma, and analyticity of harmonic functions. Uniqueness of solutions of the Dirichlet and Neumann problems. Green’s function. Explicit computation of Green’s function in the half-space and in the ball. Dirichlet’s principle. Heat equation: fundamental solution, mean value theorem and maximum principle. Uniqueness and backward uniqueness. Energy methods. Transport equation. Wave equation. Solution formula in dimensions 1, 2 and 3. Characteristic cone and finite speed of propagation. Energy methods. The method of separation of variables and its application to solving the Poisson, heat and porous medium equations. Fourier transform and applications to solving the Poisson, heat, wave and telegraph equations. Laplace transform and applications. Sobolev spaces: definitions and first properties, H=W, approximation by smooth functions, extension domains, traces, embeddings and compact embeddings, Poincaré inequality. Weak solution of an elliptic equation. Existence of weak solutions and their regularity. Compact operators. Fredholm theory. Eigenvalues and spectrum of a compact operator. Eigenvalues of the Laplacian.

Expected learning
outcomes

By the end of the course, students must demonstrate that they

know and understand the classification and main features of partial differential equations as well as the techniques for computing their solutions;
are able to apply the knowledge acquired to the study and solution of concrete examples, making correct use of proof techniques;
are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
are able to identify the most appropriate methods to analyse and solve a problem relating to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Command of the knowledge acquired, clarity of presentation, rigour in the use of language, confidence in using the notions acquired.

Partial Differential Equations