Entry requirements

BSc, Introduction to Partial Differential Equations, Functional Analysis I

Learning outcomes

The course objective is to further develop the theory of hyperbolic, parabolic and elliptic partial differential equations in connection to physical, technical and scientific problems. Main themes are well-posedness for various initial and boundary value problems and properties of solutions of the wave equation, the heat equation and the Laplace equation. Functional analysis is used as a tool for proving existence of weak solutions and studying regularity of solutions. Important areas of application are found in numerical analysis, optimisation theory, control theory, signal processing, image analysis, mechanics, solid mechanics and quantum mechanics

In order to pass the course (grade 3) the student should

be able to use geometric methods;

be able to use Kirchhoff's solution and Huygen's principle for wave propagation in one and several space variables;

know and be able to use fundamental solutions and Green functions for various types of equations;

be familiar with Perron–Wiener's method;

know how to use the maximum principle for elliptic partial differential equations;

be familiar with distributions, Sobolev spaces and embedding theorems;

know variation formulations of elliptic problems, and existence theorems for general problems; Lax–Milgram's theorem, energy estimates, Fredholm alternatives, Poincaré's inequality and error estimates (FEM);

be familiar with existence theorems for general parabolic and hyperbolic partial differential equations and related solution methods such as Galerkin's method;

have successfully carried out a minor project in some area of applications (e.g. as an introduction to the Master's thesis).

Content

Characteristics. Symbols of partial differential operators. The maximum principle. Sobolev spaces. Linear elliptic equations. Energy methods for Cachy problems for parabolic and hyperbolic equations. Fredholm theory and eigen function expansions. Potential theory.

Instruction

Lectures and problem solving sessions.

Assessment

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.

Reading list

• Reading list valid from Autumn 2013
• Reading list valid from Autumn 2008, version 2
• Reading list valid from Autumn 2008, version 1
• Reading list valid from Autumn 2007