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401-3352-09L 6 Credits BSC , MSC D-MATH
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Partial Differential Equations

Lecturers & Examiners: Prof. Dr. Joaquim Serra
Be aware that there is a large intersection, although with significant differences in material, with 401-3352-09L An Introduction to Partial Differential Equations taught in the Spring Semester 2024 as an elective course and therefore counts as the same course unit. The course unit is planned to be offered as 401-3352-09L An Introduction to Partial Differential Equations in the Spring Semester 2026.
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Last Updated: 2026-06-01 11:33:48

Abstract

Introduction to first and second-order PDE: transport, wave, Laplace, and heat equations. PDE methods: superposition, representation formulae, Duhamel, separation of variables, etc. Introduction to existence and regularity theories. Some example results for nonlinear PDE.

Objective

At the end of the course, students should have a good understanding of: - First-order PDE: transport equation. Method of characteristics: theory and computations. - Standard theory heat, Laplace, and wave equations. In particular, standard methods to represent/compute solutions. - Separation of variables, Laplace's eigenfunctions, and applications. - Challenges of existence theory and how to solve them.

Content

1. First-order equations. The Method of characteristics: linear and nonlinear, transport equation, Hamilton-Jacobi equation, scalar conservation laws. 2. Wave equation. D'Alembert formula, solutions by spherical means. 3. Laplace equation. Fundamental solution, harmonic functions and main properties, maximum principle. Poisson equation. Green functions. Regularity of solutions to the Poisson Equation. 4. Heat equation, fundamental solution, existence of solutions to the Cauchy problem and representation formulas, main properties, uniqueness and maximum principle, regularity. Methods with broad applicability (not specific to a single PDE): Duhamel principle. Separation of variables: "theory and practice". Eigenfunctions and eigenvalues of Laplace operator. Fourier transform. Representation formulae. Introduction to the existence and regularity theory: energy and viscosity approaches. Focus on estimates versus qualitative regularity. Extra topics (time permitting): Functional analytic aspects of PDE; Calculus of Variations; Weyl law; etc.

Resources

Literature

L. Evans, Partial Differential Equations, AMS 2010 (2nd edition) Q. Han, F. Lin, Elliptic Partial Differential Equations: Second Edition, Courant Lecture Notes. W. Strauss, Partial Differential Equations: An Introduction. 2nd ed. Hoboken, NJ: John Wiley & Sons, 2007. F. John, Partial Differential Equations, Springer, 1995. H. Brezis Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. Y. Pinchover & J. Rubinstein: An Introduction to Partial Differential Equations, Cambridge University Press, 2005.

General Information

Language
English
Levels
BSC , MSC
Frequency
Yearly recurring

Examination

Type
session examination
Mode
oral 20 minutes
Students who have got a definitive result for 401-3352-09S An Introduction to Partial Differential Equations (FS 2024) cannot register for the exam.

Course Components

Type Title Time & Place Hours
lecture Partial Differential Equations
  • Wed 09:15-12:00 (CAB G 51)
3 h weekly

Offered In