Vorlesung: Hamilton-Jacobi Equations (WiSe 2016/17)

News (07.01.2017): Information (sign up!) about the exam is online.

Content of the lecture

Time, room (Zeit und Ort): Tuesday 14-16 Uhr in B 040 (First meeting: October 18th).

Exercises (Übungen): There are NO exercises.

Synopsis (Kurzbeschreibung): In this course we will study classical and generalised (weak and viscosity) solutions to boundary and initial value problems for Hamilton-Jacobi Equations. The Hamilton-Jacobi Equation (a nonlinear first order Partial Differential Equation (PDE)) arises in Classical Mechanics as equivalent to the Hamiltonian or Lagrangian formalism. It also arises in Optimisation in connection with control theory for Ordinary Differential Equations (ODEs) by the method of Dynamic Programming. We will study classical solutions via the Method of Characteristics. For convex Hamiltonians depending only on the momentum p, we will study the existence and uniqueness of Lipschitz regular weak solutions via the Hopf-Lax formula. For more general Hamiltonians, we study the theory of viscosity solutions.

Topics to (possibly) be discussed: Hamilton‘s equations; (Method of) Characteristics; convex analysis; Legendre-Fenchel transformation (convex conjugate); Hopf-Lax formula; semi-concavity; viscosity solutions; Dynamic Programming (if time permits).

Audience (Hörerkreis): Master students of Mathematics (WP 17.2, 18.1, 18.2, 44.3, 45.2, 45.3), TMP-Master.

Credits: 3 ECTS.

Exam (Prüfung): There will be an oral exam of 30min (Es wird eine mündliche Prüfung von 30min geben).

Prerequisites (Vorkenntnisse): Analysis I-III. No previous knowledge of ODE, PDE, Classical Mechanics, or Convex Analysis is needed. However, some previous exposition to one or more of these topics, and a solid background in Analysis, is an advantage.

Language (Sprache): English. (Die mündliche Prüfung kan auch auf Deutsch gemacht werden).

Literature: There will be no lecture notes. Here you will find a short description of the content of the lecture (to be updated as we go along). The lecture will in part follow the book by Evans mentioned below (of which there are several copies available in the library). Further literature to come.
(Es wird kein Skript geben. Hier wird laufend eine Kurzübersicht der Vorlesung erstellt. Die Vorlesung wird zum Teil auf folgendes Buch basieren:)

[E] L. C. Evans, Partial Differential Equations: Second Edition, AMS (Graduate Studies in Mathematics), 2010.

Supplementary literatur (Ergänzende Literatur):

Here is a longer list of books (to be updated).

Office hours (Sprechstunde): Thursday 10:15-11:00 (Room B 408) or by appointment via email.


Letzte Änderung: 10 February 2017 (No more updates).

Thomas Østergaard Sørensen

Curriculum Vitae