Vorlesung: Hamilton-Jacobi Equations (SoSe 2022)
[UPDATE 18.04.2022:] This course will be entirely
live / face-to-face ("Präsenz").
Please note the present LMU Corona-rules.
To access the course material,
you need to sign up in uni2work
here.
Lecture (Vorlesung):
Wed 08:30-10:00 (in B 005). LSF
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.
Keywords: 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.
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).
Exam (Prüfung):
There will be an oral exam of 30min (Es wird
eine mündliche Prüfung von 30min geben).
See uni2work for details.
Literature:
In uni2work
you will find a copy of the notes from the lecture (to be updated as we go
along).
Supplementary literatur (Ergänzende Literatur):
Here
is a list of books.
Office hours (Sprechstunde):
See uni2work.
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Letzte Änderung: 03 August 2022 (
No more updates
).
Thomas Østergaard Sørensen