### 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