Webrelaunch 2020

Nonlinear Boundary Value Problems (Winter Semester 2020/21)

The lectures will be given online in form of pre-recorded videos.

All videos and the problem sheets will be provided in our ILIAS course.

Schedule
Lecture: Online
Problem class: Friday 8:00-9:30 Online
Lecturers
Lecturer Prof. Dr. Michael Plum
Office hours: Please get in contact by email.
Room 3.028 Kollegiengebäude Mathematik (20.30)
Email: michael.plum@kit.edu
Problem classes M.Sc. Jonathan Wunderlich
Office hours:
Room 3.026 Kollegiengebäude Mathematik (20.30)
Email: jonathan.wunderlich@kit.edu

Contents

The lecture course will be concerned with boundary value problems for nonlinear elliptic partial differential equations, mainly of second order. In contrast to the linear case, no "unified" existence theory is at hand, but various approaches for proving existence (and other properties) of solutions need to be studied. The methods investigated in the lecture course are subdivided into non-variational and variational methods.

A preliminary and incomplete list of topics:

  • Motivating examples
  • monotonicity methods
  • fixed-point methods
  • super- and subsolutions
  • non-existence results
  • radial symmetry
  • a short introduction into variational calculus
  • Euler-Lagrange equations
  • variational problems under constraints
  • critical points
  • mountain pass theorem
  • perturbation results

Prerequisites

Knowledge in functional analysis (Hilbert- and Banach spaces, weak convergence, dual space, Frechet differentiable operators) is essential, as well as the Lebesgue integral and Sobolev spaces. Knowledge in the classical theory of partial differential equations, and about weak solutions to linear problems, will be very useful.