Department of Mathematics

Introduction to nonlinear geometric PDEs

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  Lecturer: Dr. Thomas Marquardt
First Lecture: Monday, September 23
  Time: Monday 3pm - 5pm Last Lecture: Monday, December 16
  Place: HG G 26.3    
  ECTS: 4CP    

Course description

  In this course we give an introduction to the field of nonlinear geometric partial differential equations (PDEs).

We start with a short review of the geometry of hypersurfaces and the theory of linear elliptic and parabolic PDEs.

The second part of the course is devoted to the study of nonlinear elliptic PDEs. We will discuss the existence theory for the Dirichlet problem of the prescribed mean curvature equation and the capillary surface equation.

In the third part of the course we focus on nonlinear parabolic PDEs. We will develop a general framework and use it to discuss short-time existence for mean curvature flow (MCF) and inverse mean curvature flow (IMCF). Then we will analyze long-time existence and convergence for inverse mean curvature flow with a Neumann boundary condition.

Finally, we will discuss IMCF in the level set picture. This will take us to recent advances and open problems in geometry and mathematical relativity.

Structure of the exam

  Here you can find information about the structure and content of the exam: exam.pdf

Lecture notes

  The lecture notes will be updated frequently:  
     Lecture notes: LectureNotes.pdf  
     Evolution equations for flows with speed equal to powers of the mean curvature: EvolutionEquations.pdf  


  The lecture notes contain some exercises. I extracted them by topic into the files below. I encourage you to think about them. I am always happy to discuss these problems with you.  
     Exercises part I (Introduction): Exercises1.pdf  
     Exercises part II (Nonlinear elliptic PDEs): Exercises2.pdf  
     Exercises part III (Nonlinear parabolic PDEs): Exercises3.pdf  

Numerical computations

  Here are some numerical solutions for minimal, constant mean curvature, and harmonic surfaces with different boundary values. I encourage you to play with the MATLAB files to generate further examples.
      Numerical solutions: NumericalSolutions.pdf
      Corresponding Matlab files: PMC.rar

Further reading - Recent review articles and textbooks

  Here is a nice overview article by Garcke which appeared in the DMV Jahresbericht: Curvature driven interface evolution, Springer, 2013
  A nice book which gives an overview about recent developments for mean curvature flow is: Mean Curvature Flow and Isoperimetric Inequalities by Ritore and Sinestrari, Springer, 2010
  A recent textbook by Lopez: Constant Mean Curvature Surfaces with Boundary , Springer, 2013

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