Introduction to nonlinear geometric PDEs
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Lecturer:
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Dr. Thomas Marquardt
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First Lecture:
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Monday, September 23
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Time:
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Monday 3pm - 5pm
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Last Lecture:
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Monday, December 16
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Place:
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HG G 26.3
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ECTS:
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4CP
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Course description
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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.
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Structure of the exam
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Here you can find information about the structure and content of the exam: exam.pdf
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Lecture notes
Exercises
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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.
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Exercises part I (Introduction): Exercises1.pdf
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Exercises part II (Nonlinear elliptic PDEs): Exercises2.pdf
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Exercises part III (Nonlinear parabolic PDEs): Exercises3.pdf
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Numerical computations
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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.
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Numerical solutions: NumericalSolutions.pdf
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Corresponding Matlab files: PMC.rar
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Further reading - Recent review articles and textbooks