Department Mathematik
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Partial Differential Equation (Winter 2021-2022)

Phan Thành Nam, Eman Hamza, Uni2work, Piazza, Homework

All meetings will be held in person. If you cannot participate in the lectures in person, then you may try the Zoom link (Meeting ID: 923 6666 4357, Passcode: 729887).

News:

  • Final exam with solutions (15.2.2022, 10:00-14:00).
  • The retake exam will take place in the first week of April. Please send an email to register by March 15 if you are interested in.

General Information

Goal: This course gives an introduction to Partial Differential Equations (PDEs), which plays an important role in many areas of Mathematics and has applications in many other sciences. We will discuss the derivation of equations and the analysis of solutions. Among other things, we will focus on
  • The Laplace and Poisson equations of electrostatics;
  • The heat equation of diffusion processes;
  • The wave equation of the propagation of sound waves;
  • The Schrödinger equation of quantum dynamics.

Audience : Bachelor students of Mathematics (WP16), Master students of Mathematics (WP2), TMP Master, Master students of ‘Finanz und Versicherungsmathematik’ (WP49). Other participants will get "Schein" if pass the course.

Time and place: The first lecture takes place on October 19th, 2021 (Tuesday) at 14:15. All of the meetings will will be held in-person, unless otherwise notified.
  • Lectures: Tuesday 14:15-16:00 and Wednesday 12:15-14:00 (room B 004).
  • Exercises: Wednesday 10:00-11:30 (room B 047).
  • Tutorials: Thursday 14:15-16:00 (A 027).

References:
  • Lawrence C. Evans: Partial Differential Equations (Second Edition). AMS Graduate Studies in Mathematics, Volume 19, 2010.
  • Elliott Lieb and Michael Loss: Analysis (Second Edition). AMS Graduate Studies in Mathematics, Volume 14, 2001.

Exercises: There will be a homework sheet every week. Doing the homework is the best way to learn new materials and to prepare for the exams. The solutions of the homework exercises will be discussed in the exercise section.

Tutorials: The tutorial section is provided to help you in reviewing the lectures. You can bring up your questions, discuss complementary materials, and try some extra exercises.

Grade: Final grade is determined by your total performance:
  • You can get up to 100 points in the final exam.
  • You can get up to 10 bonus points in the midterm exam.
  • You can get up to 1 bonus point per week for solving exercises.
You need 50 points to pass the course and 85 points to get the final grade 1.0.

Exams: We will have the midterm exam in December 2021 and the final exam in February 2022. You can use your notes (lecture notes, homework sheets and tutorial materials).

Contents of the lectures

19 & 20.10.2021. Introduction. Chapter 1: Laplace/Poisson equation. Fundamental solution of Laplace equation in R^d. Convolution representation of solution of Poisson equation. Mean-value formula for harmonic functions. Introduction.

26 & 27.10.2021. Maximum principle for harmonic (subharmonic) functions. Uniqueness for Poisson's equation in domains. Regularity of harmonic functions. Uniqueness for Laplace equation in R^d (Liouville's theorem) and for Poisson equation in R^d. Harnack's inequality. Chapter 1.

2 & 3.11.2021. Chapter 2: Convolution, Fourier transform, and distributions. Convolution. Young's inequality. Smoothing and approximating by convolution. Basic properties of Fourier transform. Hausdorff-Young inequality. Fourier transform of |x|^{-a}.

9 & 10.11.2021. Theory of distributions. Test functions. Distributions. Examples of distributions. Distributions and convolution. Taylor expansion for distributions. Chapter 2.

16 & 17.11.2021. Equivalence of classical and distributional derivatives. Sobolev spaces W^{k,p}. Chain rule. Positive distributions are measures. Chapter 3: Weak solutions and regularity. Fundamental Laplace solution as a Green function. Poisson's equation with L^1 data. Weyl's lemma. Low regularity for Poisson's equation.

23 & 24.11.2021. Low regularity (cont.) and high regularity for Poisson's equation. Regularity on general domains. Chapter 3.

30.11 & 1.12.2021. Chapter 4: Existence for Poisson's equation on domains. Domains with C^1 boundaries. Representation of classical solutions via Green function. Green function for a half-space.

7 & 8.12.2021. Green function for a ball. Energy method and Dirichlet's principle. Sobolev spaces W^{1,p} and W_0^{1,p}.

15.12.2021. Midterm exam.

21 & 22.12.2021. Poincare inequality for H_0^1(U). Weak solution in H_0^1 of the Poisson equation. Trace theory for H^1(U).

11 & 12.1.2022. Poincare inequality for H^1(U) with non-constant traces. Weak solution in H^1 of the Poisson equation with general Dirichlet boundary conditions. Remark on H_0^1 and the kernel of the trace operator. Remark on Neumann boundary conditions. Chapter 4.

18 & 19.1.2022. Chapter 5: Heat equation. Physical interpretation. Fundamental solution of the heat equation. Non-homogeneous equation. Heat equation with L^2 data. Sobolev embedding theorem.

25 & 26.1.2022. Maximum principle. Hopf method. Bounded and unbounded versions. Well-posedness of heat equation (existence, uniqueness and stability). Backward heat equation: uniqueness, instability, and regularized solutions. Chapter 5.

1 & 2.2.2022. Chapter 6: Wave equation. d'Alembert formula in 1D. Reflection method. Euler–Poisson–Darboux equation. Kirchhoff formula in 3D. Poisson formula in 2D. Wave equation in bounded domains. Spectral method. Uniqueness. Finite speed of propagation. Chapter 6.

8 & 9.2.2022. Chapter 7: Schrödinger equation. Definition of e^{it \Delta}. Long-time behavior. Classical and weak solutions of Schrödinger equation. General spectral method. Stone theorem. Dynamics of Schrödinger operator -\Delta+V(x). Chapter 7.