Spring Semester 2019, D-MATH
Wednesday 10-12 ML H 43
Friday 10-11 HG D 7.2
Lecture | Content | Reference |
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27.02.2019 | Presentation of the course. Examples of PDEs. Classifications of PDEs. | § 1.1, § 1.2.1, § 1.2.2 of Lecture Notes. For curiosity: Navier-Stokes Equations and Minimal Surface Equation and A rapid survey of the modern theory of PDEs |
01.03.2018 (2 hours) | Well-posed problems. Examples of ill-posed problems. Hadamard Counterexample. First order linear equations with constant coefficients: solution of homogeneous and non-homogeneous Transport Equation. Method of characteristics in the case of quasilinear first order PDEs. Definition of integral surface. Geometric characterization of an integral surface in terms of the characteristic direction. Proof of Proposition 2.2.1. | § 2.2.2, § 1.2.3 and § 2.1 of Lecture Notes. |
06.03.2018 | Proof of Theorem 2.2.1, Proof of Theorem 2.2.2 about local existence and uniqueness of a smooth solution under the transversality condition. | § 2.2.2 and § 2.2.3 of Lecture Notes. It can be useful: Recall of some properties of ODES. For a review of some concepts of analysis 2 (inverse function theorem, implicit function theorem, definition of a surface in R^{3}, tangent space to a surface.) I suggest to look at the lecture notes by Prof. M. Struwe. |
08.03.2019 | End of the proof of Theorem 2.2.2, uniqueness of the maximal solution. Example 2.3.1 in Lecture notes. | § 2.3 of Lecture Notes |
13.03.2019 | Discussion of example 2.4.1 (Burgers' equation). Derivation of characteristic ODEs in the general case. | § 2.4 of Lecture Notes. It can be useful for knowledge: Geometric interpretation of the characteristics in the general case (Monge Cone). See also the book by Courant, Hilbert. Further reading of method of characteristics. |
15.03.2019 (2 hours) | Compatibility conditions. Admissible triples. Local existence: the general case. Proof of Lemma 2.4.1 and Lemma 2.4.2. Example 2.3.3 (failure of the transversality condition) | § 2.4 of Lecture Notes. |
20.03.2019 | Discussion Example 2.3.4. Characteristics for conservation laws. Introduction to Hamilton Jacobi Equations. Link between HJ equations and calculus variations problems. | & sect 2.4 and § 2.5 of Lecture Notes and § 3.3 Evans book. |
22.03.2019 | Legendre Transform. Value function and Hopf-Lax Formula. Proof of Theorem 2.5.2 | § 2.5 of Lecture Notes and § 3.3 Evans book. It can be useful for knowledge: for link between Hopf-Lax Formula and solution given by method of characteristic, look for instance the book by Cannarsa and Sinestrari. |
27.03.2019 | Proof of Theorem 2.5.3 (the Hopf-Lax formula is an a.e. solution of the HJ equation). Two counter-examples to the the uniqueness of Lipschitz continuous solutions. Introduction to the Laplace equation. | § 2.5 of Lecture Notes and § 3.3 Evans book. |
29.03.2019 (2 hours) | Link between holomorphic and harmonic functions. Conformal invariance of the Laplacian (Proof of Prop. 3.1.1 and Prof 3.1.2). Fundamental solution of the Laplace equation and its meaning. | § 3.2 of Lecture Notes. |
03.04.2019 | Proof of Theorem 3.2.1 about the solution of the Poisson equation in R^{n}. Definition of sub- and super-harmonic functions. Proof of Theorem 3.3.1 about Weak Maximum Principle. Consequences of Weak Maximum Principle: Proof of Corollaries 3.3.1,3.3.2,3.3.3,3.3.4. Non validity of Maximum Principle in unbounded domains (Remark 3.3.1). | § 3.3 and § 3.4 of Lecture Notes. For a repetition of divergence theorem and the conversion of a n-integral into integrals over spheres see e.g. Appendix C.2 of Evans 's book. It can be useful: : Exercise 3.3.2 |
05.04.2019 | Consequences of Weak Maximum Principle: Proof of Corollaries 3.3.1,3.3.2,3.3.3,3.3.4. Non validity of Maximum Principle in unbounded domains (Remark 3.3.1). Example 3.3.2. Mean-value formulas (Definition) | § 3.3 and § 3.4 of Lecture Notes. |
10.04.2019 | Proof of Theorems 3.4.1. 3.4.2, 3.4.3. Strong Maximum Principle (Proof of Theorem 3.4.4). Regularity properties: Koebe Theorem, Liouville Theorem, analiticity of harmonic functions, (proof of Theorems 3.4.5, 3.4.6, 3.4.7 ) | § 3.4 of Lecture Notes. |
12.04.2019 | Estimates derivatives harmonic functions, analiticity of harmonic functions, (proof of Theorems 3.4.8,3.4.9 3.4.10) | § 3.4 of Lecture Notes. |
17.04.2019 | Harnack Inequality, Harnack convergence theorem, Green's identities, Definition of Green function (proof of Theorems 3.4.11, 3.5.1, Exercise 3.4.3, n.6)) | § 3.4 and § 3.5 of Lecture Notes. |
19.04.2019 | Holidays between 19.04.2019 and 28.04.2019 | Frohe Ostern! |
03.05.2019 | Derivation of the Green function of the unit ball and Poisson kernel. | § 3.5.2 of Lecture Notes. |
08.05.2019 | Proof of Theorem 3.5.3 (Poisson formula for a ball). Presentation (by three students) of Exercise 3.4.2 and Exercise 3.4.3 (#3: Reflection principle, #7: Removable singularity result) | § 3.5.1 & 3.5. 2 of Lecture Notes. |
10.05.2019 | Proof of Proposition 3.5.1 (Properties of the Green functions of bounded domains). Definition of generalized sub- and super-harmonic functions. Proof of Proposition 3.6.1. | § 3.6 of Lecture Notes. |
15.05.2019 | Strong Maximum Principle for generalized sub-harmonic functions. Harmonic lifting. Perron's solution. Solvability of the Dirichlet problem for the Laplacian. Proof of Proposition 3.6.2, 3.6.2, 3.6.3, 3.6.4, 3.6.5, Corollary 3.6.1, and Theorem 3.6.1. | § 3.6 of Lecture Notes. |
17.05.2019 | Proof of Theorem 3.6.1. Exterior Sphere Condition. Example of non-solvability of the Dirichlet Problem for the laplacian. | § 3.6 of Lecture Notes. |
22.05.2019 | Exercise 3.7.1, Proof of Lemma 3.7.1 and Theorem 3.7.1 | § 3.7 of Lecture Notes. |
24.05.2019 | Discussion of Example 3.7.2. Proof of the fact that the Poisson equation with f given in the Example 3.7.2 cannot have C^{2} solutions. Proof of Theorem 3.8.2. Energy Methods. | § 3.7 & § 3.8 of Lecture Notes. |
29.05.2019 | Derivation of the Fundamental Solution of the Heat Equation. Homogeneous Cauchy Problem. Proof of Theorem 4.3.1 Presentation of Example 2.5.2 by a student. | § 4.1, §4.2, §4.3 of Lecture Notes , Example 2.5.2 |
31.05.2019 | End of the proof of Theorem 4.3.1, Tychonov’s counterexample. | §4.3 of Lecture Notes. |
Schöne Ferien und viel Erfolg! |