Spring Semester 2018, D-MATH
Wednesday 10-12 HG.E.21
Friday 11-12 HG.E.21
Lecture | Content | Reference |
---|---|---|
28.02.2018 | 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 |
02.03.2018 (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 | § 1.2.3 and § 2.1 of Lecture Notes. |
07.03.2018 | 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. and proof of Theorem 2.2.1. Statement 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. |
09.03.2018 | Proof of Theorem 2.2.2 | § 2.2.3 of Lecture Notes |
14.03.2018 | Discussion of Examples 2.3.1 (example of blow-up of the solution), 2.3.3 (example of what may happen if the transversality condition is not satisfied), 2.4.1 (Burger's equation). | § 2.3 of Lecture Notes and § 3.2 of Evans's book. It can be useful for knowledge: Further reading of method of characteristics. |
16.03.2018 (2 hours) | 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. |
21.03.2018 | Local Existence: the General case. Compatibility conditions. Proof of Lemma 2.4.1 and Lemma 2.4.2. Applications. Example 2.3.4. | § 2.4 of Lecture Notes. |
23.03.2018 (2 hours) | Introduction to Hamilton Jacobi Equations. Link between HJ equations and calculus variations problems. Legendre Transform. Value function and Hopf-Lax Formula. Proof of Theorem 2.5. | § 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. |
28.03.2018 | Proof of Theorem 2.5.3 (the Hopf-Lax formula is an a.e. solution of the HJ equation). A counter-example to the smoothness of the value function and to the uniqueness. Introduction to the Laplace equation. Invariance properties of the Laplace operator. Link between holomorphic functions and harmonic functions. | § 2.5 of Lecture Notes and § 3.3 Evans book. |
30.03.2018 | Holidays between 30.04.2018 and 7.04.2018 | Frohe Ostern! |
25.04.2018 | Fundamental solution of the Laplace equation and its meaning. Proof of Theorem 3.2.1 about the solution of the Poisson equation in R^{n}. | § 3.2 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. |
27.04.2018 | 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). Mean Value Formulas. | § 3.3 and § 3.4 of Lecture Notes. It can be useful: : Exercise 3.3.2 |
02.05.2018 | Proof of Theorems 3.4.1,3.4.2,3.4.3. Properties of Harmonic Functions. Proof of Theorem 3.4.3 (Strong Maximum Principle). Regularity of harmonic functions. | § 3.4 of Lecture Notes. It can be useful: : Exercise 3.4.3 (1-5). |
04.05.2018 | Proof of Theorems 3.4.5 (Koebe Theorem), 3.4.6 (Liouville Theorem). Proof of Theorems 3.4.7, 3.4.8, 3.4.8 | § 3.4 of Lecture Notes. |
9 & 11.05.2018 | Proof of Theorem 3.4.9. Analyticity of harmonic functions. Proof of Theorem 3.4.10. Theorem 3.4.11 (Harnack Inequality-no proof). Green Functions. Proof of Theorem 3.5.1. Representation formula using Green Function (Proof of Theorem 3.5.2). | § 3.4 and § 3.5 of Lecture Notes. It can be useful as exercise: : Proof of Harnack Convergence Theorem (Exercise 3.4.3 (part 6)). |
16 & 18.05.2018 | Green function for a ball. Green function for a half space. Properties of the Green function. Introduction of Perron's Method. | § 3.5 and &3.6 (until Corollary 3.6.1) of Lecture Notes. |
23 & 25.05.2018 | Description of Perron 's Method. Harmonic Lifting. Barrier Function. Proof of Prop 3.6.3, Theorem 3.6.1, Prop. 3.6.5, Theorem 3.6.2, Lemma 7.1 | § 3.6 and &3.7 of Lecture Notes. It can be useful as exercise: : Proof of Corollary 3.6.2. |
30.05 & 01.06.2018 | Poisson's equation: classical solutions. Energy methods. | § 3.7 and &3.8 of Lecture Notes. |