An Introduction to Partial Differential Equations

Spring Semester 2018, D-MATH

Lecturers: Prof. Francesca Da Lio & Dr. Cornelia Busch
Schedule Lectures:

Wednesday 10-12 HG.E.21

Friday 11-12 HG.E.21

Plan of the Course

  • The method of characteristics for first order equations, linear and nonlinear, transport equation, Hamilton-Jacobi equation.

  • Laplace equation, fundamental solution, harmonic functions and main properties, maximum principle. Poisson equation. Green functions. Perron method for the solution of the Dirichlet problem. Regularity of the solution of the Poisson equation.

  • Heat equation, fundamental solution, existence of solutions to the Cauchy problem and representation formulas, main properties, uniqueness by maximum principle, regularity.

  • Time permitting: Wave equation, existence of the solution, D'Alembert formula, solutions by spherical means, main properties, uniqueness by energy methods.

Lecture Notes

(These notes will be continuously upadated during the course)

Diary of the Lectures

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 R3, 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 Rn. § 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.

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