An Introduction to Partial Differential Equations

Spring Semester 2019, D-MATH

Lecturer: Prof. Francesca Da Lio
Schedule Lectures:

Wednesday 10-12 ML H 43

Friday 10-11 HG D 7.2

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
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 R3, 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 Rn. 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 C2 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!

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