An Introduction to Partial Differential Equations

Spring Semester 2024, D-MATH

Lecturer: Prof. Francesca Da Lio
Schedule Lectures:

Wednesday 10-12 HG F 26.5

Donnerstag 9-10 HG F 26.5

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

#Week Date Content Reference
1 28-29.02.2024 (4 hours) Presentation of the course. Examples of PDEs. Classifications of PDEs. 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. § 1.1, § 1.2.1, § 1.2.2, § 2.2.2, § 1.2.3 and § 2.1 of Lecture Notes. For curiosity: Navier-Stokes Equations and Minimal Surface Equation and A rapid survey of the modern theory of PDEs
2 06-07.03.2024 Definition of integral surface. Geometric characterization of an integral surface in terms of the characteristic direction. Proof of Proposition 2.2.1. 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.
3 13-14.03.2024 Discussion of example 2.4.1 (Burgers' equation). Derivation of characteristic ODEs in the general case. 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. 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.
4 20-21.03.2024 Discussion Example 2.3.4. Characteristics for conservation laws. 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 § 2.4 § 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.
5 27-28.03.2024 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 (see example 2.5.3 and example page 349 in the following paper ). Introduction to the Laplace equation. 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. Proof of Theorem 3.2.1. § 2.5, § 3.2 of Lecture Notes and § 3.3 Evans book. For curiosity: Uniqueness for First-Order Hamilton Jacobi Equations and Hopf Formula (Guy Barles).
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6 10-11.04.2024 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).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. § 3.3 and § 3.4 of Lecture Notes.
7 17-18.04.2024 Liouville Theorem, analiticity of harmonic functions, (proof of Theorems 3.4.5, 3.4.6, 3.4.7 ) 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)). Derivation of the Green function of the unit ball and Poisson kernel. § 3.3, § 3.4 and §3.5 of Lecture Notes. For curiosity: M. Kassman, Harnack Inequalities. An Introduction.
8 24-25.04.2024 Proof of Theorem 3.5.3 (Poisson formula for a ball). Definition of generalized sub- and super-harmonic functions. Strong Maximum Principle and Comparison result for generalized sub-harmonic functions. Harmonic lifting (definition). Proof of Propositions 3.6.1, 3.6.2, 3.6.3, Corollary 3.6.1. § 3.6 of Lecture Notes.
9 8.05.2024 Harmonic lifting (proof). Perron's solution. Solvability of the Dirichlet problem for the Laplacian. Proof of Proposition 3.6.3, 3.6.4, 3.6.5 and Theorem 3.6.1. § 3.6 of Lecture Notes.
10 15-16.05.2024 Exercise 3.4.7 (#7: Removable Singularity Theorem). Proof of Theorem 3.6.2. Exterior Sphere Condition. Barrier. Classical solutions of Poisson equation. Newton Potential. Proof of Lemma 3.7.1. Proof of Theorem 3.7.1 and Theorem 3.7.2. Example 3.7.1. § 3.6 and § 3.7 of Lecture Notes.
11 22-23.05.2024 Derivation of the Fundamental Solution of the Heat Equation. Homogeneous Cauchy Problem. Proof of Theorem 4.3.1 Tychonov’s counterexample. § 4.1, §4.2, §4.3 of Lecture Notes.
12 29-30.05.2024 Nonhomogeneous Cauchy Problem for the heat equation. Proof of Theorem 4.4.1. Remark 4.4.1, Proof of Theorem 4.5.1 (Weak Maximum Principle). Proof of Theorem 4.5.2 (Uniqueness of bounded solutions). Mean Value Formula for solutions to heat equations. Proof of Theorem 4.6.1. Proof of Theorem 4.6.2 (Strong Maximum Principle). § 4.4, §4.5, §4.6 of Lecture Notes.
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