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Professor: Hugo Tavares, office 5.44, Pavilhão de Matemática, IST
Schedule: Tuesdays, 13h30-15h30 (room V1.10) and Friday, 14h00-17h (room V1.32).
Office hours: after each lesson, or in a different schedule (contact directly the professor)
The first class of the week is dedicated to theory (with a 5-10minutes break in the middle). The second class is flexible: the first 1h-1h30 is dedicated to theory; after a (good break), the rest of the class is dedicated to solving and discuss exercises.
Assesment methods
The final grade is a combination of the following elements.
Final exam (50%). I let the students bring to the exam six sheets of paper with whatever material they wish. The student should have at least 8 out of 20 in the exam.
Four “Homework assignments” (10% each) during the semester. They are assigned at weeks 4, 7, 10 e 13, each with a deadline of two weeks.
Evaluation in class (10%): students should go to the board and solve some exercises from the “Problem sheets”.
Época especial and trabalhadores-estudantes: the maximum between the grade of the exam and the grade obtained in the previous formula.
Office hours: after the classes or other appointments arranged by email.
Material:
Video of the extra exercise class of June 23; written material (NEW)
Video of the class of May 29; written material.
The following sheet with formulas will be provided during the exam
Lecture notes of the PDE course: download them here
Complementary readings (to give some context): articles Partial Differential Equations in the 20th century and Remarks on the pre-history of Sobolev spaces
Overview of the first 7 weeks: see picture above
Context and objectives
Mathematical modelling is one of the most efficient ways of understanding observable phenomena and of predicting future events. Many of these models are formulated with the aid of Partial Differential Equations (PDEs), mathematical tools that capture the essence of change with respect to certain continuous variables (such as time, space or price) of certain phenomena subject to laws such as diffusion, reaction, transport, competition/cooperation, etc. Therefore, PDEs can be used to model a high variety of situations, with applications to biology (population dynamics, transmission of neural impulses,...), to physics (waves, quantum mechanics, general relativity, electromagnetism, elasticity, heat transfer,...), to engineering (vibration of bridges, fluid dynamics,...) and finance. Moreover, PDEs are also naturally connected with other theoretical mathematical fields, such as Geometry and Calculus of Variations.
With so many phenomena of different nature there is no general theory about PDEs (nor would it be expected). This curricular unit intends to be a fundamental introduction to the discipline. Our aim is to develop the theory (from the most classical one with regular solutions to the modern theory with weaker notions of solution), never losing sight of the applications and always remembering where the models come from.
One of my main goals is to give a good overview of the amazing world of PDEs; another aim is to explain that they appear in several applications and other fields. I put a large focus explaining the most modern theories.
I focus on first order PDEs (linear and quasilinear) and second order (linear) PDEs, explaining deeply the phenomena of transport, diffusion and waves.
Program
First order equations
The transport equation: the homogeneous and nonhomogeneous cases.
Linear and quasilinear equations: the characteristic methods for the Cauchy problem.
One dimensional conservation laws. Application to traffic dynamics (shock waves). Reference to the notion of weak solution in this context.
2. Second order linear equations
Classification of linear second order PDEs in dimension two: parabolic, elliptic and hyperbolic equations. Brief notions about characteristics and solutions to the Cauchy problems: statement and applications of Cauchy-Kovalevskaya’s theorem.
The Laplace and Poisson equations: fundamental solution, solution in the whole space, maximum principles, mean value property, uniqueness of solution to boundary value problems. Harmonic functions: estimates for the derivatives and analyticity. Green’s function and Poisson’s formula in the ball. Harnack inequality. The Dirichlet principle.
The heat equation: fundamental solution and heat kernel, maximum principle and uniqueness of solution to initial-boundary value problems. Mass conservation. Duhamel’s formula.
The wave equation: D’Alembert’s solution in dimension one. Duhamel’s formula. Domains of dependence and influence. Solution to the Cauchy problem in dimensions two and three.
Energy methods for the Poisson, heat and wave equations.
3. Distributions and Sobolev spaces
Brief introduction to the theory of distributions.
Sobolev spaces and their basic properties. Sobolev inequalities. The space W^{m,p}_0(Ω), where Ω is an open set in R^N. Poincaré’s inequality. The notion of trace. The extension theorem Characterization of the dual space H^{−1}(Ω).
4. Theory of weak solutions for elliptic problems
Elliptic problems: second order elliptic operators, weak formulation. Application of Riesz’ and Lax-Milgram’s theorems to solve elliptic problems with different types of boundary conditions.
Regularity results: inner and boundary L^2–regularity. Statements for the general case.
* Possible extra topics, according to the students background and interests, and available time
Fully nonlinear first order PDEs. Entropy solutions for conservation laws.
Laplacian eigenvalues with Dirichlet or Neumann boundary conditions. Generalization of the concept of Fourier series to an arbitrary dimension. Separation of variables in bounded domains of R^N: application to boundary value problems in the Poisson, heat and wave equations.
Tempered distribution and Fourier transform. Application to PDEs and to Sobolev spaces.
Evolution problems and semigroup theory. Semigroup theory and Hille-Yosida theorem. Application to the heat and wave equations.
Main bibliography
[1] H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer 2011.
[2] L. Evans, Partial Differential Equations, 2nd edition, AMS 2010.
[3] S. Salsa and G. Verzini, Partial Differential Equations in Action - From Modelling to Theory, 4th edition, Springer 2022.
[4] H. Tavares, Partial Differential Equations, lecture notes.
Secondary bibliography
[5] D. Bleecker, G. Csordas, Basic Partial Differential Equations, International Press 1997
[6] W. Craig, A course on Partial Differential Equations, AMS 2018.
[7] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer 2011.
[8] F. John, Partial Differential Equations, 4th edition, New York: Springer-Verlag, 1982.
[9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer 1983.
Commented bibliography
The course mostly follows reference [4]. These are lecture notes written by myself over the years and that are freely available in this homepage. These notes are mostly inspired by my three favourite books about (basic) PDEs, references [1, 2, 3], and they are an attempt to capture what I find better about each of them.
Reference [1] is, in my opinion, the best book making the connection between Functional Analysis and PDEs. It is also a place where all the details regarding Sobolev Spaces are presented in a very clear and rigorous way. Reference [2] is very complete and exhaustive, covering all basic PDEs with some degree of generality. Book [3] is wonderful in terms of the connection between theory and applications.
Reference [5] is more basic, aimed mainly at students with less mathematical background; it has a simpler language and has many references to applications and several examples (on the other hand, it does not cover the more modern material). Reference [6] has a nice and original perspective; it is focused more on waves and Fourier Transform. References [7,9] are more advanced and go beyond the scope of this course; I recommended to students who wish to understand more than what I teach in the lectures. Reference [8] is a classic, which I use to complement some parts of the first half of the course.
Concretely, I use and recommend the following:
0. Review of basic material: Chapter 1 of [4] and references therein. This chapter contains almost every topic that students should master in order to understand the course. Most of this material is needed for the second half (Items 3-4 of the program). This material will be presented in the fist lesson and, during the first month of lectures, the plan is to help students with less mathematical background to understand all the material.
First order equations: [4, Chapter 2]; the introduction of Fourier Transform at this early stage is inspired by [6, pp. 10-12]. For complements, I recommend [3, Chapter 4] and [8, Chapter 1] to the students.
2. Second order linear equations: [4, Chapter 3], combined with the approach of [6] (where the Fourier Transform is used to deduce the shape of the fundamental solutions) and parts of [8] (for the classification of second order linear PDEs and Cauchy-Kovalevskaya's theorem). For more details, I recommend [3, Chapters 2, 3 & 5].
3. Distributions and Sobolev spaces: [4, Chapter 4]. For more details, I recommend [1, Chapters 8 & 9] and [2, Chapter 5].
4. Theory of weak solutions for elliptic problems: [4, Chapter 5]. For more details, I recommend [3, Chapter 8] and [7] (the latter regarding the topic of elliptic regularity).
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