Fall 2026
Sagun Chanillo
Course Description:
Some of the topics I will cover:
1. Cauchy-Kowaleskaya Theorem.
2. Holmgren uniqueness Theorem.
3. Sobolev spaces. Basic properties, Hardy-Littlewood-Sobolev embedding theorem.
4. Sobolev extension theorem from a half-space.
5. Rellich-Kondrachov compactness theorem.
6. Traces of Sobolev spaces.
7. Maximum principles and Hopf boundary point lemma for second order elliptic equations.
8. The Laplacian, Green function, and Poisson kernel.
9. Schauder theory.
10. Wave equation, finite propagation speed, domain of influence.
11. Duhamel’s theorem.
12. Huyghen’s principle and solutions of wave equations and Schrodinger equation.
13. Some dispersive estimates for wave and Schrodinger equations.
Text:
I shall post notes for the lectures on the Canvas page for the course. No textbook exists that covers everything I am proposing to do. A textbook that you might wish to consult is Partial Differential Equations by Lawrence Craig Evans, Graduate Texts in Math. Vol. 19, American Mathematical Society.
Prerequisites:
501 or Instructor's permission
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Fall 2025
Zheng-Chao Han
Course Description:
This is the first half of a year-long introductory graduate course on PDEs, and should be useful for students with a variety of research interests: physics and mathematical physics, applied analysis, numerical analysis, differential geometry, complex analysis, and, of course, partial differential equations. The beginning weeks of the course aim to develop enough familiarity and experience with the basic phenomena, approaches, and methods in solving initial/boundary value problems in the contexts of the classical prototype linear PDEs of constant coefficients: the Laplace equation, the D'Alembert wave equation, the heat equation and the Schroedinger equation. A variety of tools and methods, such as Fourier series/eigenfunction expansions, Fourier transforms, energy methods, and maximum principles will be introduced. More importantly, appropriate methods are introduced for the purpose of establishing quantitative as well as qualitative characteris tic properties of solutions to each class of equations. It is these properties that we will focus on later in extending our beginning theories to more general situations, such as variable coefficient equations and nonlinear equations.
Text:
Notes by the instructor
Prerequisites:
A solid background in analysis at the level of first seven chapters of baby Rudin, basics of Fourier series, and the main theorems of vector calculus (e.g., divergence theorem).
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FALL 2024
Yanyan Li
Course Description:
Standard First Semester PDE, materials taken from first 6 to 7 chapters of the book of Evans
Text:
Evans' book "Partial Differential Equations"
Prerequisites:
501 or Instructor's permission
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FALL 2023
Zheng-Chao Han
Course Description:
This is the first half of the year-long introductory graduate course on PDE. PDE is an enormously vast field, and is an important tool for studying problems in many areas of mathematics, in particular in mathematical physics, geometry, and applied mathematics. We will introduce the most useful methods and techniques through studying some prototype equations ---to build up experience and intuition--- rather than emphasizing the most general theories. This first semester course will be more basic, emphasizing the study of constructions and properties of solutions using relatively elementary tools; but the approaches that we will take will motivate and naturally lead to more sophisticated generalizations in more advanced treatment.
Text:
Course notes
Prerequisites:
A solid background in analysis at the level of first seven chapters of baby Rudin, basics of Fourier series, and the main theorems of vector calculus (e.g., divergence theorem).
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FALL 2022
Zheng-Chao Han
Course Description:
This is the first half of the year-long introductory graduate course on PDE. PDE is an enormously vast field, and is an important tool for studying problems in many areas of mathematics, in particular in mathematical physics, geometry, and applied mathematics. We will introduce the most useful methods and techniques through studying some prototype equations ---to build up experience and intuition--- rather than emphasizing the most general theories. This first semester course will be more basic, emphasizing the study of constructions and properties of solutions using relatively elementary tools; but the approaches that we will take will motivate and naturally lead to more sophisticated generalizations in more advanced treatment.
Text:
Course notes
Prerequisites:
A solid background in analysis at the level of first seven chapters of baby Rudin, basics of Fourier series, and the main theorems of vector calculus (e.g., divergence theorem).
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FALL 2021
Natasa Sesum
Course Description:
This is the first half of a year-long introductory graduate course on PDEs, and should be useful for students with a variety of research interests: physics and mathematical physics, applied analysis, numerical analysis, differential geometry, complex analysis, and, of course, partial differential equations. The beginning weeks of the course aim to develop enough familiarity and experience with the basic phenomena, approaches, and methods in solving initial/boundary value problems in the contexts of the classical prototype linear PDEs of constant coefficients: the Laplace equation, the D'Alembert wave equation, the heat equation and the Schroedinger equation. A variety of tools and methods, such as Fourier series/eigenfunction expansions, Fourier transforms, energy methods, and maximum principles will be introduced. More importantly, appropriate methods are introduced for the purpose of establishing quantitative as well as qualitative characteris tic properties of solutions to each class of equations. It is these properties that we will focus on later in extending our beginning theories to more general situations, such as variable coefficient equations and nonlinear equations.
Text:
Lecture notes plus Partial Differential Equation by Evans
Prerequisites:
basic undergraduate course in analysis
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Schedule of Sections:
16:640:517 Schedule of Classes
Previous Semesters:
- Fall 2020 Prof. Z.C. Han
- Fall 2019 Prof. Z.C. Han
- Some information about the Fall 2017 semester's offering is posted at here.