## Final exam

The final exam is scheduled on Thursday December 5, starting at 9:00 in Burnside Hall 1214.

Please make sure you review the following topics before the exam.

• Basic properties of harmonic functions
• Mollifiers
• Weak and strong derivatives
• Weyl's lemma
• Sobolev space approach to Poisson equations
• Variational characterizations of eigenvalues
• Some explicit solutions (in the form of series or integrals) on rectangles, balls etc

There will be 5 problems, 4 of which are heavily inspired by the homework problems and by these practice problems.

## Lecture notes

• Harmonic functions [updated on 13.09.16]
• The Dirichlet problem [updated on 13.10.21]
• Poisson's equation [updated on 13.11.28]
• The Laplacian [incomplete draft]

## Assignments

• Assignment 1 [tex] due Monday September 23
• Assignment 2 [tex] due Monday October 7
• Midterm 1 [tex] due Wednesday October 23
• Assignment 3 [tex] due Monday November 11
• Midterm 2 [tex] due Monday December 2

## Class schedule

• MW 13:35–14:55, Burnside Hall 1234 (+ lecture on Dec 3)

 Date Topics W 9/4 Green's identities. Fundamental solutions M 9/9 Green's formula. Mean value property. Harnack inequality. Maximum principles W 9/11 Green's function. Poisson kernel M 9/16 Poisson's formula. Leibniz rule. Schwarz's theorem W 9/18 Removable singularity. Koebe's theorem. Derivative estimates. Taylor series M 9/23 Real analyticity. Harnack's first theorem W 9/25 Harnack's inequality and second theorem. Heine-Borel property. The Dirichlet problem M 9/30 Perron's method. Barriers. Boundary regularity W 10/2 Lebesgue's example. Osgood's criterion. Dirichlet energy M 10/7 Strong derivatives. Sobolev spaces. Weak solutions (lecture by dr. Topaloglu) W 10/9 Friedrichs inequality. Weak solution to the Dirichlet problem M 10/14 Thanksgiving W 10/16 Mollifiers. Du Bois-Reymond's lemma. Leibniz rule. Weyl's lemma M 10/21 Weak derivatives. Meyers-Serrin theorem. Trace maps W 10/23 Trace maps. Poisson equations. Variational method M 10/28 Natural boundary conditions. Traces of H1 functions W 10/30 Moduli of continuity. Interior regularity: Basic case M 11/4 Interior regularity: Higher order case. Sobolev's lemma W 11/6 Analyticity M 11/11 Regularity up to the boundary W 11/13 The Newtonian potential M 11/18 C2 regularity. Friedrichs extension. Resolvent W 11/20 Rellich's lemma. Hilbert-Schmidt theory M 11/25 Dirichlet and Neumann eigenproblems. Examples. A variational characterization W 11/27 Courant's nodal domain theorem. Courant's minimax principle. Comparison theorems M 12/2 Weyl's law. Functional calculus T 12/3 The heat equation R 12/5 Final exam

## Student seminar

• Fridays 10:30–11:30, Burnside Hall 1120

 Date Topics Speaker F 9/13 Lebesgue integration Justine F 9/20 Lp spaces Olivier F 9/27 Density Almaz F 10/4 Extension Matthew F 10/11 Thanksgiving ~ F 10/18 Embedding Gabriel F 10/25 (no seminar) ~ F 11/1 Embedding Gabriel F 11/8 Compactness Yariv F 11/15 (no seminar) ~ F 11/22 Problem solving ~ F 11/29 Problem solving ~

## Reference books

• Lawrence Craig Evans, Partial differential equations. AMS 1998.
• Qing Han, A basic course in partial differential equations. AMS 2011.
• Fritz John, Partial differential equations. Springer 1982.
• Jürgen Jost, Partial differential equations. Springer 2007.

## Online resources

• Stuff from last year: Math 580, Math 581
• From the year before: Math 580, Math 581
• Intermediate PDE Math 527 pages by Xinwei Yu (Alberta)
• Lecture notes by Arjen Doelman (Leiden)
• Harmonic function theory by Sheldon Axler, Paul Bourdon, and Wade Ramey
• Qing Han's page (Notre Dame)

## Course outline

Instructor: Dr. Gantumur Tsogtgerel

Prerequisite: MATH 375 (Honours PDE) or equivalent

Topics: The main focus of the course is going to be on linear second order equations. If time allows, we will discuss systems and some nonlinear problems. Rather than trying to build everything in full generality, we will study prototypical examples in detail to establish good intuition. Weak derivatives, Sobolev spaces, and functional analytic methods will be introduced. This essentially means that we will end up covering the topics from the calendar description of Math 580, some from that of Math 581, and some additional topics. More precisely, the planned topics are

• Green's identities, harmonic functions, Harnack inequality
• Fundamental solution, Green's function, Poisson's formula
• Dirichlet problem: Perron's method, barriers, boundary regularity
• Poisson equations: potential estimates, Kellog's theorem
• Sobolev spaces, weak and strong derivatives, Dirichlet principle
• Variational formulation, boundary conditions
• Elliptic regularity, Sobolev embedding, Rellich compactness
• Spectral theory of compact self-adjoint operators
• Laplace eigenvalues and eigenfunctions
• Spectral resolution of heat and wave equations
• Cauchy problem for heat and wave equations
• Duhamel's, Huygens, and maximum principles
• Initial-boundary value problems
• Classification of second order equations
• Elements of scattering theory (if time allows)
• Second order elliptic and parabolic systems (if time allows)
• Symmetric hyperbolic systems (if time allows)

Calendar description: Classification and wellposedness of linear and nonlinear partial differential equations; energy methods; Dirichlet principle. Brief introduction to distributions; weak derivatives. Fundamental solutions and Green's functions for Poisson equation, regularity, harmonic functions, maximum principle. Representation formulae for solutions of heat and wave equations, Duhamel's principle. Method of Characteristics, scalar conservation laws, shocks.

Calendar description of Math 581: Systems of conservation laws and Riemann invariants. Cauchy-Kowalevskaya theorem, powers series solutions. Distributions and transforms. Weak solutions; introduction to Sobolev spaces with applications. Elliptic equations, Fredholm theory and spectra of elliptic operators. Second order parabolic and hyperbolic equations. Further advanced topics may be included.

Homework: Assigned and graded roughly every other week.

Weakly seminars: We will organize weekly seminars on problem solving, standard results from analysis and geometry, and other stuff related to the course. Attendance is optional.

Exams: There will be two take-home midterms and a final exam.

Grading: The final grade will be the weighted average of homework 20%, the take-home midterm exams 30%, and the final exam 50%.