Course Description
This course is an introduction to the qualitative theory of partial
differential equations (PDEs). It is offered every year, and should be useful to students with interests
in applied
mathematics, differential geometry, mathematical physics, probability,
harmonic analysis, dynamical systems, and other areas, as well as to
PDEfocused students. We will review a few analytic tools
along the way, e.g., Fourier transform and weak convergence.
Prerequisites and corequisites
Undergraduate DE, vector calculus,
Lebesgue integral and Lp spaces. (215/255, 257/316, 317,
420/507).
Topics
Here is the
tentative outline. It can be adjusted according to audience background
and interests.

Classical linear equations
 Laplace, heat, and wave equations; their solution formulas
 mean value properties and maximum principles, applications to
uniqueness and regularity
 existence: Perron's subsolution method and Dirichlet principle
 regularity of weakly
harmonic functions, analyticity (sketch)
 Classical solutions of second order elliptic equations
 weak and strong maximal principles
 Hölder spaces and Schauder's a priori estimates
 existence by the method of continuity
 Sobolev spaces
 weak derivatives and Sobolev spaces
 inequalities of Sobolev, Morrey, Poincare, and GagliardoNirenberg
 approximations, extensions, trace, compactness, and dual spaces
 Weak solutions of elliptic equations in divergence form
 weak solutions and maximal principle
 existence and eigenvalues by LaxMilgram theorem and Fredholm
alternative
 regularity
 application to semilinear elliptic problems
 analogous results for 2ndorder parabolic equations
 Semigroup theory and evolution equations
 semigroups
 applications to the existence of evolution equations
(parabolic, wave, Schrödinger)
References
We will mainly follow Evans' book and also use
materials from the others.
 Partial Differential Equations, 2nd ed., by L. C. Evans,
American Math Society, 2010. See author's homepage
https://math.berkeley.edu/~evans
for errata. This is a general
text suitable for a first course and also for reference.
 Partial Differential Equations, 4th ed., by Fritz John,
SpringerVerlag. This is a classic textbook and contains materials
not found elsewhere, e.g. Weyl's lemma and
extended treatise on wave equation.
 Elliptic Partial Differential Equations, by Qing Han and Fanghua Lin,
Courant Lecture Notes 1, AMS/CIMS. This book is specialized in elliptic equations and
is a standard reference. It is a good intermediate book after Evans and before
GilbargTrudinger.
 Elliptic Partial Differential Equations of Second Order, 2nd ed.,
by David Gilbarg and Neil S. Trudinger, SpringerVerlag, Classics in
Mathematics series. This book is specialized in elliptic equations and
is a standard reference.
Evaluation
The evaluation is based on six (6) homework assignments and class
participation.
Owncloud webfolder
Assignments and their solutions will be posted in an owncloud
webfolder hosted by the math department. The link of the webfolder
will be given to the audience.
Also in the webfolder you will find my lecture notes
and some references.
Instructor and lectures
Instructor:
Dr. TaiPeng Tsai, Math building
room 109, phone 6048222591, ttsai at math.ubc.ca.
Lectures:
MWF, 13:00  13:50, MATH 204
Office hours:
TBA, and by appointment
(Tsai's schedule).