MATH 516 Fall 2014 Lecture Summary

[E] = Evans' book, [GT] = [Gilbarg-Trudinger], [J]=[John]

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Date Contents References
1 0902
Imagine UBC Day
0904 outline, examples of DEs, basic questions, concepts of solutions
Part I. Classical linear equations
§2.1 transport equation with constant velocity, Duhamel's principle


[E,§2.1]
2 0909 §2.2 Laplace equation and Poisson equation, fundamental solution, theorem on solution formula
[E,§2.2]
0911 proof of solution formula, mean value property, maximal principle
3 0916 MP for subharmonic function, uniqueness, comparison, smoothness, derivative estimate, real-analyticity
0918 Liouville theorem, Harnack inequality, existence of BVP, Green's function
4 0923 Green's function for balls, Dirichlet principle for BVP
0925 Perron's method of subsolutions for BVP
§2.3 Heat equation fundamental solution and solution formula, maximal principle in bounded domains
[GT, §2.8]
[E, §2.3]
5 0930
proof of maximal principle in bounded domains, mean value property and Hopf's boundary lemma, maximal principle in whole space
[E, §2.3]
1002
uniqueness, Tychonoff example, smoothness and derivative estimates
[J], [E]
6 1007
§2.4 wave equation time reversibility, energy, D'Alembert's formula, spherical means
[E, §2.4]
1009
solution formulas in 2d and 3d, domains of dependence and influence, nonhomogeneous problem
7 1014
Part II. Sobolev spaces
Motivation with elliptic PDE, Banach spaces, Holder spaces, Lebesgue spaces, weak derivative
[E, Ch.5]
1016
Sobolev spaces, Leibniz rule and as Banach spaces, approximations by smooth functions
8 1021
approximations by smooth functions II, extension, trace
1023
Imbedding inequalities
9 1028
Imbedding continued, compactness of imbedding, Poincare inequality.
1030
H^{-1}
Part III. Weak solutions of elliptic equations
Weak solutions, Lax-Milgram theorem
[E, Ch6]
10 1104
First existence theorem, Fredholm Alternative, second existence theorem
1106
Second and third existence theorems, regularity
11
1111
Remembrance Day
1113
Interior, boundary and global H^2 regularity
12
1118
maximal principle, Hopf lemma, strong maximal principle
1120
Part IV. Linear evolution equations
weak formulation, Banach-space valued Sobolev spaces, Galerkin method
[E. Ch.7]
13
1125
Existence and energy bound of weak solutions
1127
semigroup approach