We ek |
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 |
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4 | 0923 | Green's function for balls, Dirichlet principle for BVP |
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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 |
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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 |
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11 |
1111 |
Remembrance Day | |
1113 |
Interior, boundary and global H^2 regularity |
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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 |