| We ek |
Date | Contents | References |
| 1 | 0905 |
Labour Day | |
| 0907 | outline, examples of DE, basic questions, concepts of solutions | ||
| 0909 |
Part I. Classical linear
equations §2.1 transport equation with constant velocity §2.2 Laplace equation: fundamental solution and solution formula | [Evans] 2.1, 2.2 [John] 4.4, 4.5 [GT] 2.8 |
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| 2 | 0912 | proof of solution formula using Green's identity mean value theorem |
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| 0914 | properties of harmonic functions:
maximal principle, uniqueness and smoothness |
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| 0916 | more properties: derivative estimates, Liouville theorem,
Harnack inequality |
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| 3 | 0919 | Green's fucntion |
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| 0921 | Green's fucntions for half space and balls |
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| 0923 h1 due |
sketch of existence theory: Dirichlet Principle and Perron's method of subsolutions
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| 4 | 0926 | §2.3 heat equation:
fundamental solution and solution
formula, Duhamel's principle |
[Evans] 2.3 [John] Ch.7 |
| 0928 | maximal principle and uniqueness in bounded domains and in
whole space |
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| 0930 |
proof of MP in whole space, Tikhonov's example, scaling | ||
| 5 | 1003 |
regularity and derivative estimates |
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| 1005 |
§2.4 wave equation:
solution formula in 1D spherical means and Euler-Poisson-Darboux equation |
[Evans] 2.4 |
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| 1007 h2 due |
solution formulas in higher dimensions domains of dependence and influence |
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| 6 | 1010 |
Thanksgiving |
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| 1012 |
Part II. Sobolev
spaces Banach space, Holder spaces, Lebesgue spaces, weak derivative |
[Evans] Ch.5 |
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| 1014 |
examples of weak derivative, Sobolev spaces |
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| 7 | 1017 |
examples and basic properties, 3 stages of approximations |
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| 1019 |
Proof of approximation theorems |
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| 1021 |
Sobolev imbedding, Sobolev and Morrey's
inequalities |
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| 8 | 1024 |
Proof of Morrey's
inequality, Sobolev imbedding in bounded domains |
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| 1026 h3 due |
extension theorem |
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| 1028 |
compactness | ||
| 9 | 1031 |
Poincare inequality; Imbedding of W^{1,n} into BMO;
H^{-1}; trace |
|
| 1102 |
Part
III.
Weak
solutions of elliptic equations in divergence form overview, weak formulation, Lax-Milgram theorem |
[Evans] 6.1-6.4 |
|
| 1104 h4 due |
an example of nonexistence/nonuniqueness, first existence
theorem |
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| 10 | 1107 |
Fredholm Alternative, second existence
theorem |
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| 1109 |
spectrum of a compact linear operator and third
existence
theorem (eigenvalues); statement of interior H^2 regularity |
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| 1111 |
Remembrance Day | ||
| 11 | 1114 |
Lemma on difference quotient and Sobolev space, proof of
interior H^2
estimate |
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| 1116 |
Higher regularity and boundary regularity, statement of maximal
principle
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| 1118 |
proof of weak maximal principle, Hopf's Lemma |
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| 12 | 1121 h5 due | strong maximal principle Part IV. Linear evolution equations weak solution and semigroup approaches, weak formulation of parabolic equations, vector valued Sobolev spaces (§5.9.2) |
[Evans] 7.1, 7.4 |
| 1123 |
Galerkin method, energy estimate |
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| 1125 |
energy estimate continued, existence and
uniqueness of weak solutions by Galerkin method |
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| 13 |
1128 |
uniqueness continued; semigroup: examples and definition,
generator |
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| 1130 |
basic properties of generators, Hille-Yosida theorem |
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| 1202 |
examples of semigroups from evolution PDE |
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| h6 due Christmas | |||