MAT-INF4300 - Autumn 2010

Teaching plan

Date	Teacher	Place	Topic	Lecture notes / comments
24.08.2010	Ujjwal Koley	B71	Introduction	Some repetition (?); Integration by part in R^n and formulas which follow from it; Multidimensional chain rule; Convolution; Laplace equation
27.08.2010	Ujjwal Koley	B70	Laplace equation	Poisson's equation; Harmonic functions, Fundamental solution.
31.08.2010	Ujjwal Koley	B71	Mean Value theorem	Mean-value formulas; Maximum principle
03.09.2010	Ujjwal Koley	B70	Regularity	Regularity theorem, Liouville's theorem, Harnack's inequality, Introduction to Green's function
07.09.2010	Nils Henrik Risebro	B71	Green's functions	Symmetry of the Green's function. Green's function for a half plane and a sphere.
10.09.2010	NIls Henrik Risebro	B70	Green's functions, energy methods	The students show that the formula for the sphere is correct. I give some energy arguments.
14.09.2010	Achenef Tesfahun	B71	H?lder spaces. Mollifiers	H?lder spaces. Mollifiers; smooth approximation of locally integrable functions
17.09.2010	Achenef Tesfahun	B70	Weak derivatives, Sobolev spaces	Weak derivatives and Sobolev spaces: L.C. Evans Page 242-245. Fundamental Theorem of calculus of Variation.
21.09.2010	Achenef Tesfahun	B71	Sobolev spaces	Example, properties of weak derivatives and Sobolev spaces, Sobolev spaces as function spaces: L.C. Evans, page 246-249.
24.09.2010	Achenef Tesfahun	B70	Exercises	We will do the exercises (2.5.2-2.5.8 ) in L. C. Evans, page 85-87.
28.09.2010	Achenef Tesfahun	B71	Sobolev spaces, approximation	Approximating a function in Sobolev spaces by smooth functions. L. C. Evans page 250-254
01.10.2010	Achenef Tesfahun	B70	Sobolev spaces: Extension, Traces	Extension and Trace Theorems for Sobolev spaces. L. C. Evans page 254-259
05.10.2010	Achenef Tesfahun	B71	Sobolev spaces: Traces, Sobolev inequalities	We continue on the proof of Trace Theorem. I will also give the proof for Gagliardo-Nirenberg-Sobolev inequality. L.C.Evans page 258-264
08.10.2010	Achenef Tesfahun	B70	Sobolev Spaces: Sobolev inequalities	Estimates in W^{1,p} and W_0^{1,p} (when p<n). We will also see Morrey's inequality (when n<p). L.C. Evans (page 265-269)
12.10.2010	Achenef Tesfahun	B71	Sobolev Spaces: Sobolev inequalities	We finish proof of Morrey's inequalities, and see Theorem 5 (page 269 of the book). We will also do the proofs for the general Sobolev inequalities (page 270-271 of the book).
15.10.2010	Achenef Tesfahun	B70	Exercises	We will do the exercises 3, 5, 6, 7, 8 and 13 (if we get time) in L. C. Evans, page 290-291.
19.10.2010	Xavier Raynaud	B71	Compactness. Arzela-Ascoli and Rellich-Kondrachov theorems	section 5.7
22.10.2010	--	B70	Rellich-Kondrachov theorem - Poincare inequality	sections 5.7, 5.8.1. NB: lecture finishes at 13:15.
26.10.2010	--	B71	Fourier transform	Exercises 5.10.10, 5.10.14 section 4.3.1
29.10.2010	--	B70	Fourier transform - H^s spaces	Section 5.8.4. Appendix D. Exercise on Fourier transform of a Gaussian . Exercises 5.10.15
02.10.2010	--	B71	Some results and definitions from linear functional analysis	Some parts of appendix D Riesz representation theorem Elliptic equation (intro)
05.10.2010	--	B70	Weak solutions to elliptic equations and Lax-Milgram theorem	Exercise on weak convergence . Exercise 5.10.18
09.11.2010	--	B71	Existence of weak solutions - Regularity theory
12.11.2010	--	B70	Regularity theory - Difference quotient	Exercise on Poincare's inequality Oppgave 1 and 2 from (i) to (v) in exam 2008
16.11.2010	--	B71	Interior regularity
17.11.2010	--	B70	Boundary regularity	Exercise on higher interior regularity Problem 1 in exam 2009 Oppgave 1 in exam 2005
23.11.2010	--	B71	Maximum principle
26.11.2010	--	B70	Maximum principle	Exercise on regularity End of oppgave 2 in exam 2008
30.11.2010	--	B71	Maximum principle - Weak solution (rep.)
03.12.2010	--	B70		Exercise 6.6.2 Problem 2 and 3 in exam 2009 Problem 1 in exam 2006 There is an error in that problem. We will just shortly discuss it. Problem 2 in exam 2004

Published July 5, 2010 10:33 AM - Last modified Oct. 29, 2024 1:56 PM