Class Diary for M536, Spring 2008, Jochen Denzler
Wed Jan 09: Introduction to quasilinear PDEs of 1st order; overview, with
focus on the linear case yet.
Fri Jan 11: One example carried through; pertinent thms from ODEs quoted.
Mon Jan 14:
Global existence and uniqueness for linear 1st order PDEs (classical solutions)
Wed Jan 16:
Local existence and uniqueness for quasilinear 1st order PDEs. Crossing of characteristics.
Inviscid Burgers equation.
Fri Jan 18: Integral solutions defined. Nonuniqueness. Shocks and Rarefaction waves
Mon Jan 21: --- MLK DAY ---
Wed Jan 23: Hopf's transformation between viscous Burger and Heat
equation; viscosity regularization of Burgers equation.
Fri Jan 25:
Viscosity regularization finished. Some general handwaving remarks about
what insight this gives us into the material (not covered) in 3.3-3.4
Mon Jan 28:
characteristic equations for 1st order fully nonlinear PDEs.
Wed Jan 30:
Local solutions for fully nonlinear 1st oorder PDEs with noncharacteristic
initial data. --- Begun multi-variable Taylor in prep' for
Cauchy Kovalevskaya thm
Fri Feb 01:
Real analytic functions; majorant method; noncharacteristic surfaces for a PDE
Mon Feb 04:
Noncharacteristic surfaces; examples for popular equations; recursive construction
of Taylor coeffs. Hwk: Pblms 2 and 9 from ch. 4.7
Wed Feb 06:
Cauchy Kovalevskaya finished.
Fri Feb 08:
Holmgren uniqueness
Mon Feb 11:
briefly stated existence of Lewy's example. Classification 2nd order def'd.
Normal form in nD for const coefficients (linear algebra of quadratic forms)
Wed Feb 13:
Normal form in 2D; mainly hyperbolic.
Fri Feb 15:
An example of calculating a NF in the elliptic case.
Mon Feb 18: Conversion of linear wave eqn into integral equation.
Linear wave eqn in char coordinates: iterative solution leading to power series
Wed Feb 20: (writing ahead of time:) Banach's fixed point theorem, and an
application proving existence for the Cauchy problem of
a nonlinear (semilinear) wave equation
Fri Feb 22:
Characteristic IVP for the heat eqn also solvable via Banach's FPT. Overview over
Riemann function method.
Mon Feb 25:
Riemann function calculation. --- review (sub)harmonic functions and max principle.
Existence theorem for Laplace eqn stated and reviewed what further consequences it
entails (Green's fct; solvability of Poisson eqn, and, with more functional
analysis even further elliptic equations)
Wed Feb 27:
Perron method.
Fri Feb 29:
Perron method proof finished. Barrier.
Mon Mar 03:
Lebesgue's counterexample (domain where the Laplace eqn with Dirichlet
BCs has no classical solution)
Wed Mar 05: Crash course on Lebesgue integral: examples of functions
that ought to be integrable, but are not, in the sense of Riemann;
outline of construction via Daniell approach.
Fri Mar 07:
Properties of the Lebesgue Integral; measurable functions and sets.
Mon Mar 10:
Some key counterexamples.
Wed Mar 12: EXAM
Fri Mar 14: SPRING BREAK
Mon Mar 17: SPRING BREAK
Wed Mar 19: SPRING BREAK
Fri Mar 21: GOOD FRIDAY
Mon Mar 24:
Review of Banach Fixed Point Theorem: How to use it for various equations
Hwk: 30-40
Wed Mar 26:
Notion of a weak derivative; it is NOT classical differentiability almost
everywhere (and why it shouldn't be). Key examples, counterexamples, and
surprises.
Fri Mar 28:
The Sobolev spaces Wk,p.
Mon Mar 31:
Completeness of L^p
Wed Apr 02:
L^p convergence implies pointwise ae convergence of a subsequence.
Completeness of W^{k,p}. Density of smooth fcts in Sobolev space.
Fri Apr 04:
Informal discussion of traces: it makes sense to talk about `L^p boundary values'
of a W^{1,p} function'. Overview of Sobolev imnbedding
Mon Apr 07:
Gagliardo-Nirenberg-Sobolev inequality for p=1 in R^n.
Wed Apr 09:
Sobolev inequality in bounded domains: statement and basic ideas
Fri Apr 11:
Morrey estimate; Sobolev into H\"older
Mon Apr 14:
some rmks on little and big H\"older spaces. C^1 is not dense in big-H"older
Wed Apr 16:
weak solutions of elliptic equations defined; Lax Milgram
Fri Apr 18:
existence of weak solutions and Lax Milgram (estimate)
Mon Apr 21:
inner regularity of weak solutions: overview of method. --- Teaching Evaluation
Wed Apr 23:
Courant Hilbert variational problem (CHVP) for the eigenvalue problem
Fri Apr 25:
(writing ahead time) CHVP finished up
Mon Apr 28: STUDY DAY
Fri May 02: FINAL EXAM 10:15-12:15
(scheduled by university policy)
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