Class Diary for M535, Fall 2007, Jochen Denzler
Wed Aug 22: Review of multi-variable calculus. Divergence theorem
Fri Aug 24:
Gradient, divergence, Laplacian, and divergence theorem
Hwk: Read notes handed out and solve sample problems integrated
therein (see main webpage for handout)
Mon Aug 27:
Partition of unity (as a tool to prove Gauss' theorem without annoying
hypotheses on the domain geometry).
Wed Aug 29:
Smooth partition of unity via mollifier.
The transport eqn, and the IVP (Cauchy problem) for it. Characteristic
lines.
Fri Aug 31:
The desire for a generalized (`weak') notion of solutions to a PDE
forshadowed in the transport problem. General philosophy of defining weak
solutions by means of integration by parts with a smooth test function.
--- Laplace equation begun.
Hwk: pblm 1,2 on p 85; and the last 2 problems on p 11 of the calculus
notes handed out. Due Wed. if possible
Mon Sep 03: LABOR DAY
Wed Sep 05:
Fundamental sol'n of Laplacian and sol'n of Poisson equation in all of space
Fri Sep 07:
Technicalities re fundamental solution. Informal notion of `delta distribution'.
Hwk: #5 (handed out)
Mon Sep 10:
Mean value property and strong maximum principle for harmonic functions.
Wed Sep 12:
Hwk hints; A multivariable calc proof for the weak max principle. Uniqueness
for BVP of Poisson eqn. Existence thm for BVP of Poisson eqn stated without
proof.
Hwk: book p86, Pblm 4 due Mon
Fri Sep 14:
Smoothness of harmonic functions; point estimates for derivatives. Analyticity
proof sketched (multi-index notation omitted); outline only.
Mon Sep 17:
Multi-index notation explained on request; C^k norm estimated by L^1 norm in
larger domain. Harnack inequality for balls
Wed Sep 19:
Harnack chain; Green's function introduced; how it reduces the task to solve
many BVPs to the task of solving not quite so many.
Fri Sep 21:
Symmetry of Green's function. Green's function in half space via reflection.
Mon Sep 24:
Proof details for Green's function in half space.
Wed Sep 26: Lemma on Dirac sequences; Green's function for ball.
Hwk: 7,8 as handed out/posted
Fri Sep 28:
Proof details for Green's fct on ball. --- Thm on Dirichlet principle stated.
Mon Oct 01:
Dirichlet Principle proofs, and philosophy towards existence proof via calc
of variations (minus technicalities). Hwk 7,8 due Fri, 9+10 due Mon
Wed Oct 03:
A rough overview over Perron's method for an existence proof. (Details to be
filled in later in the course.) --- Heat equation and fundamental solution
introduced (claimed).
Fri Oct 05:
Duhamel's principle: stated and analogy with variation of parameters explained
Mon Oct 08:
Questions about pending homework. Notation about domains, cylinders,
parabolic boundaries. Scaling observations for the heat equation and the
C^2_1 space.
Wed Oct 10:
Rigorous proof of the solution formula for the inhomogeneous heat equation.
Fri Oct 12: FALL BREAK
Mon Oct 15:
Comments on hwk; questions re exam. Hints for hwk 10 (and its purpose in
particular).
Wed Oct 17: EXAM: We'll use the library room Ayres 300 so we can go
90 minutes
Fri Oct 19:
Discussion of exam. Maximum principle for heat equation from heat-ball mean
value property (skipped calc for heat ball mean value property).
Mon Oct 22:
Maximum principle for heat eqn Friedmann style
Wed Oct 24:
Uniqueness of IBVP and Cauchy-Problem for heat eqn (which latter needs a growth
condition on the solution).
Fri Oct 26:
Smoothness of solutions to heat equation, including the logic of an a-priori
estimate.Hwk 11,12,14 by Mon if possible
Mon Oct 29:
energy method and differential inequalities for Heat eqn; uniqueness and
backwards uniqueness
Wed Oct 31:
Fourier Series: Approximation in the L^2 norm; Parseval inequality;
Fri Nov 02:
Fourier series: Convergece of the FS to its function: uniformly for C^1 or PL
fcts; in mean fo L^2 functions. Dirichlet kernel
Mon Nov 05:
Extensions of fct on interval to periodic functions.
Heat equation on interval, via Fourier series.
Wed Nov 07:
Fourier series as eigenfunction expansion. (Conceptual; no proofs)
Fri Nov 09:
Eigenfunction expansion theorem for Laplace with various BCs; Nodal theorem,
variational principle and minimax / maximin characterizations; Weyl's law.
Illustrated and ideas sketched, no proofs.
Mon Nov 12:
Fourier transform.
Wed Nov 14:
Fourier transform.
Fri Nov 16:
Wave Equation (Modelling; Cauchy problem in 1D solved)
Mon Nov 19:
Wave eqn 3D
Wed Nov 21:
Wave eqn 3D finished. Descent to 2D idea outlined
Fri Nov 23: THANKSGIVING BREAK
Mon Nov 26:
Wave eqn (2k+1)D outlined; algebraic core of argument. Motivation to
seek semiconjugacy of 1D wave
operator and (2k+1)D radial wave operator.
Wed Nov 28:
Wave eqn in nD, overview; domain of dependence in even vs odd dimensions;
Fri Nov 30:
Inhomogeneous wave eqn; energy method begun
Mon Dec 03:
proofs of uniqueness and of domain of dependence via energy argument
Wed Dec 05: STUDY DAY
Mon Dec 10: FINAL EXAM 10:15-12:15
(scheduled by university policy)
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