Class Diary for M535, Fall 2017, Jochen Denzler
Wed Aug 23:
Intro. Derivative vs gradient, divergence, Laplacian; Divergence thm stated and
explained.
Fri Aug 25:
Proof of div theorem for `simple' domains; Def of C^k domain; partition of
unity and proof of div thm in this setting. Coninuoust partition of unity
constructed.
Mon Aug 28:
Smooth partition of unity; convolution. Examples of PDEs; names of things and
key problems.
Wed Aug 30:
Transport equation. Laplace + Poisson equations started: harmonic functions
Fri Sep 01:
genesis of the Laplace, Poisson and heat equations. Transformation of the
Laplacian into curvilinear coordinates. Hwk handed out; due next Friday
Mon Sep 04:
LABOR DAY
Wed Sep 06:
Fundamental Solution of the Laplace equation: calculated, intuition with
Dirac-delta explained and critiqued. Hints at how delta would be made rigorous
Fri Sep 08:
Rigorous proof for formula about fundamental solution, and its implications
Mon Sep 11:
Properties of harmonic fcts stated, explained; proof of Mean Value Properties
and converse; regularity of harmonic functions.
Wed Sep 13:
Strong maximum principle for harmonic functions. Also weak maximum principle
argument applicable for more general equations; uniqueness for Boundary Value
Problem of Poisson eqn.
Fri Sep 15:
Subharmonic functions briefly defined: Hwk 6 due next Friday (to find it, see
handed-out sols for #1-5).
Quantitative inner regularity estimate; its interpretation, analyticity of
harmonic functions as a consequence; Liouville theorem.
Mon Sep 18:
Proof of analyticity; Harnack inequality started.
Wed Sep 20:
Harnack inequality finished (construction of Harnack chain). Greens function:
idea and purpose
Fri Sep 22:
Green's function: construction (modulo finding harmonic corrector function)
Mon Sep 25:
Green's function in half space; proof of representation formula for harmonic
fct in terms of Poisson kernel. Hwk 7-10 due Wed 10/4
Wed Sep 27:
Green's function and Poisson kernel in ball. Warning: Laplace u in C^0 does not
imply u in C^2.
Fri Sep 29:
A bit more on Hölder continuous functions. Detailed existence statement about
Poisson eqn. Poisson eqn and calculus of variations.
Mon Oct 02:
General outline of an existence proof by calculus of variations. (Technical
details will be discussed at a later time.) ___ Fund'l Sol'n of Heat eqn barely
started.
Wed Oct 04:
Fund'l solution properties proved; used and explained Dominated Convergence
Theorem.
Fri Oct 06:
FALL BREAK
Mon Oct 09:
Variation of Constants in ODEs - Duhamel's Principle.
Wed Oct 11:
Duhamel's principle for HE proved rigorously.
Fri Oct 13:
Mean value property; and parabolic maximum principle (for HE) via heat balls
Mon Oct 16:
Uniqueness for the HE, from max principle; also uniqueness of Cauchy problem
(subject to assuming a 2-sided Gaussian bund on the solution). A nonuniqueness
example, absent some bound on the solution.
Wed Oct 18:
One-sided bound sufficient for uniqueness of CP of HE
Fri Oct 20:
One-sided bound sufficient for uniqueness of CP of HE
Mon Oct 23:
general parabolic maximum principle
Wed Oct 25:
`MID'TERM EXAM
Fri Oct 27:
finished up general par max principle; local inner regularity estimates for
HE.
Mon Oct 30:
Local inner regularity finished. Nonanalyticity in time. Uniqueness by energy
method.
Wed Nov 01:
Backwards uniqueness for HE; Intor to Fourier Series by ODE analog $\dot u = Au$
Fri Nov 03:
Intro Fourier series cont'd
Mon Nov 06:
Rigorous approach: Best approximations of periodic fcts in mean-square by trig
polynomials. Parseval inequality and L^2 convergence of FS. Completeness
equivalent to convergence of FS to fct equivalent to Parseval equality.
A quick completeness pf quoting Weierstrass approx theorem.
Wed Nov 08:
Uniform convergence of FS to C^1 functions.
Fri Nov 10:
odd and even extensions. Solving HE and inhom HE with hom. Dirichlet or Neumann
BCs.
Mon Nov 13:
Without proofs: Eigenfunctions of Dirichlet Laplacian in Lipschitz domains; and
expansions in terms of these fcts. Weyl asymptotics of eigenvalues; Courant
Nodal domain theorem. ~~~ Fourier Transform motivated as limit of Fourier series.
Wed Nov 15:
Hwk comments; Plancherel's theorem for Fourier transform
Fri Nov 17:
Plancherel's thm finished. Other equations
Mon Nov 20:
Wave equation; overview and 1dim
Wed Nov 22:
Wave eqn in 3dim by spherical means
Fri Nov 24:
THANKSGIVING BREAK
Mon Nov 27:
wave eqn in 2 dim by descent
Wed Nov 29:
Wave eqn in odd dimensions
Fri Dec 01:
Mon Dec 04:
Wed Dec 06: STUDY DAY
Wed Dec 13: FINAL EXAM 10:15-12:15
(scheduled by university policy)
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