### Class Diary for M535, Fall 2017, Jochen Denzler

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Wed Aug 23: **
Intro. Derivative vs gradient, divergence, Laplacian; Divergence thm stated and
explained.

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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.

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Mon Aug 28: **
Smooth partition of unity; convolution. Examples of PDEs; names of things and
key problems.

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Wed Aug 30: **
Transport equation. Laplace + Poisson equations started: harmonic functions

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Fri Sep 01: **
genesis of the Laplace, Poisson and heat equations. Transformation of the
Laplacian into curvilinear coordinates. *Hwk handed out; due next Friday*

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Mon Sep 04: **
LABOR DAY

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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

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Fri Sep 08: **
Rigorous proof for formula about fundamental solution, and its implications

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Mon Sep 11: **
Properties of harmonic fcts stated, explained; proof of Mean Value Properties
and converse; regularity of harmonic functions.

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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.

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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.

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Mon Sep 18: **
Proof of analyticity; Harnack inequality started.

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Wed Sep 20: **
Harnack inequality finished (construction of Harnack chain). Greens function:
idea and purpose

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Fri Sep 22: **
Green's function: construction (modulo finding harmonic corrector function)

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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*

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Wed Sep 27: **
Green's function and Poisson kernel in ball. Warning: Laplace u in C^0 does not
imply u in C^2.

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Fri Sep 29: **
A bit more on Hölder continuous functions. Detailed existence statement about
Poisson eqn. Poisson eqn and calculus of variations.

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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.

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Wed Oct 04: **
Fund'l solution properties proved; used and explained Dominated Convergence
Theorem.

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Fri Oct 06: **
FALL BREAK

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Mon Oct 09: **
Variation of Constants in ODEs - Duhamel's Principle.

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Wed Oct 11: **
Duhamel's principle for HE proved rigorously.

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Fri Oct 13: **
Mean value property; and parabolic maximum principle (for HE) via heat balls

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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.

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Wed Oct 18: **
One-sided bound sufficient for uniqueness of CP of HE

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Fri Oct 20: **
One-sided bound sufficient for uniqueness of CP of HE

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Mon Oct 23: **
general parabolic maximum principle

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Wed Oct 25: **
`MID'TERM EXAM

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Fri Oct 27: **
finished up general par max principle; local inner regularity estimates for
HE.

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Mon Oct 30: **
Local inner regularity finished. Nonanalyticity in time. Uniqueness by energy
method.

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Wed Nov 01: **
Backwards uniqueness for HE; Intor to Fourier Series by ODE analog $\dot u = Au$

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Fri Nov 03: **
Intro Fourier series cont'd

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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.

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Wed Nov 08: **
Uniform convergence of FS to C^1 functions.

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Fri Nov 10: **
odd and even extensions. Solving HE and inhom HE with hom. Dirichlet or Neumann
BCs.

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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.

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Wed Nov 15: **
Hwk comments; Plancherel's theorem for Fourier transform

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Fri Nov 17: **

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Mon Nov 20: **

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Wed Nov 22: **

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Fri Nov 24: **
THANKSGIVING BREAK

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Mon Nov 27: **

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Wed Nov 29: **

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Fri Dec 01: **

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Mon Dec 04: **

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Wed Dec 06: ** STUDY DAY

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Wed Dec 13: ** FINAL EXAM 10:15-12:15
(scheduled by university policy)

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