### 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:
Mon Nov 20:
Wed Nov 22:
Fri Nov 24: THANKSGIVING BREAK
Mon Nov 27:
Wed Nov 29:
Fri Dec 01:
Mon Dec 04:
Wed Dec 06: STUDY DAY
Wed Dec 13: FINAL EXAM 10:15-12:15 (scheduled by university policy)

- - - - - - - - - - - - - - - - - - - - - -