## Course material for M534, Jochen Denzler

I do not closely follow a textbook, but recommend, as an affordable reference
with significant intersection with the course, `Calculus of Variations'
by Gelfand and Fomin, which is available at Dover for a price that makes
you smile with joy.
**Course announcement in previous semester: **
calcvar.txt

**Syllabus for M534: **(various file formats)
syllabus.dvi --
syllabus.ps --
syllabus.pdf

**Homework Problems: **

These files will be continuously updated (last update Apr 11: hooray,
finally some new hwk pblms have arrived...)

Hwk.dvi --
**Hwk.ps** --
**Hwk.pdf** --
Hwk00.pdf

Note: representation of imbedded graphics is software dependent in Hwk.dvi
and will not work in Hwk00.pdf; so if possible use the boldfaced formats

**Reference Sheet**
different notions of relative minima in Calc'Var':
ps --
pdf

**Reference Sheet**
necessary and sufficient conditions for minima:
ps --
pdf

**Proof Details** Formalism of derivatives in Banach spaces, and sufficient
conditions for weak minima (whole segment vs sufficiently short subsegments):
dvi --
ps --
pdf
*(These proof details are provided for synergetic effect with more advanced
classes, to see the formalism in action; they are not required course material)
*

### Class diary and homework assignments:

**
Wed Jan 11: ** Intro and Overview

**
Fri Jan 13: ** Some sample problems --
*Hwk: begin handed-out pblms; try to get 1-3 by Wed*

**
Mon Jan 16: ** MLK DAY

**
Wed Jan 18: ** Notions of differentiability; a first sample problem.

**
Fri Jan 20: ** Fundamental Lemma, and DuBois-Reymond. Sample problem
finished.
*Hwk: Can we move to 4+5 soon? Typo in number 4: The function I
(rather than g, which doesn't occur anywhere) is unbounded below*

**
Mon Jan 23: ** A simple ad-hoc minimum proof for our sample problem;
Different notions of relative minima (weak vs strong)

**
Wed Jan 25: ** Local vs global: minimality of short vs long segments.
The Euler-Lagrange eqn is a necessary condition for locally weak minima.

**
Fri Jan 27: ** Proof for Euler-Lagrange eqn; a brief remark on locality of
DiffEqs vs nonlocality of the minimum question.
Erdmann's corner condition; free boundaries; The brachystochrone example
started. *Hwk: hard deadline Monday for 1-5. -- 9 by Monday,
next priority 10-12. 8 as time permits.*

**
Mon Jan 30: ** Brachystochrone ODE solved. C^2 regularity of solutions.

**
Wed Feb 01: ** Discussion on Hwk; Energy integral.

**
Fri Feb 03: ** Legendre condition; convexity in MV-Calc.

**
Mon Feb 06: ** BVP for Brachystochrone solved. Absolute minimality
proved by a convexity argument in new coordinates v=sqrt(2y). A few general
remarks on convexity in Calc of Var. *Hwk: 13 asap*

**
Wed Feb 08: ** Lagrange multiplier method; the catenary.

**
Fri Feb 10: ** The Lagrangian form of mechanics and elimination of
confining forces

**
Mon Feb 13: ** Mechanics finished up; the Dirichlet principle
(technical difficulties pointed out but not resolved)

**
Wed Feb 15: ** Facts re Laplace equation and its variational principle.

**
Fri Feb 17: ** Intro direct methods: the R^n case, and an
outline of modifications needed in Calc'of Var's.

**
Mon Feb 20: ** Geodesics without integrals; existence proof via
Arzela Ascoli

**
Wed Feb 22: ** Geodesics concluded. Discussion of pending hwk: #14,#15
(on farewell minimals, and on catenoid)

**
Fri Feb 24: ** Hwk discussion. Derivative and integral preliminaries for
direct methods.

**
Mon Feb 27: ** Hwk discussion. Crash course: basic facts about the
Lebesgue integral.

**
Tue Feb 28: ** (4:30-5:45 makeup seesion)
Lebesgue integral, weak derivatives and Sobolev spaces defined and
explained informally. Compactness theorem in W^{1,2}

**
Wed Mar 01: ** Sturm Liouville Eigenvalue problem

**
Fri Mar 03: ** Sturm Liouville and Courant Hilbert variational problem
and their analogy to the eigenvalue problem for symmetric matrices.

**
Mon Mar 06: ** Geodesics revisited; Finite elements

**
Wed Mar 08: ** Lavrentiev phenomenon

**
Fri Mar 10: ** Lavrentiev phenomenon

**
Mon Mar 13: ** Discussion of catenoid and Goldschmidt solution

**
Wed Mar 15: ** Intro to sufficient conditions: Jacobi equation
and conjugate points

**
Fri Mar 17: ** (optional material for those present:) a quick intro to
saddle point methods (minimax and Palais-Smale without proofs)

**
Mon Mar 20: ** SPRING BREAK

**
Wed Mar 22: ** SPRING BREAK

**
Fri Mar 24: ** SPRING BREAK

**
Mon Mar 27: ** Review; proof of theorem about conjugate points begun.

**
Wed Mar 29: ** Proof of theorem about conjugate points; Riccati equation

**
Fri Mar 31: ** Outline of extremal field method in a simple example

**
Mon Apr 03: ** Hilbert's invariant integral approach

**
Wed Apr 05: ** Hilbert's invariant integral approach

**
Fri Apr 07: ** Discussion of exam

**
Mon Apr 10: ** Weierstrass E-function

**
Wed Apr 12: **

**
Fri Apr 14: ** GOOD FRIDAY

**
Mon Apr 17: ** cancelled (makeup was 75 min on Feb 28)

**
Wed Apr 19: ** cancelled (makeup was 75 min exam out of class time)

**
Fri Apr 21: ** cancelled (2 x 75 = 3 x 50)

**
Mon Apr 24: **

**
Wed Apr 26: **

**
Fri Apr 28: **

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

**
Mon May 01: ** STUDY DAY: regular class time will be extra office hour

**
Tue May 09: ** final exam

**
**