## 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:
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Mon May 01: STUDY DAY: regular class time will be extra office hour
Tue May 09: final exam