Syllabus

Course Outline

Topics in L.C.Evans' PDE text

8/20 Overview of the course- examples

8/25 Euler-Lagrange equation, convexity, examples (2.2)

8/27 Weierstrass example, rot symm minimal surfaces, brachistochrone (2.2)

9/1: Second variation, Jacobi fields, Legendre criterion, conjugate points (Jost 1.3)

9/3 minimal graphs: first and second variation, Jacobi fields. Jacobi's theorem on conjugate points

9/8 Presentation (Till) regularity in 1 dimension, proof of Jacobi's theorem.

9/10 Regularity in one dimension (proofs); Legendre transform

9/15 class canceled

9/17 class canceled

9/22 Hamiltonian formulation

9/25 Hamilton-Jacobi equation: derivation (ref: Jost& Jost 4.2), characteristics for 1st order PDE (ref: Evans)

9/29 Hamilton-Jacobi equation: Hamilton's eqns as characteristic system for (HJ) [ref. Evans], Jacobi's theorem on

complete soluttions (ref: Dacorogna), examples (ref. Jost&Jost)

10/1 Hamilton-Jacobi equation: solution via Hopf-Lax formula in autonomous case

10/6 Weak solutions of Hamilton-Jacobi equations

10/8 Presentations: problems on convexity (Joshua Mike, Stefan Schnake, Kevin Sonnanburg)

10/13 Class canceled

10/15 FALL BREAK

10/20 Viscosity solutions of the Hamilton-Jacobi-Bellman equations

10/22 Rademacher's theorem (presentation by Caleb Castleberry)

Rademacher's Theorem (notes by C. Castleberry)

10/27 Existence by the direct method: main theorem

10/29 Existence by the direct method: weak continuity of determinants

11/3 Student presentations (Raoufat, Zhang), direct method/ vector-valued case.

11/5 harmonic maps to manifolds/ interior regularity of minimizers

11/10 Plateau's problem: Douglas-Rado solution via the energy functional (from Lawson)

11/12 Presentation (Pan) Plateau's problem (end): Courant-Lebesgue lemma, equicontinuity.

11/17 Minimal surface equation: existence for small data.

11/19 Minimal surface equation: Bermstein's theorem. (from Colding-Minicozzi)

11/24 student presentation (Zhu): Wirtinger's ineq./ 2-dim'l case of isop ineq/isop const=Sobolev const. (from Chavel)

11/26 Thanksgiving (no lecture)

12/1 isoperimetric inequality in R^n: Sobolev ineq. w/ sharp constant, uniqueness via Alexandrov's theorem. (from Chavel)