MATH 534- CALCULUS OF VARIATIONS-FALL 2015
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)