Math 237A (Topics in Differential Equations)

Course Topics: Mathematical and Computational Geometric PDE
Instructor: Prof. Michael Holst (5739 AP&M, mholst@math.ucsd.edu)
Term: Winter 2007
Lecture: 9:30a-10:50a Tu-Th, APM 5829
Textbook(s):

N. Straumann, General Relativity: With Applications to Astrophysics, Springer, 2004.
R. Wald, General Relativity, Univ. of Chicago Press, 1984.
A. Cohen, Numerical Analysis of Wavelet Methods, JAI Press, 2003.
Class webpage: http://ccom.ucsd.edu/~mholst/teaching/ucsd/237a_w07

CATALOG DESCRIPTION: 237A-B-C. Topics in Differential Equations (4-4-4)
May be repeated for credit with consent of adviser. Prerequisite: consent of instructor.
GRADES, EXAMS, DATES: Your grade in the course is based on attending the lectures and participating in the class.
  • Final: TBA
  • No Class: 1/9, 1/11, 2/20
  • First lecture: Tuesday 9 Jan 2007.
  • Last lecture: Thursday 15 Mar 2007.
  • Finals week: 19-24 Mar 2007.
ANNOUNCEMENTS, NOTES, ETC:
  • IPAM Workshop Notes on Elliptic PDE and Approximation theory
  • Notes on Calculus in Banach Spaces, Variational Methods, and Mechanics
SCHEDULE FOR THE LECTURE TOPICS: The overall plan is to first develop the basic mathematical framework for understanding a class of PDE arising in geometric settings, and based on this develop a general approximation theory framework for designing provably good computational methods for this class of PDE.

Relevant applications include models of biomembranes, models of fluid interactions, geometric flows in material science, flows of metrics in geometry, models of black hole collisions in astrophysics, and physics-based reality models in computer animation.

Since we only have ten weeks, we will build the lectures primarily around one particular geometric PDE model (general relativity) describing the gravitational interaction of massive objects such as black holes and neutron stars. However, the notation, theory, and techniques we develop will be useful for other applications.

The outline below will be updated as the quarter proceeds, based on how the interests of the participants and the lecturer evolve.

Lecture Topics Covered
1 (1/16/07, week 2) Metric spaces, vector spaces, normed spaces, inner-product spaces; A two-point boundary-value problem, weak formulation, the natural appearance of L2 and the L2-based Sobolev space H1 via the Cauchy-Schwarz inequality; Lp spaces on bounded open sets in Rn, the Lp-based Sobolev spaces W[k,p].
2 (1/18/07, week 2) Linear elliptic PDE on bounded open sets in Rn, weak formulation; Linear functionals and bilinear forms; The Riesz representation theorem, the Bounded Operator Theorem, and the Lax-Milgram Theorem; Linear operators on normed and inner-product spaces, the operator norm, the (Hilbert) Adjoint Theorem.
3 (1/23/07, week 3) Normed, metric, and inner-product spaces again; Cauchy sequences and completion of metric spaces; Banach and Hilbert spaces; Lp and W[k,p] as Banach spaces; Linear functionals on Banach spaces, the dual norm, the dual space as a Banach space; Holder, Lipschitz, contraction conditions on maps; The Contraction Mapping Theorem (CMT) in normed spaces; Proof of the Lax-Milgram Theorem via the CMT.
4 (1/25/07, week 3) Continuity, equivalent definitions, uniform continuity; Weak/strong convergence of sequences, strong implies weak; Gateaux and Frechet differentiation of nonlinear mappings between Banach spaces; Generalized Taylor expansion; Newton's Method for nonlinear operator equations on Banach spaces; Convergence properties of Newton's method.
5 (1/30/07, week 4) A general variational problem for a nonlinear function on a Banch spaces (P1), the condition for stationarity problem (P2), and the linearized condition for stationarity problem (P3); Exploiting the connection between problems P1-P3 in Newton's method to enlarge basin of attaction; Galerkin methods for problems P1-P3.
6 (2/1/07, week 4) Two examples of P1-P3: A nonlinear energy functional giving rise to a scalar semi-linear PDE as condition for stationarity, and a nonlinear energy giving rise to the nonlinear elasticity equations as condition for stationarity; The trace theorem in the Sobolev spaces W[k,p] and use in formulating and solving Dirichlet problems; General approximation theorems for Petrov-Galerkin methods for abstract linear operator equations in Banach spaces, Cea's Lemma as a special case; General approximation theorems for Petrov-Galerkin methods for abstract nonlinear equations in Banach spaces.
7 (2/6/07, week 5) Ck-manifolds, charts, an atlas of charts, transformation properties of scalar fields and differentials; multilinear forms; intrinsic and axiomatic definitions of (contravariant) vectors and (covariant) covectors (at a point on the manifold); intrinsic and axiomatic definitions of general rank (r,s) tensors; vector, covector, and general rank (r,s) tensor fields over a manifold.
8 (2/8/07, week 5) Covariant differentiation; affine connections; covariant differention of vectors, covectors, and general rank (r,s) tensors; Tangent and co-tangent spaces, tangent and co-tangent bundles, fibers of tangent and co-tangent bundles; Natural projection and its inverse; covariant differential. Non-commutativity of covariant differentiation: Riemann and torsion tensors; Flatness of the metric tensor and vanishing Riemann tensor.
9 (2/13/07, week 6) The metric tensor, Christoffel symbols of the metric tensor as affine connections; Symmetries of Christoffel symbols and vanishing torsion tensor; Symmetric positive definite (SPD) tensors and long discussion on inner-products on tangent spaces; Riemannian manifolds; Symmetrics of the Riemann tensor, the Bianchi identities; The Ricci tensor, scalar curvature, the Einstein equations.
10 (2/15/07, week 6) Calculus on Banach spaces, constraints, implicitly defined manifolds.
11 (2/22/07, week 7) Submersions, the Ljusternik result, The Lagrange Functional Theorem for constrained stationarity.
12 (2/27/07, week 8) Lagrangians for unconstrained and constrained variational problems in Banach spaces; Example of a quadratic energy with linear constraints in a Banach space; Example of a general energy functional over a bounded domain in Rn with PDE constraints; The Euler-Lagrange equations.
13 (3/1/07, week 8) Lagrangian and Hamiltonian mechanical and dynamical systems; The Principle of Least Action (or rather the Principle of Stationary Action); The equations of motion of mechanical systems as stationary points of an action integral built from a Lagrangian (density); The Legendre Transformation and the Hamiltonian; The equations of motion of mechanical systems as stationary points of an action integral built from a Hamiltonian; Examples from particle mechanics (ODE) and fields (PDE).
14 (3/6/07, week 9) The Laplacean on a Riemannian manifold; Sobolev spaces on a Riemannian manifold; Embedding, compactness, trace, density, and extension; General linear and nonlinear elliptic theory; Approximation theory for Petrov-Galerkin methods.
15 (3/8/07, week 9) The Heat Equation on a Riemannian manifold; Mean curvature and related flows; Flows of Riemannian metrics; Ricci flow; Petrov-Galerkin methods for computing flows.
16 (3/13/07, week 10) The Einstein equations as stationary points of an action integral; Hamiltonian formulation of the Einstein equations; The constraints in the Einstein equations.
17 (3/15/07, week 10) Analysis of the Einstein constraints as a coupled nonlinear elliptic system on a Riemannian manifold; Petrov-Galerkin methods for computing solutions.