
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: 9:30a10:50a TuTh, 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:
237ABC. Topics in Differential Equations (444)
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: 1924 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 physicsbased 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,
innerproduct spaces;
A twopoint boundaryvalue problem, weak formulation,
the natural appearance of L2 and the L2based Sobolev
space H1 via the CauchySchwarz inequality;
Lp spaces on bounded open sets in Rn, the Lpbased 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 LaxMilgram Theorem;
Linear operators on normed and innerproduct spaces,
the operator norm, the (Hilbert) Adjoint Theorem.

3 (1/23/07, week 3) 
Normed, metric, and innerproduct 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 LaxMilgram 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 P1P3 in
Newton's method to enlarge basin of attaction;
Galerkin methods for problems P1P3.

6 (2/1/07, week 4) 
Two examples of P1P3: A nonlinear energy functional giving
rise to a scalar semilinear 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 PetrovGalerkin methods
for abstract linear operator equations in Banach spaces,
Cea's Lemma as a special case;
General approximation theorems for PetrovGalerkin methods
for abstract nonlinear equations in Banach spaces.

7 (2/6/07, week 5) 
Ckmanifolds, 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 cotangent spaces, tangent and cotangent
bundles, fibers of tangent and cotangent bundles;
Natural projection and its inverse;
covariant differential.
Noncommutativity 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 innerproducts 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 EulerLagrange 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 PetrovGalerkin 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;
PetrovGalerkin 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;
PetrovGalerkin methods for computing solutions.

