MAP6437 (01) Applied Analysis I

Fall 2006

Instructor: Dr. Xiaoming Wang
Office: 312 LOV
Phone: 644-6419
Hour: TR 12:40-1:55PM
Place: DSL468
Office Hours: Tuesday 9:30-10:20 & Thursday 2:05-2:55 and/or by appointment
Textbook: Methods of Applied Mathematics by Todd Arbogast and Jerry Bona, lecture notes (available from Dr. Arbogast's webpage) to be published. See also Supplementary Textbook
Prereq: Advanced Calculus, Linear Algebra, Measure and Integration, or authorization by the instructor
In case you have any questions, please do not hesitate to contact me (644-6419 (O), wxm@math.fsu.edu (e-mail)) or drop by my office at 312 LOV

Course Description

The objective of the sequence is to introduce the students to some of the basic (functional) analytic tools needed in modern applied mathematics research.
This is a new core sequence in the Applied and Computational Mathematics area, and is one of the topics of qualify exams in Applied and Computational Mathematics. Waivers to the Applied Analysis Core (Qualify) Exam may be granted by the end of the successful completion of the sequence.

Course Topics (I&II)

Banach/Hilbert spaces Introduction to Banach and Hilbert spaces, with examples (C^k and L^p); continuous functionals on Banach/Hilbert spaces, extension and representation (Hahn-Banach theorem, Riesz representation); contraction mapping theorem and application to IVP for ODEs; best approximation and orthogonal projections in Hilbert space, compactness, Arzela-Ascoli theorem, weak/strong convergence
Distributions test functions, calculus of distributions, convergence
Fourier analysis Discrete and continuous Fourier transform in classical and distributional settings, convolution; application to linear PDE with constant coefficients and von Neumann analysis of numerical schemes
Sobolev spaces approximation, extension, imbedding, interpolation and traces; application to PDEs (Lax-Milgram theorem)
Differential calculus and variational methods differential calculus in Banach spaces, Euler-Lagrange equations, constrained extrema and Lagrange multiplier, inverse and implicit function theorems; application to least action principle, and other variational problems.
Linear operators linear operators on Banach spaces, adjoint operators, self-adjoint and symmetric operators, Fredholm alternative, spectral theory for compact and compact, self-adjoint operators in a Hilbert space, Hilbert-Schmidt operators, interpolation theorems; application to Sturm-Liouville theory.
Applied stochastic analysis: law of large number and central limit theorem, Markov chains, Wiener process, invariance principle, stochastic differential equations, Ito calculus, Fokker-Planck equations
Asymptotic methods: matched asymptotic expansion, WKB, homogenization.

Supplementary Textbooks

Applied Analysis by John K. Hunter and Bruno Nachtergaele, published by World Scientific. The book is available online at Dr. Hunter's home page.
Functional Analysis by Peter D. Lax, published by Wiley, New York, c2002.
Methods of Mathematical Physics by Richard Courant and David Hilbert, published by Wiley-Interscience, 1962
Analysis , by Elliot H. Lieb and Michael Loss, published by AMS.

Required Work and Grading Policy

20% Classroom participation.
40% Problem Sets. You may work with at most one partner on problem sets.
20% midterm exam
20% final exam

Homework Assignments

	Assignment	Due	Key
AA1HW1	pp.26-27, #20, 21, 22, 24	Sept. 19th	available from the instructor
AA1HW2	pp.55, Ch.2, #20, 22, 24, 32, 40 and the optional problems	Oct. 19th	available from the instructor
AA1HW3	pp. 105, Ch.3, #7(b), 8, 10, 18, 20 and the optional problems	Nov. 9	available from the instructor
AA1HW4 double as final	pp.138-140, ch.4, #10,13,17,19,20 and the optional problem	Dec. 12,	available from the instructor

AA1 HW2 optional problems

(1): Let &isin C_b ([0, &infin)). Show that there exists a Banach limit (generalized limit) LIM such that LIM (g) = limsup_{t &rarr &infin} g(t).

(2) Consider the following subsapce M of the Banach space C([0,1]) equipped with the sup-norm. M={f&isin C([0,1]) | f(0)=0, ∫₀¹ f(x) dx = 0}. For u(x)=x, show that d(x, M)=1/2, but that the distance is not attained for any m&isin M.

(3) Show that the weak solution of the Stokes equations a(u,v)+b(p,v)=F(v), for all v, b(q, u)=0, for all q, has a unique solution (u,p)∈ (H¹₀)²×L², where a(u,v)=(∇ u, ∇ v), b(p,v)=-(p,∇·v), and F is a continuous linear functional on (H¹₀)². Hint:first show the existence of v within the divergence free space.

AA1 HW3 optional problems

(1): Let {u_n, n=1,2, ... } be an orthnormal set in a Hilbert space H. Show that u_n→ 0 weakly.

(2): Consider the real Hilbert space of vector valued square integrable function on a two dimensional torus H=(L²(T²))². Let V be the closure of {v∈ (C^&infin(T²))² | ∇·v=0}. Let W be the closure of {w∈ (C^&infin(T²))² | w=&nablaφ for some φ∈C^&infin(T²)}. Show that (L²(T²))² is the direct sum of V and W, i.e., H =V⊕W. In other words, the space V and W are orthogonal to each other, and for each element u∈ H, there exist a unique element v∈V and a unique element w∈W, such that u=v+w.

AA1 HW4 optional problem: (a), Show that a test function on the unit square is the divergence of a 2D vector valued test function on the unit square if and only if the integral of it over the square is zero. (b) Show that a 2D vector valued distribution on the unit square is the gradient of a scalar distribution if and only if it's action on all divergence free (divergence zero) 2D vector valued test functions vanish. (c) What can you say about higher spatial dimension?

AA2 HW2 optional problem: solve the linear Schrodinger equation i &part_tu+ &Delta u = 0 using Fourier transform. What can you say about the solutions?

AA2 HW3 optional problems

1. Let u ∈ C¹ ([0,1]x[0,1]) satisfying u(x₁, 0)=0. Prove the Poincare inequality via direct method.

2. Show for the unit square [0,1]X[0,1], the conclusion of 5(b) on page 192 holds.

3. Verify the common properties on the regular Sturn-Liouville problem (p(x)u'(x))'+q(x)u(x) = λw(x)u(x), u(0)=u(1)=0, where the weight function w and p are smooth and positive on [0,1] and q is bounded.

AA2 HW4 optional problem: Show that the Euler equation in the Lagrangian coordinates for inviscid incompressible homogeneous fluid is the same as geodesics on the space of volume preserving diffeomorphisms with the metric given by kinetic energy.

Americans Disabilities Act

register with and provide documentation to the Student Disability Resource Center; and
bring a letter to the instructor indicating the need for accommodation and what type. This should be done during the first week of class.

Academic Honor Policy

http://dof.fsu.edu/honorpolicy.htm