Material for Preliminary Exam B --- Real Analysis & PDE
The following list gives topics on which the Preliminary Examination B in Real Analysis & PDE will be based. Math 5317 & Math 5321 cover many, but not necessarily all, of these topics. Students (even those who have taken Math 5317 and/or Math 5321) are advised to prepare for the examination using many resources, including (but not limited to) the books suggested below.
It should be noted that a student who has taken only Math 5317, not Math 5321, should be able to pass this examination without necessarily answering any questions on the PDE material.
1. Measure Theory
· Fields, σ-Fields, Additive Set Functions
· Measures, Measure Extension Theorem
· σ-Algebras of Borel and Lebesgue Sets
· Construction of non-Measurable Set
· Lebesgue-Stieltjes Measures on R
· Measurable Functions
· Convergence in μ (measure)
· Convergence Almost Everywhere (μ-a.e. )
· Almost Uniform Convergence -- Egorov Theorem
· Simple Functions
· Definition of ∫ f dμ and Properties
· Monotone Convergence, Fatou Lemma, Lebesgue Dominated Convergence Theorem
· Convergence in Lp(μ)
4. Signed Measures
· Hahn-Jordan Decomposition
· Radon-Nikodym Theorem/Applications
· Lebesgue Decomposition Theorem
5. Product Measures
· Construction of the Product Measure μxν
· Fubini Theorem
6. Four important linear PDEs as follows:
· Transport Equation
· Laplace’s equation, fundamental solution, strong maximum principle, Harnack’s inequality, Green’s function, energy methods, uniqueness, regularity
· Heat equation, fundamental solution, strong maximum principle, uniqueness, regularity, energy methods
· Wave equation, d’Alembert’s formula, energy methods, uniqueness, domain of dependence.
7. Other ways to represent PDE solutions as follows:
· Separation of variables
· Similarity solutions
· Transform methods
· Converting nonlinear PDEs into linear PDEs.
H.L. Royden, Real Analysis (or any other source addressing topics 1-5 with detailed proofs).
Lawrence C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.
Fritz John, Partial Differential Equations, 4th Edition, Springer-Verlag, New York 1991.