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 Lebesque Sets

·        Construction of non-Measurable Set

·        Lebesgue-Stieltjes Measures on R

2.      Convergence

·        Measurable Functions

·        Convergence in μ (measure)

·        Convergence Almost Everywhere (μ-a.e. )

·        Almost Uniform Convergence -- Egorov Theorem

3.      Integration

·        Simple Functions

·        Definition of  ∫ f dμ and Properties

·        Monotone Convergence, Fatou Lemma, Lebesque Dominated Convergence Theorem

·        Convergence in L^p(μ)

4.      Signed Measures

·        Hahn-Jordan Decomposition

·        Radon-Nikodym Theorem/Applications

·        Lebesque 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.

REFERENCES

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, 4^th Edition, Springer-Verlag, New York 1991.