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.