Some of the topics I will cover:
1. Cauchy-Kowaleskaya Theorem.
2. Holmgren uniqueness Theorem.
3. Sobolev spaces. Basic properties, Hardy-Littlewood-Sobolev embedding theorem.
4. Sobolev extension theorem from a half-space.
5. Rellich-Kondrachov compactness theorem.
6. Traces of Sobolev spaces.
7. Maximum principles and Hopf boundary point lemma for second order elliptic equations.
8. The Laplacian, Green function, and Poisson kernel.
9. Schauder theory.
10. Wave equation, finite propagation speed, domain of influence.
11. Duhamel’s theorem.
12. Huyghen’s principle and solutions of wave equations and Schrodinger equation.
13. Some dispersive estimates for wave and Schrodinger equations.
I shall post notes for the lectures on the Canvas page for the course. No textbook exists that covers everything I am proposing to do. A textbook that you might wish to consult is Partial Differential Equations by Lawrence Craig Evans, Graduate Texts in Math. Vol. 19, American Mathematical Society.
501 or Instructor's permission