Math 819–Partial Differential Equations

Text book

Course outline

Conservation Laws
1. Transport equations: the pde $u_t+au_x=0$. Evans: §2.1.1 (page 18). Method of characteristics for quasilinear first order scalar pde $u_t+a(x, u)u_x = b(x, u)$. §3.2.2 (page 99—). Evans treats the fully nonlinear case. Look here for the computation about characteristics in the much simpler quasilinear case.
2. Conservation laws, and Burger's equation in particular: solution by characteristics, non-existence of global solutions.
3. Integral solutions for conservation laws: Evans §3.4.1 (page 136—137)
4. Rankine–Hugoniot condition. Non uniqueness of integral solutions (still Evans §3.4.1)
   Homework #1
5. Entropy condition.
6. Existence of entropy solutions for Burger’s equation using finite differences. We give a different proof from Evans’s (bacause we skipped the section on HJ equations). Here are class notes for the alternative proof.
7. Uniqueness of the entropy solution (time permitting).
Laplace and Poisson's equations
8. Fundamental solutions; Newton potential and Poisson kernel.
9. Mean Value property of harmonic functions.
10. Maximum principle; harmonic functions are $C^\infty$ smooth; two proofs of the interior gradient estimate for harmonic functions–one using the Mean Value Property, and one using Bernstein’s method.
    Homework #2
11. Dirichlet’s principle.
Sobolev spaces and Second order elliptic equations
12. Lecture notes for math 725 from spring 2000. These contain a quick introduction to distributions, Functional analysis, and Sobolev spaces.
13. Definition of Sobolev spaces $W^{1,p}(U)$, $W_0^{1,p}(U)$, $W^{-1,p}(U)$. Density of smooth functions. Boundary values of $f\in W^{1,p}(U)$.
14. Sobolev and Poincaré inequalities.
15. Weak formulation of Poisson’s equation. General divergence form elliptic equations.
16. Existence and uniqueness theorem; Lax–Milgram lemma.
17. Regularity of weak solutions.
18. Compactness: eigenvalues and eigenfunctions.
19. Regularity of the “solution” to $-\Delta u = f$.
Heat equation on $\R^n$
20. Fundamental solution, existence for the initial value problem on $\R^n$ .
21. Uniqueness and non–uniqueness: Tychonov’s example. Widder’s theorem.
22. A paradox involving the heat equation and the Fourier transform.
23. The maximum principle for parabolic equations.
24. Final homework assignment.

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