Partial Differential Equations

Basic theory

• First order PDE

  • Characteristic method, Hamilton-Jacobi equation 

• Laplace equation

• Heat Equation

Advanced analysis

• Maximum principle

• gradient estimate

• Sobolev space

• Schauder estimate

• Weak derivative and Sobolev embedding

實際進度

1. 0929 Characteristic method

2. 1006 Bergers' equation

3. 1013 Weak solutions, shock curves, Rankine-Hugoniot condition

4. 1020 Riemann problems

5. 1027 Hamilton-Jacobi equation, entropy conditions

6. 1103 [The first Exam}

7. 1110 Hopf-Lax formula. Fundamental solution of Laplace equation

8. 1117 MVP, strong maximum principle,  and smoothness of harmonic functions 

9. 1124 skip

10. 1201 Derivative estimate, Liouville Theorem, Harnack inequality 

11. 1208 Poisson equation on R^n,  Representation formula (I)

12. 1215  Representation formula (II). Green's function for half-plane. 

13. 1222 Discussion on Exam 1. Energy method. Norms and inequalities(GNS, Morrey, Sobolev).

14. 1229 [The second Exam] 

15. 0105 Schauder estimate

16. 0112  Weak derivative and Sobolev embedding

17. 0119 (補1124)