| Dato | Undervises av | Sted | Tema | Kommentarer / ressurser |
| 22.08.2011 | SHC | B91 | Introduction | Prerequisites. Goals. Notations. Examples of PDEs. Laplace equation. Fundamental solution. §2.1 and 2.2.1.a. |
| 23.08.2011 | SHC | B62 | Poisson equation | Integral formula for Poisson equation on R^n and mean value property. §2.2.1.b and 2.2.2. |
| 29.08.2011 | Lebesgue integral. | Facts about Lebesgue integral. Appendix E. | ||
| 30.08.2011 | Regularity and maximum principle. | Smoothing by convolution. Appendix C.4. Regularity. Maximum principle, uniqueness. §2.2.3.a-b. | ||
| 05.08.2011 | Estimates on derivatives. Exercises. | Exercises in Notes_1. §2.2.3.c. | ||
| 06.09.2011 | Analyticity. Exercises. | Exercises in Notes_2. §2.2.3.d-e. | ||
| 12.09.2011 | Green's functions, Dirichlet's principle. | Motivation, definition and examples of Green's functions. Dirichlet's principle. §2.2.4 and §2.2.5. | ||
| 13.09.2011 | Weak derivatives | Weierstrass' critique of Dirichlet's principle. §5.2.1. | ||
| 19.09.2011 | Sobolev spaces | Weak derivatives, definition of Sobolev spaces, elementary properties. §5.2. | ||
| 20.09.2011 | Density of smooth functions | Density of smooth functions in Sobolev spaces. §5.3, §C.1 | ||
| 26.09.2011 | Density of smooth functions | Density of smooth functions continued. Absolutely continuous functions on R. | ||
| 27.09.2011 | Traces | Traces on the boundary. §5.5. | ||
| 03.10.2011 | Integration on boundaries | Background material: C^1 boundaries, partitions of unity, integration on graphs. | ||
| 04.10.2011 | Stokes theorem | Background material: Stokes theorem on C^1 domains, with proof. | ||
| 10.10.2011 | Hilbert spaces | Projection on closed convex sets. Riesz representation theorem. §D.1-3. | ||
| 14.10.2011 | B62 | NB: fredag 12-14 | Poincaré inequality (for Dirichlet boundary condition). Lax-Milgram. Second order elliptic equations on domains. §6.1 and 6.2.1. | |
| 17.10.2011 | Exercises | On projections in Hilbert spaces. For next time, Ex §5: 4,8,9,14. | ||
| 18.10.2011 | Extensions and traces | Extension by zero, Sobolev preserving extension. §5.4, 5.5. | ||
| 24.10.2011 | Compact operators | Definition (§D.5). The operator norm limit of compact operators is compact. Compactness of the injection of W^{1,p} into L^p. §5.7. | ||
| 25.10.2011 | Neumann boundary condition | Poincaré inequality §5.8.1. Laplace equation with Neumann boundary condition. | ||
| 30.11.2011 | Help with the compulsory exercises (oblig). | |||
| 01.11.2011 | Work on the compulsory exercise! No teaching. | |||
| 07.11.2011 | Exercises, Poincaré inequalities. | Exercises. A general theorem yielding various Poincaré inequalities. | ||
| 08.11.2011 | Fredholm alternative. | Energy estimates, Fredholm alternative. §6.2. | ||
| 14.11.2011 | Sobolev inequality | §5.6.1 | ||
| 15.11.2011 | Sobolev inequality | continued. Remarks on the compulsory exercise. | ||
| 21.11.2011 | Elliptic regularity | §5.8.2 Difference quotients. §6.3 Regularity. | ||
| 22.11.2011 | Exercises | |||
| 28.11.2011 | Year summary | |||
| 29.11.2011 | Last lecture |
Undervisningsplan
Published Aug. 19, 2011 7:33 PM
- Last modified Nov. 25, 2011 4:21 PM