| 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