Sumários
Lesson 19: Elliptic Problems, part I
4 Maio 2018, 11:00 • Hugo Tavares
Variational Formulation of Elliptic Problems:
-definition of second order elliptic operator in divergence and non-divergence form.
- brief commants about several definitions of solution: classical, strong, distributional solutions.
The notion of weak solution in the particular case Lu=-Delta u + u:
- homogeneous Dirichlet boundary conditions; the Dirichlet's principle.
Exercise class 10
3 Maio 2018, 11:00 • Hugo Tavares
Sobolev Spaces. Solution of exercises 1d)e), 2 of Exercise Sheet 7. Hints about exercise 3.
Lesson 18: Sobolev Spaces, part V
2 Maio 2018, 13:00 • Hugo Tavares
The extension theorem: ideas about the proof in the general case using a partition of the unity. Consequences: density results and Sobolev embeddings for W^{1,p}(Omega) with Omega a C^1-domain.
The trace operator. Green's identity in H^1.
Lesson 17: Sobolev Spaces, part IV
27 Abril 2018, 11:00 • Hugo Tavares
Poincaré's inequality: proof and consequences (equivalent norm in H^1_0(Omega)).
Characterization of H^{-1}(Omega), the dual space of H^1_0(Omega).
Statement of the extension theorem for W^{1,p}(Omega), with Omega a C^1 domain. Proof when Omega is a cube.
Exercise class 9
26 Abril 2018, 11:00 • Hugo Tavares
Fourier Transform and applications to PDE. Solution of the exercises 2, 3, 4 of Exercise Sheet 6. Solution of exercise 1a)b)c) of Exercise Sheet 7.