Sumários
Lesson 19: Elliptic Problems, part I
7 Maio 2019, 10:30 • 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.
Lesson 18: Sobolev Spaces, part V
3 Maio 2019, 12:00 • 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
30 Abril 2019, 10:30 • 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.