Sumários
Lesson 21: Elliptic Problems, part II
14 Maio 2019, 10:30 • Hugo Tavares
Lax-Milgram's theorem and its application to the existence and uniqueness of weak solutions for a general second order elliptic operator in divergence form (the case of homogeneous Dirichlet boundary conditions).
Lesson 20: Elliptic Problems, part II
10 Maio 2019, 12:00 • Hugo Tavares
Conclusion of the proof of the Dirichlet's principle.
The notion of weak solution for L=-Delta u+u: non homogenous Dirichlet boundary conditions; Neumann boundary conditions.Motivations for the need of generalizing Riesz's theorem.
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.