Weak Formulation of Linear Elliptic PDEs

This lecture covers the weak formulation of linear elliptic partial differential equations, starting with an introduction to dual spaces and density arguments. It then explores corollaries and classical selections, discussing the possibility of allowing rougher functions. The lecture delves into the concept of weak solutions of PDEs, focusing on the Rodee problem and classical selections. It concludes with the abstract variational formulation and the Lax-Milgram theorem, emphasizing the existence and uniqueness of solutions. The presentation includes applications to the Poisson equation, highlighting the importance of boundedness and coercivity in the context of linear and bounded forms.