Assess students' ability to derive weak formulations for second-order linear elliptic boundary value problems (Dirichlet or Neumann), choose the correct Hilbert space (e.g. H1_0 or H1), verify bilinear form continuity and coercivity from ellipticity and coefficient bounds, identify bounded linear functionals, and apply the Lax-Milgram theorem to prove existence and uniqueness of weak solutions; include at least one worked example where these assumptions are checked and the energy estimate is derived.