Assess the student's ability to formulate variational (weak) formulations for second-order linear elliptic boundary value problems (e.g. −div(A∇u)+cu=f with homogeneous Dirichlet data on a bounded domain), verify boundedness and coercivity of the associated bilinear form on H1_0(Ω), apply the Lax–Milgram theorem to conclude existence and uniqueness of the weak solution, and derive the standard a priori energy estimate; include one concrete PDE for which the student must perform the verification and conclude existence/uniqueness.