Assess students' ability to formulate the weak form of a second-order linear elliptic boundary value problem on a bounded domain, select the appropriate Hilbert space (e.g. H0^1(Ω)), verify boundedness and coercivity of the associated bilinear form using uniform ellipticity and the Poincaré inequality, check boundedness of the linear functional, apply the Lax–Milgram theorem to prove existence and uniqueness of the weak solution, and derive basic a priori (energy) estimates showing dependence on coefficients and boundary data.