Assess the learner's ability to formulate second-order linear elliptic boundary value problems in a variational (weak) form on suitable Hilbert spaces (e.g. H0^1), verify boundedness and coercivity of the associated bilinear form, and apply the Lax–Milgram theorem to prove existence, uniqueness, and continuous dependence of weak solutions (including deriving energy estimates). Scope includes typical Dirichlet problems with uniformly elliptic, bounded coefficients and source terms in H−1 or L2; learners should check hypotheses, cite functional-analytic assumptions, and explain how each hypothesis leads to the theorem's conclusion.