Assess students' ability to formulate weak problems for second‑order elliptic boundary-value problems in appropriate Sobolev spaces (e.g. H^1_0), verify boundedness and coercivity of the associated bilinear form (using Poincaré and Cauchy–Schwarz), apply the Lax–Milgram theorem to obtain existence and uniqueness of weak solutions, and derive basic energy estimates and regularity conclusions (local H^2 or global H^2 under smoothness assumptions).