Assess graduate students' ability to derive weak formulations for second‑order linear elliptic boundary value problems on bounded domains (Dirichlet and Neumann), verify boundedness and coercivity of the associated bilinear form to apply the Lax–Milgram theorem for existence and uniqueness of weak solutions, and then apply Sobolev embedding and elliptic regularity results to deduce higher H^s and C^{k,α} regularity (with norm estimates) under standard hypotheses on coefficients, domain regularity, and data compatibility.