Assess students' ability to formulate a linear second-order elliptic boundary value problem as a variational problem in the appropriate Sobolev space, verify bilinear form continuity and coercivity (using uniform ellipticity and Poincaré inequality as needed), check continuity of the linear functional, apply the Lax–Milgram theorem to prove existence and uniqueness of the weak solution, and derive the standard a priori energy estimate; include discussion of Dirichlet and Neumann boundary conditions and precise hypotheses on coefficients and domain regularity.