Assess students' ability to derive and analyze weak (variational) formulations of second-order linear elliptic boundary value problems (e.g. −div(A∇u)+cu=f with Dirichlet or mixed boundary conditions). Require: choose the appropriate Sobolev space (H0^1 or H^1), define the bilinear form and linear functional, prove boundedness and coercivity using uniform ellipticity and bounded coefficients, apply the Lax–Milgram theorem to conclude existence and uniqueness, and state the a priori energy estimate and relevant compatibility/regularity assumptions.