Assess students' ability to apply Sobolev embedding theorems to establish regularity of weak solutions of second-order elliptic PDEs on bounded domains. Tasks include verifying solution membership in Sobolev spaces (H^1, W^{2,p}), using embeddings to deduce continuity or Hölder regularity (C^{0,α}), deriving W^{2,p} or C^{1,α} conclusions from source-term integrability (e.g. f∈L^p) and spatial dimension, constructing bootstrapping arguments with elliptic estimates, and identifying limitations or counterexamples.