Assess graduate students' understanding of Sobolev spaces W^{k,p}(Ω): definition via weak derivatives, equivalent norms and completeness, formulation and applicability of Sobolev embedding theorems (continuous vs. compact embeddings, critical exponents, and typical counterexamples), and the theory of trace operators on Lipschitz domains (existence, uniqueness, continuity to boundary spaces). Items should mix concise proofs, conceptual explanations, and short problem-solving (verifying embeddings for specific domains, constructing counterexamples, and computing or characterizing traces).