What are the two hypotheses on the bilinear form a(u,v) in the Lax-Milgram theorem?
AContinuity (|a(u,v)| ≤ M||u||||v||) and coercivity (a(u,u) ≥ α||u||²)
BSymmetry (a(u,v) = a(v,u)) and positivity (a(u,u) > 0)
CLinearity and boundedness
DCompactness and injectivity
Continuity ensures the bilinear form defines a bounded operator, and coercivity ensures this operator is invertible with a bounded inverse. Together they guarantee unique solvability. Symmetry is NOT required—this is the key advantage over the Riesz representation theorem.
Question 2 True / False
The Lax-Milgram theorem requires the bilinear form to be symmetric.
TTrue
FFalse
Answer: False
Lax-Milgram works for non-symmetric bilinear forms, which arise in convection-diffusion equations and other PDEs with first-order terms. When the form IS symmetric, the solution also minimizes the associated energy functional, giving additional structure.
Question 3 Short Answer
How does coercivity of a(u,v) = ∫∇u·∇v dx on H¹₀(Ω) follow?
Think about your answer, then reveal below.
Model answer: From the Poincaré inequality: a(u,u) = ∫|∇u|²dx ≥ C||u||²_{H¹₀} because ||u||_{L²} ≤ C_P||∇u||_{L²} on bounded domains
The Poincaré inequality on bounded domains states that the L² norm of u is controlled by the L² norm of ∇u for functions vanishing on the boundary. Therefore ||∇u||²_{L²} controls the full H¹ norm ||u||²_{H¹₀} = ||∇u||²_{L²}, establishing coercivity.
Question 4 True / False
The Lax-Milgram theorem also provides a stability estimate for the solution.
TTrue
FFalse
Answer: True
The proof yields ||u|| ≤ (1/α)||F||, where α is the coercivity constant. This says the solution is bounded in terms of the data, and small changes in F produce proportionally small changes in u—continuous dependence on data.