To derive the weak formulation of -Δu = f, we multiply by a test function v and apply:
AIntegration by parts (Green's first identity)
BThe Fourier transform
CThe method of characteristics
DTaylor expansion
Multiplying -Δu = f by v ∈ H¹₀ and integrating by parts gives ∫∇u·∇v dx = ∫fv dx. The boundary term vanishes because v = 0 on ∂Ω. This moves one derivative from u to v, requiring only u ∈ H¹ rather than u ∈ H² (which the classical formulation Δu would need).
Question 2 True / False
A weak solution of an elliptic PDE is always a classical (C²) solution.
TTrue
FFalse
Answer: False
A weak solution is an element of a Sobolev space and need not be twice differentiable. However, elliptic regularity theory shows that if the data (domain, coefficients, right-hand side) are sufficiently smooth, then the weak solution is in fact classical. This is a nontrivial theorem, not automatic.
Question 3 Short Answer
What advantage does the weak formulation have over the classical formulation?
Think about your answer, then reveal below.
Model answer: It requires fewer derivatives on the solution, allowing existence proofs in Sobolev spaces using functional analysis (Lax-Milgram, compactness, variational methods)
The classical formulation of -Δu = f requires u to be twice differentiable, which is hard to establish directly. The weak formulation only needs u ∈ H¹, and existence in H¹ can be proved by abstract functional analysis. Regularity theory then bootstraps: if f and the domain are smooth, the weak solution is also smooth.
Question 4 Multiple Choice
In the weak formulation, the bilinear form a(u,v) = ∫∇u·∇v dx is coercive on H¹₀(Ω), meaning:
Aa(u,u) ≥ α||u||²_{H¹₀} for some α > 0
Ba(u,v) = a(v,u) for all u,v
Ca(u,v) ≤ M||u|| ||v|| for some M
D|a(u,v)| = 0 implies u = 0
Coercivity (also called ellipticity) means the bilinear form controls the norm: a(u,u) = ∫|∇u|²dx ≥ α||u||²_{H¹₀} by the Poincaré inequality. Coercivity is the key hypothesis of the Lax-Milgram theorem that guarantees existence and uniqueness of the weak solution.