If u ∈ H¹₀(Ω) is a weak solution of -Δu = f with f ∈ L²(Ω), what regularity does u have?
Au ∈ H²(Ω)
Bu ∈ H¹(Ω) only
Cu ∈ C^∞(Ω)
Du ∈ L^∞(Ω)
The basic elliptic regularity theorem states that if f ∈ L²(Ω) and ∂Ω is smooth (or Ω = ℝⁿ), then u ∈ H². This is a gain of two Sobolev derivatives beyond f, matching the order of the Laplacian. If f ∈ H^k, then u ∈ H^{k+2} by induction.
Question 2 True / False
Elliptic regularity is a local property: u is smooth wherever f is smooth, regardless of boundary regularity.
TTrue
FFalse
Answer: True
Interior regularity holds independently of boundary conditions or boundary smoothness. If f ∈ C^∞(ω) for an open set ω ⊂⊂ Ω, then u ∈ C^∞(ω). Boundary regularity requires additional assumptions on the smoothness of ∂Ω and the boundary data.
Question 3 Short Answer
What is the Weyl lemma?
Think about your answer, then reveal below.
Model answer: Every distribution satisfying Δu = 0 is a smooth (C^∞) function, and in fact real-analytic
The Weyl lemma is the simplest case of elliptic regularity: if u is merely a distribution solving Laplace's equation, it must be an infinitely differentiable (and even real-analytic) function. This shows that the elliptic operator Δ cannot have genuinely distributional solutions.
Question 4 True / False
For a domain with a corner (non-smooth boundary), the H² regularity theorem for -Δu = f may fail.
TTrue
FFalse
Answer: True
Corners and edges in the boundary cause singularities in the solution that limit its Sobolev regularity. Near a reentrant corner of angle > π, the solution may be only in H^s for s < 1 + π/angle, even if f is smooth. Understanding these corner singularities is crucial for numerical methods on non-smooth domains.