Questions: Sobolev Spaces for PDEs

Question 1

The Sobolev space H¹(Ω) = W^{1,2}(Ω) consists of L² functions whose:

• A. First-order weak derivatives are also in L²
• B. Classical derivatives exist and are in L²
• C. Fourier transform is in L²
• D. First-order weak derivatives are bounded

Answer: (A)

H¹(Ω) = {u ∈ L²(Ω) : ∂u/∂x_i ∈ L²(Ω) for all i}, where ∂u/∂x_i is the weak (distributional) derivative. The function need not be classically differentiable. The norm is ||u||² = ∫|u|² + ∫|∇u|².

Question 2

In dimension n = 3, the Sobolev embedding W^{1,2}(Ω) ⊂ L^p(Ω) holds for all p ≤ 6.

• T. True
• F. False

Answer: True

The Sobolev embedding theorem states that W^{1,p}(Ω) ⊂ L^{p*}(Ω) where p* = np/(n-p) when p < n. For n = 3 and p = 2, p* = 6. So H¹ functions in 3D are automatically in L^p for all p ≤ 6, but not necessarily for p > 6.

Question 3

What does the trace theorem for Sobolev spaces establish?

Model answer: Functions in W^{1,p}(Ω) have well-defined boundary values (traces) in L^p(∂Ω), even though they are defined only almost everywhere in Ω

Since L^p functions are only defined up to sets of measure zero, and ∂Ω has zero n-dimensional measure, boundary values are not directly meaningful. The trace theorem shows that the restriction map u → u|_∂Ω extends continuously from smooth functions to all of W^{1,p}, with the trace lying in the fractional Sobolev space W^{1-1/p, p}(∂Ω).

Question 4

H¹₀(Ω) is the closure of C_c^∞(Ω) in the H¹ norm. Functions in H¹₀(Ω) have what boundary behavior?

• A. Zero trace (vanish on ∂Ω in the trace sense)
• B. Bounded on ∂Ω
• C. Smooth on ∂Ω
• D. No condition on the boundary

Answer: (A)

H¹₀(Ω) = {u ∈ H¹(Ω) : trace(u) = 0 on ∂Ω}. This is the natural space for Dirichlet boundary conditions in the weak formulation—functions that 'vanish on the boundary' in the generalized sense.