Math 295a Fall Term 2001

Assignment 5

1. Let Ω ⊂ ℝ^{n} be a bounded domain with smooth boundary and prove Poincaré’s Inequality

∥u∥_{L_{2}(Ω)} ≤ c∥∇u∥_{L_{2}(Ω)} , u ∈ H^{∘}^{1}(Ω) .

What does the constant c depend on? Is the boundedness assumption really necessary?

2. Let Ω ⊂ ℝ^{n} and p ∈ (1, ∞) and define

W_{p}^{m}(Ω) = {u ∈ D′(Ω) | ∂^{α}u ∈ L_{p}(Ω) , |α| ≤ m} .

Show that W_{p}^{m}(Ω) is a Banach space if endowed with the norm ∥·∥_{m,p} defined by

∥u∥_{m,p} = ( ∑_{|α|≤m} ∥∂^{α}u∥_{L_{p}(Ω)}^{p})^{1/p} , u ∈ W_{p}^{m}(Ω) .

Prove that W_{p}^{1}(0, 1) ↪ BUC^{1−1/p}([0, 1]). [Hint: Use the fact that C^{1}([0, 1]) is dense in W_{p}^{1}(0, 1)]

3. Prove that S(ℝ^{n}) is dense in H^{s}(ℝ^{n}) for s ∈ ℝ.

4. Let Ω = 𝔹(0, 1/2) and the function u be defined through

u(x, y) = log(log(\frac{2}{\sqrt{x^{2} + y^{2}}} )) , (x, y) ∈ Ω .

Then u is obviously not continuous in (x, y) = (0, 0). Prove that, however, u ∈ H^{1}(Ω) . Let now

u(x, y) = xy [log | log |(x, y)|| − log log 2] , (x, y) ∈ Ω.

Then

u ∈ C^{1}(Ω̄) and ∂_{j}^{2}u ∈ C(Ω̄) , j = 1, 2

but u ∉ C^{2}(Ω̄), that is, u is a solution of the Dirichlet problem in Ω for a continuous datum but is not twice continuously differentiable.

5. Let Ω ⊂ ℝ^{n} be a bounded domain with smooth boundary and prove that the norm ∥| · ∥|_{L_{2}(Ω)} = ∥△ · ∥_{L_{2}(Ω)} is an equivalent norm on

H^{∘}^{2}(Ω). Find the natural weak formulation of the bvp

△^{2}u = f ∈ L_{2}(Ω) , u = 0 , ∂_{ν}u = 0 on ∂Ω

and prove that it has a unique solution u ∈ H^{∘}^{2}(Ω).

The Homework is due by November 30 2001