Math295c Spring Term 2002

Assignment 2

A linear operator A : dom(A) ⊂ E ⟶ E is invertible on a normed vector space iff there is B ∈ L(E) with AB = id_{E} and BA = id_{dom(A)}. The resovent set ρ(A) and the spectrum σ(A) of A are then given by

ρ(A) := {λ ∈ ℂ | λ − A is invertible}

and

σ(A) = ℂ \ ρ(A) ,

respectively. The spectrum of an operator A is usually divided into point spectrum σ_{p}(A), continuous spectrum σ_{c}(A) and residual spectrum σ_{r}(A) where

σ_{p}(A) = {λ ∈ σ(A) | ker(λ − A) ≠ {0}}
σ_{c}(A) = {λ ∈ σ(A) | (λ − A)^{−1} is densely defined but not bounded}
σ_{r}(A) = {λ ∈ σ(A) | range(λ − A) is not dense in E}

Elements of σ_{p}(A) are called eigenvalues.

1. Produce simple examples of operators showing that any of the spectral sets σ_{p}(A), σ_{c}(A) and σ_{r}(A) can be non empty.

2. Let E be a normed vector space and A ∈ L(E) a compact operator. Show that σ_{p}(A) is countable and the only possible accumulation point is λ = 0. Also 0 ∈ σ(A) if dim(E) = ∞. It can be shown that any spectral value λ ≠ 0 is an eigenvalue. [Hint: Show that for any given ε > 0 there are only finitely many eigenvalues with |λ| ≥ ε.]

3. Let E be a normed vector space and A ∈ L(E) a compact operator. Show that

dim(ker[(λ − A)^{n}]) < ∞ for any λ ≠ 0 and n = 1, 2, …

[Use the fact that \overline{𝔹}_{E}(0, 1) is compact iff dim(E) < ∞ and that eigenvectors to different eigenvalues are linearly independent.]

4. (Pohožaev’s identity) Assume that g ∈ C(ℝ, ℝ), G(u) = ∫_{0}^{u} g(v) dv and that Ω ⊂ ℝ^{n} is bounded with smooth boundary. Let u be a classical solution of

\begin{cases}−△u = g(u) & in Ω \\ u = 0 & on ∂Ω \end{cases}

and show that it satisfies

n ∫_{Ω} G(u) dx + \frac{2 − n}{2} ∫_{Ω} u g(u) dx = \frac{1}{2} ∫_{∂Ω}(∇u · ν)^{2}(x · ν) dσ

[Hint: Use Gauss theorem with the vector field V (x) = (x · ∇u)∇u.]

5. Use Pohožaev’s identity to prove that no nontrivial solution can exist

for

\begin{cases}−△u = |u|^{p} & in Ω \\ u = 0 & on ∂Ω \end{cases}

if p > \frac{n+2}{n−2} and Ω is a star-shaped bounded Lipschitz domain in ℝ^{n}. The Homework is due by May 3 2002.