Download Assignment 2 with 11 Problems - Geometric Structure | MATH 549 and more Assignments Mathematics in PDF only on Docsity!
Math 549 Geometric Structures Spring 2008 Assignment # Due Wednesday, June 11, 2008
The point value of each problem is shown in parentheses. For full credit, do at least 80 points worth of problems.
- (10) Let k be a nonnegative integer and Ω ⊆ Rn^ be an open set. Show that Hk(Rn) ∩ E ′(Ω) ⊆ H^ ˚k(Ω).
- (10) Let u be the distribution on Rn^ defined by the following locally integrable function:
u(x^1 ,... , xn) =
1 , xn^ > 0 , 0 , xn^ ≤ 0.
Determine the distributional derivatives ∂iu, ∂i∂j u, i, j = 1,... , n.
- (10) Suppose n > 2, and let u be the distribution on Rn^ defined by the locally integrable function u(x) =
|x|n−^2
Show that the following equation holds in the distribution sense: ∑^ n
j=
∂j ∂j u = cnδ 0 ,
where δ 0 is the distribution defined by (δ 0 , ϕ) = ϕ(0), and cn is a constant. Determine cn.
- (25) For any nonnegative integer k, show that C c∞ (Rn) is dense in the Sobolev space Hk(Rn), as follows.
(a) Show that compactly supported elements of Hk(Rn) are dense. (b) Choose ρ ∈ C c∞ (Rn) such that
Rn^ ρ(x)^ dx^ = 1, and set^ ρε(x) =^ ε
−nρ(x/ε). Show that ∫ Rn^ ρε(x)^ dx^ = 1, and that^ ρε^ →^ δ^ as distributions (this means that (ρε, ϕ)^ →^ (δ, ϕ) for each ϕ ∈ C c∞ (Rn)). (c) If u ∈ Hk(Rn) is compactly supported, define uε = u ∗ ρε. Show that uε ∈ C c∞ (Rn). (d) If u ∈ C^0 c (Rn), show that uε → u uniformly. (e) If u ∈ L^2 (Rn), show that uε ∈ L^2 (Rn), and
‖uε‖L 2 ≤ ‖u‖L 2.
[Hint: Write |u(x − y)ρε(y)| = (|u(x − y)||ρε(y)|^1 /^2 )(|ρε(y)|^1 /^2 ) and use the Cauchy-Schwartz inequality.] (f) Using the fact that C^0 c (Rn) is dense in L^2 (by standard measure theory), and interchang- ing limits appropriately, show that uε → u in L^2 norm.
(g) Show that, if u ∈ Hk(Rn) and m ≤ k, ∂muε ∂xi^1... ∂xim^
∂mu ∂xi^1... ∂xim^ ∗ ρε.
Use this to show that ∂muε ∂xi^1... ∂xim^
∂mu ∂xi^1... ∂xim in L^2 norm. (h) Prove the result.
- (10) Give a counterexample to the result of Problem 4 when Rn^ is replaced by the unit ball in Rn.
- (10) Let E be any smooth vector bundle over a compact smooth manifold M. For any nonnegative integer k, show that Γ(E) is dense in Hk(M, E).
- (10) For any nonnegative integer k and any open set Ω ⊆ Rn, show that compactly supported elements of Hk(Ω) are in ˚Hk(Ω).
- (15)
(a) For any real number s, show that a compactly supported distribution u ∈ E ′(Rn) is in Hs(Rn) if and only if it satisfies an estimate of the form
(u, ϕ) ≤ C‖ϕ‖H−s , ϕ ∈ C c∞ (Rn).
(b) Show that every compactly supported distribution on Rn^ is in Hs for some s.
- (10) Compute the formal adjoint of each of the following differential operators. (In each case, the metric involved is the standard Euclidean one.)
(a) ∇ : Γ(T ∗Rn) → Γ(T 2 Rn). (b) P : C∞(S^1 ) → C∞(S^1 ), given by P u(θ) = u′(θ) cos θ.
- (20) This problem outlines an alternate approach to computing the symbol of the Laplace- Beltrami operator on an oriented manifold. Suppose V is an n-dimensional vector space endowed with an inner product and an orientation. Let (ei) be any oriented orthonormal basis for V and (ei) the dual basis. Let μ denote the volume element determined by the inner product and the orientation (thus μ = e^1 ∧ · · · ∧ en). In terms of this basis, define a linear operator ∗ : ΛkV → Λn−k^ V , the Hodge star operator, by setting
∗(ei^1 ∧ · · · ∧ eik^ ) = sgn(σ)ej^1 ∧ · · · ∧ ejn−k^ ,
where (j 1 ,... , jn−k) is a the unique increasing multi-index of length n − k such that
{i 1 ,... , ik, j 1 ,... , jn−k} = { 1 ,... , n},
and σ is the permutation sending (i 1 ,... , ik, j 1 ,... , jn−k) to (1,... , n); then extend ∗ to all of ΛkV by linearity.
(a) Show that, for any k-form ω, ∗ω is the unique (n − k)-form such that η ∧ ∗ω = 〈η, ω〉 μ for all k-forms η. This proves that ∗ is well-defined independently of choice of basis.
