SECOND ASSIGNMENT, DUE SEPTEMBER 18 IN CLASS18.155 FALL 2001RICHARD MELROSEIn the main these questions form theorems in H¨ormander’s book [1], so the proofsare available there. I suggest that you try to work them out on your own and inany case I expect written proofs, even if you need to get the idea from the book.Of course, at the very least, you will have to translate the notation.Problem 1. [H¨ormander, Theorem 3.1.4] Let I ⊂ R be an open, non-empty interval.i) Show (you may use results from class) that there exists ψ ∈ C∞c(I) withRRψ(x)ds = 1.ii) Show that any φ ∈ C∞c(I) may be written in the formφ =˜φ + cψ, c ∈ C,˜φ ∈ C∞c(I) withZR˜φ = 0.iii) Show that if˜φ ∈ C∞c(I) andRR˜φ = 0 then there exists µ ∈ C∞c(I) such thatdµdx=˜φ in I.iv) Suppose u ∈ C−∞(I) satisfiesdudx= 0, i.e.u(−dφdx) = 0 ∀ φ ∈ C∞c(I),show that u = c for some constant c.v) Suppose that u ∈ C−∞(I) satisfiesdudx= c, for some constant c, show thatu = cx + d for some d ∈ C.Problem 2. [H¨ormander Theorem 3.1.16]i) Use Taylor’s formula to show that there is a fixed ψ ∈ C∞c(Rn) such thatany φ ∈ C∞c(Rn) can be written in the formφ = cψ +nXj=1xjψjwhere c ∈ C and the ψj∈ C∞c(Rn) depend on φ.ii) Recall that δ0is the distribution defined byδ0(φ) = φ(0) ∀ φ ∈ C∞c(Rn);explain why δ0∈ C−∞(Rn).iii) Show that if u ∈ C−∞(Rn) and u(xjφ) = 0 for all φ ∈ C∞c(Rn) and j =1, . . . , n then u = cδ0for some c ∈ C.iv) Define the ‘Heaviside function’H(φ) =Z∞0φ(x)dx ∀ φ ∈ C∞c(R);show that H ∈ C−∞(R).v) ComputeddxH ∈ C−∞(R).12 RICHARD MELROSEProblem 3. Using Problems 1 and 2, find all u ∈ C−∞(R) satisfying the differentialequationxdudx= 0 in R.References[1] L. H¨ormander, The analysis of linear partial differential operators, vol. 1, Springer-Verlag,Berlin, Heidelberg, New York, Tokyo, 1983.Department of Mathematics, Massachusetts Institute of TechnologyE-mail address:
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