EIGHT ASSIGNMENT, DUE NOVEMBER 6 IN CLASS18.155 FALL 2001RICHARD MELROSEProblem 1. Wavefront set computations and more – all pretty easy, especially ifyou use results from class.i) Compute WF(δ) where δ ∈ S0(Rn) is the Dirac delta function at the origin.ii) Compute WF(H(x)) where H(x) ∈ S0(R) is the Heaviside functionH(x) =(1 x > 00 x ≤ 0.Hint: Dxis elliptic in one dimension, hit H with it.iii) Compute WF(E), E = iH(x1)δ(x0) which is the Heaviside in the firstvariable on Rn, n > 1, and delta in the others.iv) Show that Dx1E = δ, so E is a fundamental solution of Dx1.v) If f ∈ C−∞c(Rn) show that u = E ? f solves Dx1u = f.vi) What does our estimate on WF(E ? f ) tell us about WF(u) in terms ofWF(f)?Problem 2. The wave equation in two variables (or one spatial variable).i) Recall that the Riemann functionE(t, x) =(−14if t > x and t > −x0 otherwiseis a fundamental solution of D2t− D2x(check my constant).ii) Find the singular support of E.iii) Write the Fourier transform (dual) variables as τ, ξ and show thatWF(E) ⊂ {0} × S1∪ {(t, x, τ, ξ); x = t > 0 and ξ + τ = 0}∪ {(t, x, τ, ξ); −x = t > 0 and ξ = τ} .iv) Show that if f ∈ C−∞c(R2) then u = E ? f satisfies (D2t− D2x)u = f.v) With u defined as in iv) show thatsupp(u) ⊂ {(t, x); ∃ (t0, x0) ∈ supp(f ) with t0+ x0≤ t + x and t0− x0≤ t − x}.vi) Sketch an illustrative example of v).vii) Show that, still with u given by iv),sing supp(u) ⊂ {(t, x); ∃ (t0, x0) ∈ sing supp(f) witht ≥ t0and t + x = t0+ x0or t − x = t0− x0}.viii) Bound WF(u) in terms of WF(f).12 RICHARD MELROSEProblem 3. A little uniqueness theorems. Supp os e u ∈ C−∞c(Rn) recall that theFourier transform ˆu ∈ C∞(Rn). Now, suppose u ∈ C−∞c(Rn) satisfies P (D)u = 0for some non-trivial polynomial P, show that u = 0.Department of Mathematics, Massachusetts Institute of TechnologyE-mail address:
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