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THIRD (DELAYED) PROBLEM SET FOR 18.155 DUE OCTOBER 17 IN CLASS OR 2-174. These are Problems 24 - 30 of the notes on the web. Problem 1. Prove the continuity in the mean of L2 functions on Rn , that Z sup |u(x + t) − u(x)|2 dx → 0 as → 0. |t|< Rn You will probably have to go back to first principles to do this. Show that it is enough to assume u ≥ 0 has compact support. Then show it is enough to assume that u is a simple, and integrable, function. Finally look at the definition of Lebesgue measure and show that if E ⊂ Rn is Borel and has finite Lebesgue measure then lim µ(E\(E + t)) = 0 |t|→∞ where µ = Lebesgue measure and E + t = {p ∈ Rn ; p0 + t , p0 ∈ E} . Problem 2. Prove Leibniz’ formula Dxα (ϕψ) = X α β≤α β Dxβ ϕ · Dxα−β ψ ∞ for any C functions and ϕ and ψ. Here α and β are multiindices, β ≤ α means βj ≤ αj for each j and Y α αj = . β βj j I suggest induction! Problem 3. Prove the generalization of a result from class that u ∈ S 0 (Rn ), supp(u) ⊂ {0} (which means that u(φ) = 0 for all φ ∈ S(Rn ) such that φ = 0 in |x| < for some > 0) implies there are constants cα for |α| ≤ m, for some m, such that X u= cα Dα δ. |α|≤m Hint This is not so easy! I would be happy if you can show that u ∈ M (Rn ), supp u ⊂ {0} implies u = cδ. To see this, you can show that ϕ ∈ S(Rn ), ϕ(0) = 0 ⇒ ∃ ϕj ∈ S(Rn ) , ϕj (x) = 0 in |x| ≤ j where j ↓ 0 and sup |ϕj − ϕ| → 0 as j → ∞ . To prove the general case you need something similar — that given m, if ϕ ∈ S(Rn ) and Dxα ϕ(0) = 0 for |α| ≤ m then ∃ ϕj ∈ S(Rn ), ϕj = 0 in |x| ≤ j , j ↓ 0 such that ϕj → ϕ in the C m norm. 1 2 THIRD (DELAYED) PROBLEM SET FOR 18.155 DUE OCTOBER 17 IN CLASS OR 2-174. 0 Problem 4. If m ∈ N, m0 > 0 show that u ∈ H m (Rn ) and Dα u ∈ H m (Rn ) for all 0 |α| ≤ m implies u ∈ H m+m (Rn ). Is the converse true? Problem 5. Show that every element u ∈ L2 (Rn ) can be written as a sum n X u = u0 + Dj uj , uj ∈ H 1 (Rn ) , j = 0, . . . , n . j=1 Problem 6. Consider for n = 1, the locally integrable function (the Heaviside function), 0 x≤0 H(x) = 1 x > 0. Show that Dx H(x) = cδ; what is the constant c? Problem 7. For what range of orders m is it true that δ ∈ H m (Rn ) , δ(ϕ) = ϕ(0)?