22 2. FOUR IMPORTANT LINEAR PDE for i = 1,...,n, and so Δu = v (r) + n − 1 r v (r). Hence Δu = 0 if and only if (5) v + n − 1 r v = 0. If v = 0, we deduce log(|v |) = v v = 1 − n r , and hence v (r) = a rn−1 for some constant a. Consequently if r 0, we have v(r) = b log r + c (n = 2) b rn−2 + c (n ≥ 3), where b and c are constants. These considerations motivate the following DEFINITION. The function (6) Φ(x) := − 1 2π log |x| (n = 2) 1 n(n−2)α(n) 1 |x|n−2 (n ≥ 3), defined for x ∈ Rn, x = 0, is the fundamental solution of Laplace’s equation. The reason for the particular choices of the constants in (6) will be apparent in a moment. (Recall from §A.2 that α(n) denotes the volume of the unit ball in Rn.) We will sometimes slightly abuse notation and write Φ(x) = Φ(|x|) to emphasize that the fundamental solution is radial. Observe also that we have the estimates (7) |DΦ(x)| ≤ C |x|n−1 , |D2Φ(x)| ≤ C |x|n (x = 0) for some constant C 0. b. Poisson’s equation. By construction the function x → Φ(x) is har- monic for x = 0. If we shift the origin to a new point y, the PDE (1) is unchanged and so x → Φ(x − y) is also harmonic as a function of x, x = y. Let us now take f : Rn → R and note that the mapping x → Φ(x − y)f(y) (x = y) is harmonic for each point y ∈ Rn, and thus so is the sum of finitely many such expressions built for different points y.
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