2.2. LAPLACE’S EQUATION 39 Combining this calculation with estimate (36), we deduce |u(x)−g(x0)| ≤ 2ε, provided |x − x0| is suﬃciently small. c. Green’s function for a ball. To construct Green’s function for the unit ball B(0, 1), we will again employ a kind of reflection, this time through the sphere ∂B(0, 1). DEFINITION. If x ∈ Rn − {0}, the point ˜ x = x |x|2 is called the point dual to x with respect to ∂B(0, 1). The mapping x → ˜ x is inversion through the unit sphere ∂B(0, 1). We now employ inversion through the sphere to compute Green’s func- tion for the unit ball U = B0(0, 1). Fix x ∈ B0(0, 1). Remember that we must find a corrector function φx = φx(y) solving (37) Δφx = 0 in B0(0, 1) φx = Φ(y − x) on ∂B(0, 1) then Green’s function will be (38) G(x, y) = Φ(y − x) − φx(y). The idea now is to “invert the singularity” from x ∈ B0(0, 1) to ˜ x / ∈ B(0, 1). Assume for the moment n ≥ 3. Now the mapping y → Φ(y − ˜) x is harmonic for y = ˜. x Thus y → |x|2−nΦ(y − ˜) x is harmonic for y = ˜, x and so (39) φx(y) := Φ(|x|(y − ˜)) x is harmonic in U. Furthermore, if y ∈ ∂B(0, 1) and x = 0, |x|2|y − ˜|2 x = |x|2 |y|2 − 2y · x |x|2 + 1 |x|2 = |x|2 − 2y · x + 1 = |x − y|2. Thus (|x||y − ˜|)−(n−2) x = |x − y|−(n−2). Consequently (40) φx(y) = Φ(y − x) (y ∈ ∂B(0, 1)), as required.

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