2.2. LAPLACE’S EQUATION 33 Proof. Let r := 1 4 dist(V, ∂U). Choose x, y V , |x y| r. Then u(x) = B(x,2r) u dz 1 α(n)2nrn B(y,r) u dz = 1 2n B(y,r) u dz = 1 2n u(y). Thus 2nu(y) u(x) 1 2n u(y) if x, y V , |x y| r. Since V is connected and ¯ V is compact, we can cover ¯ V by a chain of finitely many balls {Bi}i=1, N each of which has radius r 2 and Bi Bi−1 = for i = 2,... , N. Then u(x) 1 2n(N+1) u(y) for all x, y V . 2.2.4. Green’s function. Assume now U Rn is open, bounded, and ∂U is C1. We propose next to obtain a general representation formula for the solution of Poisson’s equation −Δu = f in U, subject to the prescribed boundary condition u = g on ∂U. a. Derivation of Green’s function. Suppose u C2( ¯ U ) is an arbitrary function. Fix x U, choose ε 0 so small that B(x, ε) U, and apply Green’s formula from §C.2 on the region := U B(x, ε) to u(y) and Φ(y x). We thereby compute (24) u(y)ΔΦ(y x) Φ(y x)Δu(y) dy = ∂Vε u(y) ∂Φ ∂ν (y x) Φ(y x) ∂u ∂ν (y) dS(y), ν denoting the outer unit normal vector on ∂Vε. Recall next ΔΦ(x y) = 0 for x = y. We observe also ∂B(x,ε) Φ(y x) ∂u ∂ν (y) dS(y) Cεn−1 max ∂B(0,ε) |Φ| = o(1) as ε 0. Furthermore the calculations in the proof of Theorem 1 show ∂B(x,ε) u(y) ∂Φ ∂ν (y x) dS(y) = ∂B(x,ε) u(y) dS(y) u(x)
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