2.2. LAPLACE’S EQUATION 37 DEFINITION. Green’s function for the half-space R+ n is G(x, y) := Φ(y − x) − Φ(y − ˜) x (x, y ∈ R+, n x = y). Then Gyn (x, y) = Φyn (y − x) − Φyn (y − ˜) x = −1 nα(n) yn − xn |y − x|n − yn + xn |y − ˜|n x . Consequently if y ∈ ∂R+,n ∂G ∂ν (x, y) = −Gyn (x, y) = − 2xn nα(n) 1 |x − y|n . Suppose now u solves the boundary-value problem (32) Δu = 0 in R+n u = g on ∂R+.n Then from (30) we expect (33) u(x) = 2xn nα(n) ∂R+n g(y) |x − y|n dy (x ∈ R+)n to be a representation formula for our solution. The function K(x, y) := 2xn nα(n) 1 |x − y|n (x ∈ R+,y n ∈ ∂R+)n is Poisson’s kernel for R+, n and (33) is Poisson’s formula. We must now check directly that formula (33) does indeed provide us with a solution of the boundary-value problem (32). THEOREM 14 (Poisson’s formula for half-space). Assume g ∈ C(Rn−1)∩ L∞(Rn−1), and define u by (33). Then (i) u ∈ C∞(R+) n ∩ L∞(R+),n (ii) Δu = 0 in R+,n and (iii) lim x→x0 x∈Rn + u(x) = g(x0) for each point x0 ∈ ∂R+.n
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