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Unformatted text preview: We consider Dirichlet boundary value problems which are posed either inside or outside of a sphere with radius R > 0: our domain Ω is either Ω − := { x ∈ R 3  x  < R } ( squiggleright “interior problem”), or (361) Ω + := { x ∈ R 3  x  > R } ( squiggleright “exterior problem”). (362) Notice that Ω − is a bounded domain, whereas Ω + is not. In both cases, the boundary value problem has the form Δ u = 0 in Ω , (363) u ( R,ϕ,ϑ ) = f ( ϕ,ϑ ) , ϕ ∈ [0 , 2 π ) , ϑ ∈ [0 ,π ] , (364) lim r →∞ u ( r,ϕ,ϑ ) = 0 , ϕ ∈ [0 , 2 π ) , ϑ ∈ [0 ,π ] . (365) Separation of Variables We write the unknown function in the form u ( r,ϕ,ϑ ) = F ( r ) G ( ϕ,ϑ ). We separate the radial variable from the angular variables by r 2 F ′′ F + 2 r F ′ F = 1 G Δ S 2 G = k ∈ R , (366) and the separate differential equations are given by r 2 F ′′ + 2 rF ′ kF = 0 , r < R or r > R, (367) Δ S 2 G + kG = 0 , ϕ ∈ [0 , 2 π ) , ϑ ∈ [0 ,π ] . (368) We solve for G first. For that purpose, we separate again, G ( ϕ,ϑ ) = H ( ϕ ) Q ( ϑ ), and we obtain H ′′ H = sin ϑ Q (sin ϑQ ′ ) ′ + k sin 2 ϑ = μ 2 ∈ R . (369) The separate ODEs are now given by H ′′ + μ 2 H = 0 , ϕ ∈ [0 , 2 π ) , (370) sin 2 ϑQ ′′ + sin ϑ cos ϑQ ′ + ( k sin 2 ϑ μ 2 ) Q = 0 , ϑ ∈ [0 ,π ] . (371) 64 Auxiliary Conditions and Solution of Separate ODEs 2 πperiodic solutions of ( 370 ) are found for...
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This note was uploaded on 11/17/2011 for the course MATH 529 taught by Professor Staff during the Spring '08 term at UNC.
 Spring '08
 Staff
 Math

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