hw10.pdf - APM 346 HOMEWORK 10 1 Problem 2 from Strauss 10.2(postponed from HW9 2 Recall that we obtained for each triple of positive integers ~n =(nx

# hw10.pdf - APM 346 HOMEWORK 10 1 Problem 2 from Strauss...

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APM 346 HOMEWORK 10 1. Problem 2 from Strauss 10.2 (postponed from HW9). 2. Recall that we obtained for each triple of positive integers ~n = ( n x , n y , n z ) an eigenvalue- eigenfunction pair λ ~n = ( | ~n | π L ) 2 = π 2 L 2 ( n 2 x + n 2 y + n 2 z ) , X ~n ( x, y, z ) = sin ( n x πx L ) sin ( n y πy L ) sin ( n z πz L ) . Further, recall that the collection of eigenfunctions { X ~n : ~n ∈ { 1 , 2 , . . . } 3 } form an or- thonormal basis for V = { f : [0 , L ] 3 C piecewise continuous } with the inner product h f, g i = ( 2 L ) 3 Z [0 ,L ] 3 fg dxdydz. (a) Explain why there does not exist another eigenfunction Y with the eigenvalue λ (2 , 1 , 1) = 6 π 2 L 2 which is linearly independent of { X (2 , 1 , 1) , X (1 , 2 , 1) , X (1 , 1 , 2) } . (b) Determine the multiplicity of the eigenvalue λ = 11 π 2 L 2 . 3. So far we have computed the Dirichlet spectrum of - Δ on an interval (of length L ), a disk (of radius a ), and a cube (with sidelength L ). Describe in general terms how the spectrum varies with the “size” of the domain (the parameters L , a ).

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