PDE Help - Eigenfunction of a LaPlacian

[rexasaurus]

1. A function "v" where v(x,y,z)≠0 is called an eigenfunction of the Laplacian Δ (on some region Ω - with specified homogenous BC) if v satisifies the BC ad also Δv=-λv on Ω for some number λ.

Part A: Give an example of an eigenfunction of Δ when Ω is the cube [0,∏]³ with Dirichlet BCs.
Part B: Show that, for any eigenfunction v, the function u(t,x,y,z)=cos(√λt)v(x,y,z) is the solution for the problem:
u_tt=ΔU on [0,T]XΩ with Ω=[0,∏]³
u=0 on ∂Ω for 0≤t≤T
{u(0,x,y,y)=v(x,y,z) for 0≤x,y,z≤∏}
{u(0,x,y,y)=0 for 0≤x,y,z≤∏}

2. We are studying heat equation, wave equation, eigenfunctions, d'Alamberts sol., ray-tracing, etc.

3. Okay... I am not sure how to approach a problem of this nature. I sketched the cube from [0,∏]. I am assuming that the answer is given in part B. I can see how the cosine function is always 0 at the BC's. Any help with this process would be appreciated.

 

[LCKurtz]

Hint: Can you think of a function whose second derivative is minus a constant times itself and whose values at 0 and pi are zero?

 

[rexasaurus]

Sin(K*u)?? Second derivative would be -K²sin(K*u)

I am not sure which variable should be inserted for "u" since u is a function of x,y,z.

 

[LCKurtz]

Hint: Can you think of a function whose second derivative is minus a constant times itself and whose values at 0 and pi are zero?

Sin(K*u)?? Second derivative would be -K²sin(K*u)

I am not sure which variable should be inserted for "u" since u is a function of x,y,z.

Your independent variables are x,y, and z. In part A you are trying to match the boundary conditions. So u = sin(x) would take care of the x = 0 and x = π boundary conditions. What about the others?