1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Poisson's equation, properties of eigenvalues

  1. Oct 31, 2012 #1
    hello,

    Let T be a open, connected and bounded subset of ℝ3 which has a smooth boundary bd(T).

    Consider the equation Δu = -λu with either the Dirichlet condition (u=0 in bd(T)) or Neumann (where δu/δn = 0 on bd(T)).

    Define:

    [itex] \left\langle g,h \right\rangle =\iiint_{T}{g(x)h^{*}(x)}dx[/itex]
    where * means conjugate

    Show that for a Dirichlet problem the eigenvalues are positive and that for a Neumann problem they are not negative.

    So the first greens identity is

    [itex] \iint_{bd(T)}{v\frac{\partial u}{\partial n}}dS=\iiint_{T}{\nabla v\cdot \nabla u\,d\vec{x}}+\iiint_{T}{v\Delta u\,d\vec{x}} [/itex]

    using this with v=u,

    for a Dirichlet problem,

    u=0 on the boundary so

    [itex] 0=\iiint_{T}{\nabla u\cdot \nabla u\,d\vec{x}}+\iiint_{T}{u\Delta u\,d\vec{x}} [/itex]

    then

    [itex] \iiint_{T}{\nabla u\cdot \nabla u\,d\vec{x}}=\lambda \iiint_{T}{{{u}^{2}}\,d\vec{x}} [/itex]

    is that correct?

    and its the same result for the Neumann problem except the du/dn term is zero on the boundary so i cant really see how to get that the eigenvalues are positive, or non negative,

    is this not the correct approach?
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Poisson's equation, properties of eigenvalues
  1. Poisson's equation (Replies: 4)

  2. Poissons Equation (Replies: 9)

  3. Poisson equation (Replies: 1)

Loading...