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Homework Help: Poisson's equation, properties of eigenvalues

  1. Oct 31, 2012 #1

    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)).


    [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]


    [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
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