hello,(adsbygoogle = window.adsbygoogle || []).push({});

Let T be a open, connected and bounded subset of ℝ^{3}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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Poisson's equation, properties of eigenvalues

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**