Poisson's equation, properties of eigenvalues

[linda300]

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:

$\left\langle g,h \right\rangle =\iiint_{T}{g(x)h^{*}(x)}dx$
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

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

using this with v=u,

for a Dirichlet problem,

u=0 on the boundary so

$0=\iiint_{T}{\nabla u\cdot \nabla u\,d\vec{x}}+\iiint_{T}{u\Delta u\,d\vec{x}}$

then

$\iiint_{T}{\nabla u\cdot \nabla u\,d\vec{x}}=\lambda \iiint_{T}{{{u}^{2}}\,d\vec{x}}$

is that correct?

and its the same result for the Neumann problem except the du/dn term is zero on the boundary so i can't really see how to get that the eigenvalues are positive, or non negative,

is this not the correct approach?

 

[lippyka]

Thank you Yes, that is the correct approach. For the Dirichlet problem, we have: \begin{align}0 &= \iiint_{T}{\nabla u\cdot \nabla u\,d\vec{x}}+\iiint_{T}{u\Delta u\,d\vec{x}} \\&= \iiint_{T}{\nabla u\cdot \nabla u\,d\vec{x}}-\lambda \iiint_{T}{{{u}^{2}}\,d\vec{x}} \end{align}Using the definition of the inner product: \begin{align}0 &= \left\langle \nabla u, \nabla u \right\rangle - \lambda\left\langle u, u \right\rangle \\&= \| \nabla u \|^2 - \lambda \| u \|^2 \\ &= \| \nabla u \|^2 - \lambda \| u \|_2^2 \\ &\geq 0 \end{align}where $\| \cdot \|$ is the usual vector norm, and $\| \cdot \|_2$ is the $L^2$-norm. For the Neumann problem, we have: \begin{align}0 &= \iiint_{T}{\nabla u\cdot \nabla u\,d\vec{x}}+\iiint_{T}{u\Delta u\,d\vec{x}} \\&= \iiint_{T}{\nabla u\cdot \nabla u\,d\vec{x}}-\lambda \iiint_{T}{{{u}^{2}}\,d\vec{x}} \end{align}Using the definition of the inner product: \begin{align}0 &= \left\langle \nabla u, \nabla u \