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I've already posted this question on Physics.SE, but I thougth it could be useful to ask also here.

No problem if moderators will ask me to cancel this thread... But, please, have mercy! :-D


Let $\Omega \subseteq \mathbb{R}^N$ be a domain and let $V,m:\Omega \to \mathbb{R}$ be two measurable and sufficiently summable functions.

When one considers the eigenvalue problem for the operator $\mathcal{L}:=-\Delta +V$ w.r.t. the weight $m$, i.e.: $$\tag{P} \begin{cases} -\Delta u(x) + V(x)\ u(x) = \lambda\ m(x)\ u(x) &\text{, in } \Omega\\ u(x)=0 &\text{, on } \partial \Omega , \end{cases}$$ the function $V$ is usually called potential and the function $m$ is called weight.

Then, a weighted eigenvalue of $\mathcal{L}$ w.r.t. $m$ is any number $\lambda \in \mathbb{R}$ s.t. (P) has at least one nontrivial weak solution $u\in H_0^1(\Omega)$, i.e.: $$\forall \phi \in C_c^\infty(\Omega),\quad \int_\Omega \nabla u\cdot \nabla \phi\ \text{d} x + \int_\Omega V\ u\ \phi\ \text{d} x = \lambda\ \int_\Omega m\ u\ \phi\ \text{d} x\; .$$

My questions are:

  1. Is there any reasonable physical interpretation of those eigenvalues? And what is it?

  2. Why have the functions $V$ and $m$ those names?

Moreover, I heard that the $p$-laplacian (i.e., $\Delta_p u := \operatorname{div} (|\nabla u|^{p-2}\ \nabla u)$, which reduces to the usual laplacian when $p=2$) can be used to model nonlinear elasticity or something like that; therefore I have also the following question:

What about any possible physical meaning of the nonlinear weighted eigenvalues coming from the problem: $$\tag{Q} \begin{cases} -\Delta_p u(x) + V(x)\ |u(x)|^{p-2}\ u(x) = \lambda\ m(x)\ |u(x)|^{p-2}\ u(x) &\text{, in } \Omega\\ u(x)=0 &\text{, on } \partial \Omega , \end{cases}$$ where $1 < p < \infty$?

Many thanks in advance, guys!

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    In the linear case, the spectrum of $-\Delta + V(\cdot )$ is strictly connected to the study of standing waves for Schrödinger equations. There are hundreds of papers about this. For the $p$-Laplace operator, the problem is nonlinear and mostly open. As far as I know, it is chiefly a mathematical problem.2012-07-03
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    Related: http://mathoverflow.net/questions/66418/on-the-physics-background-of-p-laplacian-equation2012-07-03
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    @Siminore : Do you know if there is any paper about sufficient conditions for the first eigenvalue of $-\Delta +V$ to be $> 0$?2012-07-04
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    You could start from these lecture notes. Some conditions are hidden there :-) http://www.math.nsysu.edu.tw/~amen/posters/pankov.pdf2012-07-04
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    I'm absolutely going to take a look at those notes! Thank you.2012-07-04
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    Theorem 8.17 in the above notes says $-\Delta+V$ is symmetric for $V \in L^2+L^\infty$ on the domain of $\Delta$. So, at least it doesn't have negative eigenvalues. This makes sense in terms of springs and masses - if the eigenvalue is negative, the acceleration would be in the same direction as the position so the system would blow up.2012-07-04
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    Cross-listed from http://physics.stackexchange.com/q/31100/24512012-07-04

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