All Questions

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and $f \in L^2(\Omega)$. It is well known that if $\lambda$ is a Dirichlet Laplacian eigenvalue, then the equation $$\begin{cases} -\Delta ...

I would like to reproduce the results of Manas - The music of gold: Can gold counterfeited coins be detected by ear?, but it skips a lot of steps, and the mathematics behind it is a bit advanced for ...

Let $\Omega$ be an open bounded domain with a boundary $\partial\Omega$. Consider the following Neumann eigenvalue problem for Laplacian: find $(\phi_n,\lambda_n)\in H^1(\Omega)\times \mathbb{R}$ \...

I am interested when one can say the eigenvalues of an elliptic operator are real analytic with respect to a parameter. In particular I have seen many people say the first eigenvalue is analytic but ...

Consider $ \Omega$ a smooth bounded domain in $ \mathbb R^N$. I am interested in the gap between the first and second eigenvalues of the operator $ -\Delta + V(x)$. Let $ \phi_1>0$ and $ \phi_2$ ...

I am sure this is extremely well known but I have been digging a bit and I can't find what I need. Consider $B$ to be the unit ball in $ \mathbb R^N$ and consider the eigenvalue problem \begin{cases}...