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The Laplacian on a compact Riemannian manifold has a discrete spectrum. For example on a circle of perimeter $L$ the $n$-th eigenvalue starting at $0$ is $-\lambda_n = -(2\pi/L)^2 n^2$. On the other ...

A friend in physics asked this question, and I didn't know the answer. Are there lower bounds on the first eigenvalue of the Laplacian of a Riemann surface equipped with a metric of constant negative ...

Let $\Omega$ be a ``nice'' domain in $\mathbb{R}^2$ (bounded and with piecewise smooth boundary). Consider the clamped plate eigenvalue problem, given by $$\Delta^2 u = \lambda u $$ $$ u|_{\partial \...