Dirichlet eigenvalue

Summary

A Dirichlet eigenvalue of a bounded domain $\Omega$ is a number $\lambda$ such that $-\Delta u = \lambda u$ has a nonzero solution vanishing on the boundary. These numbers form an increasing sequence of positive numbers that tends to infinity. They are the squared frequencies of a vibrating membrane fixed along its edge. They also control the decay of heat in $\Omega$ with zero boundary temperature. How they depend on the shape of $\Omega$ is the subject of spectral geometry, with results such as the Faber–Krahn inequality.