Dirichlet eigenvalue

In mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can hear the shape of a drum is: given the Dirichlet eigenvalues, what features of the shape of the drum can one deduce. Here a "drum" is thought of as an elastic membrane Ω, which is represented as a planar domain whose boundary is fixed. The Dirichlet eigenvalues are found by solving the following problem for an unknown function u ≠ 0 and eigenvalue λ

${\displaystyle {\begin{cases}\Delta u+\lambda u=0&{\rm {{in\ }\Omega }}\\u|_{\partial \Omega }=0.&\end{cases}}}$

 

 

 

 

()

Here Δ is the Laplacian, which is given in xy-coordinates by

...
Wikipedia