I consider a vibrating membrane $D\subset {\mathbb{R}}^2 $, fixed on $\partial D$. The vertical displacement $f=f(x,y,t)$ of the membrane satisfies the wave equation. I search solutions of the form $f(x,y,t)=g(x,y)h(t)$ and I obtain: \begin{equation*} \begin{cases} \Delta\:g+ \lambda \:g=0\quad {\rm in}\;D \\ g=0\quad {\rm on} \; \partial D.\end{cases} \end{equation*} , with $\lambda>0$.
I know that I can obtain a general oscillation of the membrane as superposition of normal modes, which are determined by the eigenvalues of the Laplacian. So I have that the sound that can emit my drum can be deduced univocally from the spectrum of the Laplacian (which depends on the the domain $D$).
I would like to know if this assertion is true and e if it is expressed correctly:
a person with a perfect ear, capable of identifying all the frequencies of the modes of vibration, can identify the shape of the drum just from hearing these frequencies $iff$ the domain $D$ is uniquely determined by the eigenvalues of the Laplacian.
This is Kac's question: `Can one hear the shape of a drum?'. Is this a correct formulation of the question?
It is a correct formulation, but the answer is no: there are domains of different shapes whose Laplacians have the same spectrum. See e.g. Wikipedia