I'm working on some topics related to spectral optimization as a function of the domain. For example it is known for almost a century (lord Rayleigh and Faber, Krahn) that the shape which minimizes the first eigenvalue of the Laplacian on a domain with zero boundary condition under area constraint is a disk.
Formally the eigenvalues are the sequence of positive values for which there exist non-trivial functions $u$ such that $$ \begin{cases} -\Delta u = \lambda(\Omega) u & \text{ in }\Omega \\ \hfill u = 0 \ \ & \text{ on }\partial \Omega \end{cases} $$ What I just wrote above is that in the two dimensional case the solution of $$ \min_{|\Omega| = c} \lambda_1(\Omega) $$ is achieved when $\Omega$ is a disk. It is possible to ask what are the shapes which minimize higher eigenvalues. Little is known theoretically for $k \geq 3$ and numerical results can be found at the following links
Apart of the purely mathematical interest in the problem, there is a nice application:
Let's say that we want to produce a drum which produces a certain base frequency such that the area of the membrane is minimal. Then it is best to make a circular drum. Incidentally, the circular shape also minimizes the perimeter at a given frequency. Therefore the most cost effective drum is a circular one.
The application described above deals with the fundamental eigenvalue.
Is there any physical advantage if we build shapes which minimize the second, third, fourth eigenvalues? (I'm interested in the case where the membrane is fixed at the boundary)
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