Optimizing the second, third,... eigenvalues - applications 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


*

*area constraint

*perimeter constraint
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)  

 A: Nice question and lovely piece of optimization! I have been thinking about that for a while.
I can only answer from a point of physical music acoustics. It appears to be interesting feature for drum design and construction but my considerations end with a conclusion that it is not much useful in praxis. Here is why:
Case 1 - Directly struck drum: the sound is either "pure percussive" (i.e. just one hit, very short in time, spectrum without pronounced harmonics, almost without harmonic decay oscillations - typically a snare drum) or it is the sound of a tuned membranophone such as timpani. But in the latter case, the audible tone is produced mainly by oscillations of air volume inside the drum. Naturally, in both cases there are oscillations of higher membrane modes but either there are plenty of them (the snare drum case) or they are not the most important feature of sound production (the timpani case).
Case 2 - Driven membrane: Obviously, this could be nice feauture considering a membranophone with continual (not percussive) driving (like brazilian cuíca). But when you start to play with the drum shape, you will jeopardise the uniformity of membrane tension which would immediately send your calculations to the deepest hell cause the simple Laplace equation would not be valid any more with gradient of previously constant parameter.
