It is a classical result that the first eigenvalue of the Dirichlet Laplacian amongst open sets of fixed measure is minimal for the ball. For the second eigenvalue it is known that the union of two disjoint balls of equal measure realizes the minimum. For higher eigenvalues little is known.
In this talk we will consider a number of problems related to the following question: Does the behaviour of sets minimizing the k-th eigenvalue stabilize as k becomes large? In particular, we shall discuss the problem of minimizing the sum of the first K eigenvalues amongst open sets of fixed measure in the limit as K tends to infinity.