Minimising convex combinations of low eigenvalues

We consider the variational problem a1O+ß2O+1a- β3O.open in Rn= 1a, β[0, 1]a+β= 1, where kO is the kth eigenvalue of the Dirichlet Laplacian acting in L 2O and O is the Lebesgue measure of O. We investigate for which values of a, β every minimiser is connected. © 2014 EDP Sciences, SMAI.