On the minimization of Dirichlet eigenvalues

berg, m. van den;

Results are obtained for two minimization problems:
Ik(c) = inf{λk(Ω) : Ω open, convex in Rm, T (Ω) = c}
and
Jk(c) = inf{λk(Ω) : Ω quasi-open in Rm, |Ω| <= 1, P(Ω) <= c},
where c > 0, λk(Ω) is the kth eigenvalue of the Dirichlet Laplacian acting in L2(Ω), |Ω| denotes the Lebesgue measure of Ω, P(Ω) denotes the perimeter of Ω, and where T is in a suitable collection of functions. The latter includes the perimeter of Ω and the moment of inertia of Ω with respect to its centre of mass.