On the minimization of Dirichlet eigenvalues of the Laplace operator

We study the variational problem $$\inf \{λ_k(Ω): Ω \textup{open in}\ \R^m,\ |Ω| < \infty, \ \h(\partial Ω) \le 1 \},$$ where $λ_k(Ω)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(Ω)$, $\h(\partial Ω)$ is the $(m-1)$- dimensional Hausdorff measure of the boundary of $Ω$, and $|Ω|$ is the Lebesgue measure of $Ω$. If $m=2$, and $k=2,3, \cdots$, then there exists a convex minimiser $Ω_{2,k}$. If $m \ge 2$, and if $Ω_{m,k}$ is a minimiser, then $Ω_{m,k}^*:= \textup{int}(\overline{Ω_{m,k}})$ is also a minimiser, and $\R^m\setminus Ω_{m,k}^*$ is connected. Upper bounds are obtained for the number of components of $Ω_{m,k}$. It is shown that if $m\ge 3$, and $k\le m+1$ then $Ω_{m,k}$ has at most $4$ components. Furthermore $Ω_{m,k}$ is connected in the following cases : (i) $m\ge 2, k=2,$ (ii) $m=3,4,5,$ and $k=3,4,$ (iii) $m=4,5,$ and $k=5,$ (iv) $m=5$ and $k=6$. Finally, upper bounds on the number of components are obtained for minimisers for other constraints such as the Lebesgue measure and the torsional rigidity.

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