On the minimization of Dirichlet eigenvalues of the Laplace operator

Abstract

We study the variational problem $$\inf \{λ_k(Ω): Ω \textup{open in}\ \R^m,\ |Ω| < \infty, \ \h(\partial Ω) \le 1 \},$$ where ${\lambda }_k(\Omega )$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(\Omega )$, $\h(\partial Ω)$ is the $(m-1)$- dimensional Hausdorff measure of the boundary of $\Omega$, and $|\Omega |$ is the Lebesgue measure of $\Omega$. If $m=2$, and $k=2,3,\cdots$, then there exists a convex minimiser ${\Omega }_{2,k}$. If $m\ge 2$, and if ${\Omega }_{m,k}$ is a minimiser, then ${\Omega }_{m,k}^{\ast }:=\text{int}(\overline{{\Omega }_{m,k}})$ is also a minimiser, and ${\mathbb{R}}^m\setminus {\Omega }_{m,k}^{\ast }$ is connected. Upper bounds are obtained for the number of components of ${\Omega }_{m,k}$. It is shown that if $m\ge 3$, and $k\le m+1$ then ${\Omega }_{m,k}$ has at most $4$ components. Furthermore ${\Omega }_{m,k}$ 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.

arXiv comment16 pages

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