On the minimization of Dirichlet eigenvalues

Abstract

Results are obtained for two minimization problems: $$I_k(c)=\inf \{λ_k(Ω): Ω \textup{open, convex in}\ \mathbb{R}^m,\ \mathcal{T}(Ω)= c \},$$ and $$J_k(c)=\inf\{λ_k(Ω): Ω \textup{quasi-open in}\ \mathbb{R}^m, |Ω|\le 1, \mathcal {P}(Ω)\le c \},$$ where $c>0$, $λ_k(Ω)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(Ω)$, $|Ω|$ denotes the Lebesgue measure of $Ω$, $\mathcal{P}(Ω)$ denotes the perimeter of $Ω$, and where $\mathcal{T}$ is in a suitable class set functions. The latter include for example the perimeter of $Ω$, and the moment of inertia of $Ω$ with respect to its center of mass.