On the minimization of Dirichlet eigenvalues

Abstract

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

15 pages

Cited by in corpus (4)

• Maximising Neumann eigenvalues on rectangles
• Minimising Dirichlet eigenvalues on cuboids of unit measure
• Stretching convex domains to capture many lattice points
• Maximal spectral surfaces of revolution converge to a catenoid