# Minimising convex combinations of low eigenvalues

created by mazzoleni on 03 Jul 2012
modified on 20 Oct 2014

[BibTeX]

Published paper

Inserted: 3 jul 2012
Last Updated: 20 oct 2014

Journal: ESAIM:COCV
Volume: 20
Number: 02
Pages: 442-459
Year: 2014
Doi: http://dx.doi.org/10.1051/cocv/2013070

Abstract:

We consider the variational problem $\inf{\{\alpha \lambda_{1}(\Omega)+ \beta \lambda_{2}(\Omega)+ (1-\alpha-\beta)\lambda_3(\Omega)\;:\; \Omega \ \text{open in} \ \mathbb{R}^n, \ \mathcal{L}^n(\Omega)\leq 1 \}},$ for $\alpha,\beta\in[0,1]$, $\alpha+\beta\leq1,$ where $\lambda_k(\Omega)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(\Omega)$. We investigate for which values of $\alpha,\beta$ a minimiser is connected.