On the boundary of the attainable set of the Dirichlet spectrum

Details On the boundary of the attainable set of the Dirichlet spectrum On the boundary of the attainable set of the Dirichlet spectrum

2011-12-20

arxiv.org/abs/1112.4689v1

Mathematics

Optimization and Control

7 pages, 3 figures

Scientific paper

Denoting by $\mathcal{E}\subseteq \R^2$ the set of the pairs $(\lambda_1(\Omega),\lambda_2(\Omega))$ for all the open sets $\Omega\subseteq\R^N$ with unit measure, and by $\Theta\subseteq\R^N$ the union of two disjoint balls of half measure, we give an elementary proof of the fact that $\partial\E$ has horizontal tangent at its lowest point $(\lambda_1(\Theta),\lambda_2(\Theta))$.

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