Denoting by $\E\subseteq \R^2$ the set of the pairs $\big(\lambda_1(\Omega),\,\lambda_2(\Omega)\big)$ 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 $\big(\lambda_1(\Theta),\,\lambda_2(\Theta)\big)$.
On the boundary of the attainable set of the Dirichlet spectrum
BRASCO, Lorenzo;
2013
Abstract
Denoting by $\E\subseteq \R^2$ the set of the pairs $\big(\lambda_1(\Omega),\,\lambda_2(\Omega)\big)$ 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 $\big(\lambda_1(\Theta),\,\lambda_2(\Theta)\big)$.File in questo prodotto:
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