On the boundary of the attainable set of the Dirichlet spectrum

Abstract

Denoting by $\mathcal{E}\subseteq {\mathbb{R}}^2$ the set of the pairs $({\lambda }_1(\Omega ),{\lambda }_2(\Omega ))$ for all the open sets $\Omega \subseteq {\mathbb{R}}^N$ with unit measure, and by $\Theta \subseteq {\mathbb{R}}^N$ 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 $({\lambda }_1(\Theta ),{\lambda }_2(\Theta ))$.

arXiv comment7 pages, 3 figures

References in corpus (1)

• On the minimization of Dirichlet eigenvalues of the Laplace operator

Cited by in corpus (1)

• On Blaschke-Santaló diagrams for the torsional rigidity and the first Dirichlet eigenvalue