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On the boundary of the attainable set of the Dirichlet spectrum

Denoting by $\mathcal{E}\subseteq \R^2$ the set of the pairs $(λ_1(Ω),λ_2(Ω))$ for all the open sets $Ω\subseteq\R^N$ with unit measure, and by $Θ\subseteq\R^N$ 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 $(λ_1(Θ),λ_2(Θ))$.