On the boundary of the attainable set of the Dirichlet spectrum

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 $(\lambda_1(\Theta),\lambda_2(\Theta))$.