*Published Paper*

**Inserted:** 20 dec 2011

**Last Updated:** 16 feb 2015

**Journal:** Z. Angew. Math. Phys.

**Number:** 64

**Pages:** 591-597

**Year:** 2012

**Abstract:**

Denoting by $\mathcal{E}\subseteq \mathbb{R}^2$ the set of the pairs $\big(\lambda_1(\Omega),\,\lambda_2(\Omega)\big)$ 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 $\partial\mathcal{E}$ has horizontal tangent at its lowest point $\big(\lambda_1(\Theta),\,\lambda_2(\Theta)\big)$.

**Keywords:**
shape optimization, Dirichlet-Laplacian spectrum

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