Calculus of Variations and Geometric Measure Theory

L. Brasco - C. Nitsch - A. Pratelli

On the boundary of the attainable set of the Dirichlet spectrum

created by brasco on 20 Dec 2011
modified by pratelli on 16 Feb 2015


Published Paper

Inserted: 20 dec 2011
Last Updated: 16 feb 2015

Journal: Z. Angew. Math. Phys.
Number: 64
Pages: 591-597
Year: 2012


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