Regularity of the optimal sets for the second Dirichlet eigenvalue
Dario Mazzoleni, Baptiste Trey, Bozhidar Velichkov · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2022
This paper is dedicated to the regularity of the optimal sets for the second eigenvalue of the Dirichlet Laplacian. Precisely, we prove that if the set \Omega minimizes the functional \mathcal{F}_\Lambda(\Omega) = \lambda_{2} (\Omega) + \Lambda |\Omega| , among all subsets of a smooth bounded open set D \subset \mathbb{R}^{d} , where \lambda_{2} (\Omega) is the second eigenvalue of the Dirichlet Laplacian on \Omega and \Lambda > 0 is a fixed constant, then \Omega is equivalent to the union of two disjoint open sets \Omega_{+} and \Omega_{-} , which are C^{1,\alpha} -regular up to a (possibly empty) closed set of Hausdorff dimension at most d - 5 , contained in the one-phase free boundaries D \cap \partial\Omega_{+} \backslash \partial\Omega_{-} and D \cap \partial\Omega_{-} \backslash \partial\Omega_{+} .