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 minimizes the functional , among all subsets of a smooth bounded open set , where is the second eigenvalue of the Dirichlet Laplacian on and is a fixed constant, then is equivalent to the union of two disjoint open sets and , which are -regular up to a (possibly empty) closed set of Hausdorff dimension at most , contained in the one-phase free boundaries and .
Cite this article
Dario Mazzoleni, Baptiste Trey, Bozhidar Velichkov, Regularity of the optimal sets for the second Dirichlet eigenvalue. Ann. Inst. H. Poincaré Anal. Non Linéaire 39 (2022), no. 3, pp. 529–573DOI 10.4171/AIHPC/14