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 Ω minimizes the functional F_Lambda(Ω)= λ2(Ω)+Lambda|Ω|; among all subsets of a smooth bounded open set D ⊂ Rd , where λ2(Ω) is the second eigenvalue of the Dirichlet Laplacian on Ω and Lambda> 0 is a fixed constant, then Ω is equivalent to the union of two disjoint open sets Ω+ and Ω-, which are C^{1,α}-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 Ω+ \setminus \partial Ω– and D \cap \partialΩ-\setminus \partial Ω+.
Regularity of the optimal sets for the second Dirichlet eigenvalue
Mazzoleni D.;
2022-01-01
Abstract
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 Ω minimizes the functional F_Lambda(Ω)= λ2(Ω)+Lambda|Ω|; among all subsets of a smooth bounded open set D ⊂ Rd , where λ2(Ω) is the second eigenvalue of the Dirichlet Laplacian on Ω and Lambda> 0 is a fixed constant, then Ω is equivalent to the union of two disjoint open sets Ω+ and Ω-, which are C^{1,α}-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 Ω+ \setminus \partial Ω– and D \cap \partialΩ-\setminus \partial Ω+.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
