Regularity of the optimal sets for the second Dirichlet eigenvalue

Mazzoleni, Dario; Trey, Baptiste; Velichkov, Bozhidar

doi:10.4171/aihpc/14

Mazzoleni, Dario ; Trey, Baptiste ; Velichkov, Bozhidar

Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 3, pp. 529-573

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Résumé

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 $ℱ_{Λ} (Ω) = λ_{2} (Ω) + Λ | Ω |$ , among all subsets of a smooth bounded open set $D \subset ℝ^{d}$ , where $λ_{2} (Ω)$ is the second eigenvalue of the Dirichlet Laplacian on $Ω$ and $Λ > 0$ 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 $D \cap \partial Ω_{+}$ $∖$ $\partial Ω_{-}$ and $D \cap \partial Ω_{-}$ $∖$ $\partial Ω_{+}$ .

Accepté le : 2021-06-20
Publié le : 2022-03-09

DOI : 10.4171/aihpc/14

Classification : 35R35, 47A75, 49Q10
Keywords: Dirichlet eigenvalues, regularity of free boundaries

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     author = {Mazzoleni, Dario and Trey, Baptiste and Velichkov, Bozhidar},
     title = {Regularity of the optimal sets for the second {Dirichlet} eigenvalue},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {529--573},
     year = {2022},
     volume = {39},
     number = {3},
     doi = {10.4171/aihpc/14},
     language = {en},
     url = {https://www.numdam.org/articles/10.4171/aihpc/14/}
}

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Mazzoleni, Dario; Trey, Baptiste; Velichkov, Bozhidar. Regularity of the optimal sets for the second Dirichlet eigenvalue. Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 3, pp. 529-573. doi: 10.4171/aihpc/14

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