Extremality Conditions and Regularity of Solutions to Optimal Partition Problems Involving Laplacian Eigenvalues

Abstract

Abstract. Let Ω ⊂ R N be an open bounded domain and m ∈ N. Given k 1 , . . . , km ∈ N, we consider a wide class of optimal partition problems involving Dirichlet eigenvalues of elliptic operators, of the following formwhere λ k i (ω i ) denotes the k i -th eigenvalue of (−∆, H 1 0 (ω i )) counting multiplicities, and Pm(Ω) is the set of all open partitions of Ω, namelyWhile existence of a quasi-open optimal partition (ω 1 , . . . , ωm) follows from a general result by Bucur, Buttazzo and Henrot [Adv. Math. Sci. Appl. 8, 1998], the aim of this paper is to associate with such minimal partitions and their eigenfunctions some suitable extremality conditions and to exploit them, proving as well the Lipschitz continuity of some eigenfunctions, and regularity of the partition in the sense that the free boundary ∪ m i=1 ∂ω i ∩ Ω is, up to a residual set, locally a C 1,α hypersurface. This last result extend the ones in the paper by Caffarelli and Lin [J. Sci. Comput. 31, 2007] to the case of higher eigenvalues.

On the honeycomb conjecture for Robin Laplacian eigenvalues

“…Without any attempt of completeness, we refer to [9,10,11] for some recent papers in shape optimization involving free boundaries with Robin conditions. While there is a wide literature about optimal partitions for the first Dirichlet Laplacian eigenvalue (see for instance [3,4,6,15,16,23,24,25,31]), to the best of our knowledge the study of the same kind of problem for the first Robin Laplacian eigenvalue is a completely unexplored field. Object of this paper are the optimization problems…”

On the honeycomb conjecture for a class of minimal convex partitions

“…As a matter of fact, optimal spectral partitions have received an increasing attention in the last decade, including also the case when the energy of the partition is the maximal eigenvalue among the chambers (in particular by Helffer and coauthors); without any attempt of completeness, let us quote the papers [1,2,3,10,11,15,16,17,25]. In particular, in [17], a similar honeycomb conjecture for the maximal eigenvalue problem is attributed to Van den Berg.…”

On the Singular Set of Free Interface in an Optimal Partition Problem

“…(1) spatial segregation in reaction-diffusion systems, as well as competitive systems in population dynamics as in [17,43], (2) Bose-Einstein condensates in multiple hyperfine spin states as in [15,43], (3) optimal partition problems involving higher eigenvalues as in [37], (4) harmonic maps into singular targets of curvature bounded from above in the sense of Alexandrov and their geometric applications as in [8] and the references therein.…”