REGULARITY OF ALL MINIMIZERS OF A CLASS OF SPECTRAL PARTITION PROBLEMS

Hugo Tavares , Alessandro Zilio (1)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

We study a rather broad class of optimal partition problems with respect to monotone and coercive functional costs that involve the Dirichlet eigenvalues of the partitions. We show a sharp regularity result for the entire set of minimizers for a natural relaxed version of the original problem, together with the regularity of eigenfunctions and a universal free boundary condition. Among others, our result covers the cases of the following functional costs (ω1,. .. , ωm) → m i=1 k i j=1 λj(ωi) p i 1/p i , m i=1 k i j=1 λj(ωi) , m i=1 k i j=1 λj(ωi) where (ω1,. .. , ωm) are the sets of the partition and λj(ωi) is the j-th Laplace eigenvalue of the set ωi with zero Dirichlet boundary conditions.

Mots clés: elliptic competitive systems optimal partition problems Laplacian eigenvalues segregation phenomena extremality conditions regularity of free boundary problems blowup techniques; elliptic competitive systems; optimal partition problems; Laplacian eigenvalues; segregation phenomena; extremality conditions; regularity of free boundary problems; blowup techniques

Domaines: Equations aux dérivées partielles [math.AP]

Hugo Tavares, Alessandro Zilio. REGULARITY OF ALL MINIMIZERS OF A CLASS OF SPECTRAL PARTITION PROBLEMS. Mathematics in Engineering, 2020, 3 (1), pp.1-31. ⟨10.3934/mine.2021002⟩. ⟨hal-03884876⟩