An optimal partition problem for the localization of eigenfunctions. Issue 4 (10th October 2022)

• Record Type: Journal Article
• Authors: David, Guy; Pourmohammad, Hassan
• Is Part Of: Mathematika. Volume 68:Issue 4(2022)
• Page Start: 1302
• Page End: 1330
• Publication Date: 2022-10-10
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1112/mtk.12169
• Languages: English
• ISSNs: 0025-5793

Abstract

Abstract: We study the minimizers of a functional on the set of partitions of a domain Ω ⊂ R n $\Omega \subset \mathbb {R}^n$ into N subsets W j $W_j$ of locally finite perimeter in Ω, whose main term is ∑ j = 1 N ∫ Ω ∩ ∂ W j a ( x ) d H n − 1 ( x ) $\sum _{j=1}^N \int _{\Omega \cap \partial W_j} a(x) d\mathcal {H}^{n-1}(x)$ . Here the positive bounded function a may, for instance, be related to the Landscape function of some Schrödinger operator. We prove the existence of minimizers through the equivalence with a weak formulation, and the local Ahlfors regularity and uniform rectifiability of the boundaries Ω ∩ ∂ W j $\Omega \cap \partial W_j$ .