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Compactness and dichotomy in nonlocal shape optimization
Author(s) -
Parini E.,
Salort A.
Publication year - 2020
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800234
Subject(s) - compact space , mathematics , eigenvalues and eigenvectors , sequence (biology) , infinity , laplace operator , dirichlet boundary condition , dirichlet distribution , boundary (topology) , homogeneous , shape optimization , mathematical analysis , pure mathematics , boundary value problem , combinatorics , finite element method , physics , quantum mechanics , biology , genetics , thermodynamics
We prove a general result about the behaviour of minimizing sequences for nonlocal shape functionals satisfying suitable structural assumptions. Typical examples include functions of the eigenvalues of the fractional Laplacian under homogeneous Dirichlet boundary conditions. Exploiting a nonlocal version of Lions' concentration‐compactness principle, we prove that either an optimal shape exists or there exists a minimizing sequence consisting of two “pieces” whose mutual distance tends to infinity. Our work is inspired by similar results obtained by Bucur in the local case.