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Compactness and dichotomy in nonlocal shape optimization

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  • E. Parini
  • A. Salort

Abstract

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.

Suggested Citation

  • E. Parini & A. Salort, 2020. "Compactness and dichotomy in nonlocal shape optimization," Mathematische Nachrichten, Wiley Blackwell, vol. 293(11), pages 2208-2232, November.
    • Handle: RePEc:bla:mathna:v:293:y:2020:i:11:p:2208-2232
    DOI: 10.1002/mana.201800234
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