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Equilibrium measure for a nonlocal dislocation energy with physical confinement

  • Maria Giovanna Mora ORCID logo EMAIL logo and Alessandro Scagliotti ORCID logo
Published/Copyright: January 14, 2021
Advances in Calculus of Variations
From the journal Advances in Calculus of Variations Volume 15 Issue 4

Abstract

In this paper, we characterize the equilibrium measure for a family of nonlocal and anisotropic energies I α that describe the interaction of particles confined in an elliptic subset of the plane. The case α = 0 corresponds to purely Coulomb interactions, while the case α = 1 describes interactions of positive edge dislocations in the plane. The anisotropy into the energy is tuned by the parameter 𝛼 and favors the alignment of particles. We show that the equilibrium measure is completely unaffected by the anisotropy and always coincides with the optimal distribution in the case α = 0 of purely Coulomb interactions, which is given by an explicit measure supported on the boundary of the elliptic confining domain. Our result does not seem to agree with the mechanical conjecture that positive edge dislocations at equilibrium tend to arrange themselves along “wall-like” structures. Moreover, this is one of the very few examples of explicit characterization of the equilibrium measure for nonlocal interaction energies outside the radially symmetric case.

Keywords: Nonlocal interaction; potential theory; dislocations
MSC 2010: 31A15; 35Q70

Funding statement: The first author acknowledges support by the Università degli Studi di Pavia through the 2017 Blue Sky Research Project “Plasticity at different scales: micro to macro” and by GNAMPA–INdAM. The second author acknowledges support by GNAMPA–INdAM through the 2016 “Borsa di Avviamento alla Ricerca”.

  1. Communicated by: Frank Duzaar

References

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Received: 2020-07-23
Accepted: 2020-12-10
Published Online: 2021-01-14
Published in Print: 2022-10-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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  2. Borderline regularity for fully nonlinear equations in Dini domains
  3. Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions
  4. Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝ N
  5. A note on Kazdan–Warner equation on networks
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  9. Pairings between bounded divergence-measure vector fields and BV functions
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  15. Equilibrium measure for a nonlocal dislocation energy with physical confinement
https://doi.org/10.1515/acv-2020-0076
Keywords for this article
Nonlocal interaction; potential theory; dislocations

Articles in the same Issue

  1. Frontmatter
  2. Borderline regularity for fully nonlinear equations in Dini domains
  3. Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions
  4. Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝ N
  5. A note on Kazdan–Warner equation on networks
  6. Integral representation for energies in linear elasticity with surface discontinuities
  7. Minkowski inequalities and constrained inverse curvature flows in warped spaces
  8. Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian
  9. Pairings between bounded divergence-measure vector fields and BV functions
  10. Equivalence between distributional and viscosity solutions for the double-phase equation
  11. Vanishing John–Nirenberg spaces
  12. Extremals in nonlinear potential theory
  13. On the structure of divergence-free measures on ℝ2
  14. Solutions of the (free boundary) Reifenberg Plateau problem
  15. Equilibrium measure for a nonlocal dislocation energy with physical confinement
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