In this paper, we characterize the equilibrium measure for a family of nonlocal and anisotropic energies I-alpha that describe the interaction of particles confined in an elliptic subset of the plane. The case alpha = 0 corresponds to purely Coulomb interactions, while the case alpha = 1 describes interactions of positive edge dislocations in the plane. The anisotropy into the energy is tuned by the parameter a 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 alpha = 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.

Equilibrium measure for a nonlocal dislocation energy with physical confinement / Mora, Mg; Scagliotti, A. - In: ADVANCES IN CALCULUS OF VARIATIONS. - ISSN 1864-8258. - (2021), pp. 1-10. [10.1515/acv-2020-0076]

Equilibrium measure for a nonlocal dislocation energy with physical confinement

Scagliotti, A
2021

Abstract

In this paper, we characterize the equilibrium measure for a family of nonlocal and anisotropic energies I-alpha that describe the interaction of particles confined in an elliptic subset of the plane. The case alpha = 0 corresponds to purely Coulomb interactions, while the case alpha = 1 describes interactions of positive edge dislocations in the plane. The anisotropy into the energy is tuned by the parameter a 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 alpha = 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.
1
10
10.1515/acv-2020-0076
Mora, Mg; Scagliotti, A
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/129531
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