A maximum-principle approach to the minimisation of a nonlocal dislocation energy

Joan Mateu; Maria Giovanna Mora; Luca Rondi; Lucia Scardia; Joan Verdera; Joan Mateu; Maria Giovanna Mora; Luca Rondi; Lucia Scardia; Joan Verdera

doi:10.3934/mine.2020012

Mathematics in Engineering

2020, Volume 2, Issue 2: 253-263. doi: 10.3934/mine.2020012

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A maximum-principle approach to the minimisation of a nonlocal dislocation energy

1.
Department de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia
2.
Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy
3.
Dipartimento di Matematica, Università di Milano, via Saldini, 50, 20133 Milano, Italy
4.
Department of Mathematics, Heriot-Watt University, EH14 4AS Edinburgh, United Kingdom

Received: 05 September 2019 Accepted: 13 December 2019 Published: 05 February 2020

In this paper we use an approach based on the maximum principle to characterise the minimiser of a family of nonlocal and anisotropic energies I_α defined on probability measures in $\mathbb{R}^2$. The purely nonlocal term in I_α is of convolution type, and is isotropic for α = 0 and anisotropic otherwise. The cases α = 0 and α = 1 are special: The first corresponds to Coulombic interactions, and the latter to dislocations. The minimisers of I_α have been characterised by the same authors in an earlier paper, by exploiting some formal similarities with the Euler equation, and by means of complex-analysis techniques. We here propose a different approach, that we believe can be applied to more general energies.
- nonlocal interaction,
- potential theory,
- maximum principle,
- dislocations,
- Kirchhoff ellipses
Citation: Joan Mateu, Maria Giovanna Mora, Luca Rondi, Lucia Scardia, Joan Verdera. A maximum-principle approach to the minimisation of a nonlocal dislocation energy[J]. Mathematics in Engineering, 2020, 2(2): 253-263. doi: 10.3934/mine.2020012

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Abstract

In this paper we use an approach based on the maximum principle to characterise the minimiser of a family of nonlocal and anisotropic energies I_α defined on probability measures in $\mathbb{R}^2$. The purely nonlocal term in I_α is of convolution type, and is isotropic for α = 0 and anisotropic otherwise. The cases α = 0 and α = 1 are special: The first corresponds to Coulombic interactions, and the latter to dislocations. The minimisers of I_α have been characterised by the same authors in an earlier paper, by exploiting some formal similarities with the Euler equation, and by means of complex-analysis techniques. We here propose a different approach, that we believe can be applied to more general energies.

References

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