A maximum-principle approach to the minimisation of a nonlocal dislocation energy
In this paper we use an approach based on the maximum principle to characterise the minimiser of a family of nonlocal and anisotropic energies <em>I</em><sub><em>α</em></sub> defined on probability measures in $\mathbb{R}^2$. The purely nonlocal term in <em>...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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AIMS Press
2020-05-01
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| Series: | Mathematics in Engineering |
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| Online Access: | https://www.aimspress.com/article/10.3934/mine.2020012/fulltext.html |
| _version_ | 1818179629839548416 |
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| author | Joan Mateu Maria Giovanna Mora Luca Rondi Lucia Scardia Joan Verdera |
| author_facet | Joan Mateu Maria Giovanna Mora Luca Rondi Lucia Scardia Joan Verdera |
| author_sort | Joan Mateu |
| collection | DOAJ |
| description | In this paper we use an approach based on the maximum principle to characterise the minimiser of a family of nonlocal and anisotropic energies <em>I</em><sub><em>α</em></sub> defined on probability measures in $\mathbb{R}^2$. The purely nonlocal term in <em>I</em><sub><em>α</em></sub> is of convolution type, and is isotropic for <em>α</em> = 0 and anisotropic otherwise. The cases <em>α</em> = 0 and <em>α</em> = 1 are special: The first corresponds to Coulombic interactions, and the latter to dislocations. The minimisers of <em>I</em><sub><em>α</em></sub> 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. |
| first_indexed | 2024-12-11T21:06:55Z |
| format | Article |
| id | doaj.art-8c4ac17adfc6493bbdd726feab8c1b58 |
| institution | Directory Open Access Journal |
| issn | 2640-3501 |
| language | English |
| last_indexed | 2024-12-11T21:06:55Z |
| publishDate | 2020-05-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematics in Engineering |
| spelling | doaj.art-8c4ac17adfc6493bbdd726feab8c1b582022-12-22T00:50:50ZengAIMS PressMathematics in Engineering2640-35012020-05-012225326310.3934/mine.2020012A maximum-principle approach to the minimisation of a nonlocal dislocation energyJoan Mateu0Maria Giovanna Mora1Luca Rondi2Lucia Scardia3Joan Verdera41 Department de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia2 Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy3 Dipartimento di Matematica, Università di Milano, via Saldini, 50, 20133 Milano, Italy4 Department of Mathematics, Heriot-Watt University, EH14 4AS Edinburgh, United Kingdom1 Department de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, CataloniaIn this paper we use an approach based on the maximum principle to characterise the minimiser of a family of nonlocal and anisotropic energies <em>I</em><sub><em>α</em></sub> defined on probability measures in $\mathbb{R}^2$. The purely nonlocal term in <em>I</em><sub><em>α</em></sub> is of convolution type, and is isotropic for <em>α</em> = 0 and anisotropic otherwise. The cases <em>α</em> = 0 and <em>α</em> = 1 are special: The first corresponds to Coulombic interactions, and the latter to dislocations. The minimisers of <em>I</em><sub><em>α</em></sub> 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.https://www.aimspress.com/article/10.3934/mine.2020012/fulltext.htmlnonlocal interactionpotential theorymaximum principledislocationskirchhoff ellipses |
| spellingShingle | Joan Mateu Maria Giovanna Mora Luca Rondi Lucia Scardia Joan Verdera A maximum-principle approach to the minimisation of a nonlocal dislocation energy Mathematics in Engineering nonlocal interaction potential theory maximum principle dislocations kirchhoff ellipses |
| title | A maximum-principle approach to the minimisation of a nonlocal dislocation energy |
| title_full | A maximum-principle approach to the minimisation of a nonlocal dislocation energy |
| title_fullStr | A maximum-principle approach to the minimisation of a nonlocal dislocation energy |
| title_full_unstemmed | A maximum-principle approach to the minimisation of a nonlocal dislocation energy |
| title_short | A maximum-principle approach to the minimisation of a nonlocal dislocation energy |
| title_sort | maximum principle approach to the minimisation of a nonlocal dislocation energy |
| topic | nonlocal interaction potential theory maximum principle dislocations kirchhoff ellipses |
| url | https://www.aimspress.com/article/10.3934/mine.2020012/fulltext.html |
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