{"title":"关于具有两个、三个或四个优先梯度的奇异扰动问题的微结构形成","authors":"Janusz Ginster","doi":"10.1007/s00332-024-10067-x","DOIUrl":null,"url":null,"abstract":"<p>In this manuscript, singularly perturbed energies with 2, 3, or 4 preferred gradients subject to incompatible Dirichlet boundary conditions are studied. This extends results on models for martensitic microstructures in shape memory alloys (<span>\\(N=2\\)</span>), a continuum approximation for the <span>\\(J_1-J_3\\)</span>-model for discrete spin systems (<span>\\(N=4\\)</span>), and models for crystalline surfaces with <i>N</i> different facets (general <i>N</i>). On a unit square, scaling laws are proved with respect to two parameters, one measuring the transition cost between different preferred gradients and the other measuring the incompatibility of the set of preferred gradients and the boundary conditions. By a change of coordinates, the latter can also be understood as an incompatibility of a variable domain with a fixed set of preferred gradients. Moreover, it is shown how simple building blocks and covering arguments lead to upper bounds on the energy and solutions to the differential inclusion problem on general Lipschitz domains.\n</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"23 1","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Formation of Microstructure for Singularly Perturbed Problems with Two, Three, or Four Preferred Gradients\",\"authors\":\"Janusz Ginster\",\"doi\":\"10.1007/s00332-024-10067-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this manuscript, singularly perturbed energies with 2, 3, or 4 preferred gradients subject to incompatible Dirichlet boundary conditions are studied. This extends results on models for martensitic microstructures in shape memory alloys (<span>\\\\(N=2\\\\)</span>), a continuum approximation for the <span>\\\\(J_1-J_3\\\\)</span>-model for discrete spin systems (<span>\\\\(N=4\\\\)</span>), and models for crystalline surfaces with <i>N</i> different facets (general <i>N</i>). On a unit square, scaling laws are proved with respect to two parameters, one measuring the transition cost between different preferred gradients and the other measuring the incompatibility of the set of preferred gradients and the boundary conditions. By a change of coordinates, the latter can also be understood as an incompatibility of a variable domain with a fixed set of preferred gradients. Moreover, it is shown how simple building blocks and covering arguments lead to upper bounds on the energy and solutions to the differential inclusion problem on general Lipschitz domains.\\n</p>\",\"PeriodicalId\":50111,\"journal\":{\"name\":\"Journal of Nonlinear Science\",\"volume\":\"23 1\",\"pages\":\"\"},\"PeriodicalIF\":2.6000,\"publicationDate\":\"2024-07-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Nonlinear Science\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00332-024-10067-x\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Nonlinear Science","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00332-024-10067-x","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
本手稿研究了具有 2、3 或 4 个优先梯度的奇异扰动能量,这些能量受不相容的迪里夏特边界条件的限制。这扩展了形状记忆合金中马氏体微结构模型(\(N=2\))、离散自旋系统的\(J_1-J_3\)-模型的连续近似(\(N=4\))以及具有 N 个不同切面(一般 N)的结晶表面模型的研究成果。在单位正方形上,证明了关于两个参数的缩放定律,一个是测量不同优选梯度之间的过渡成本,另一个是测量优选梯度集与边界条件的不相容性。通过改变坐标,后者也可以理解为可变域与固定的优选梯度集的不相容性。此外,我们还展示了如何通过简单的构件和覆盖论证得出能量上限以及一般 Lipschitz 域上微分包容问题的解。
On the Formation of Microstructure for Singularly Perturbed Problems with Two, Three, or Four Preferred Gradients
In this manuscript, singularly perturbed energies with 2, 3, or 4 preferred gradients subject to incompatible Dirichlet boundary conditions are studied. This extends results on models for martensitic microstructures in shape memory alloys (\(N=2\)), a continuum approximation for the \(J_1-J_3\)-model for discrete spin systems (\(N=4\)), and models for crystalline surfaces with N different facets (general N). On a unit square, scaling laws are proved with respect to two parameters, one measuring the transition cost between different preferred gradients and the other measuring the incompatibility of the set of preferred gradients and the boundary conditions. By a change of coordinates, the latter can also be understood as an incompatibility of a variable domain with a fixed set of preferred gradients. Moreover, it is shown how simple building blocks and covering arguments lead to upper bounds on the energy and solutions to the differential inclusion problem on general Lipschitz domains.
期刊介绍:
The mission of the Journal of Nonlinear Science is to publish papers that augment the fundamental ways we describe, model, and predict nonlinear phenomena. Papers should make an original contribution to at least one technical area and should in addition illuminate issues beyond that area''s boundaries. Even excellent papers in a narrow field of interest are not appropriate for the journal. Papers can be oriented toward theory, experimentation, algorithms, numerical simulations, or applications as long as the work is creative and sound. Excessively theoretical work in which the application to natural phenomena is not apparent (at least through similar techniques) or in which the development of fundamental methodologies is not present is probably not appropriate. In turn, papers oriented toward experimentation, numerical simulations, or applications must not simply report results without an indication of what a theoretical explanation might be.
All papers should be submitted in English and must meet common standards of usage and grammar. In addition, because ours is a multidisciplinary subject, at minimum the introduction to the paper should be readable to a broad range of scientists and not only to specialists in the subject area. The scientific importance of the paper and its conclusions should be made clear in the introduction-this means that not only should the problem you study be presented, but its historical background, its relevance to science and technology, the specific phenomena it can be used to describe or investigate, and the outstanding open issues related to it should be explained. Failure to achieve this could disqualify the paper.