{"title":"各向异性伯恩斯坦问题","authors":"Connor Mooney, Yang Yang","doi":"10.1007/s00222-023-01222-4","DOIUrl":null,"url":null,"abstract":"Abstract We construct nonlinear entire anisotropic minimal graphs over $\\mathbb{R}^{4}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> , completing the solution to the anisotropic Bernstein problem. The examples we construct have a variety of growth rates, and our approach both generalizes to higher dimensions and recovers and elucidates known examples of nonlinear entire minimal graphs over $\\mathbb{R}^{n},\\, n \\geq 8$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>8</mml:mn> </mml:math> .","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"The anisotropic Bernstein problem\",\"authors\":\"Connor Mooney, Yang Yang\",\"doi\":\"10.1007/s00222-023-01222-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract We construct nonlinear entire anisotropic minimal graphs over $\\\\mathbb{R}^{4}$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> , completing the solution to the anisotropic Bernstein problem. The examples we construct have a variety of growth rates, and our approach both generalizes to higher dimensions and recovers and elucidates known examples of nonlinear entire minimal graphs over $\\\\mathbb{R}^{n},\\\\, n \\\\geq 8$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>8</mml:mn> </mml:math> .\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2023-10-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00222-023-01222-4\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00222-023-01222-4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 1
摘要
在$\mathbb{R}^{4}$ r4上构造了非线性全各向异性极小图，完成了各向异性Bernstein问题的求解。我们构建的示例具有各种增长率，并且我们的方法既可以推广到高维，也可以恢复和阐明$\mathbb{R}^{n},\, n \geq 8$ R n, n≥8上的非线性完整最小图的已知示例。
Abstract We construct nonlinear entire anisotropic minimal graphs over $\mathbb{R}^{4}$ R4 , completing the solution to the anisotropic Bernstein problem. The examples we construct have a variety of growth rates, and our approach both generalizes to higher dimensions and recovers and elucidates known examples of nonlinear entire minimal graphs over $\mathbb{R}^{n},\, n \geq 8$ Rn,n≥8 .
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