{"title":"Epsilon-Regularity for Griffith Almost-Minimizers in Any Dimension Under a Separating Condition","authors":"Camille Labourie, Antoine Lemenant","doi":"10.1007/s00205-023-01935-z","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we prove that if (<i>u</i>, <i>K</i>) is an almost-minimizer of the Griffith functional and <i>K</i> is <span>\\(\\varepsilon \\)</span>-close to a plane in some ball <span>\\(B\\subset {\\mathbb {R}}^N\\)</span> while separating the ball <i>B</i> in two big parts, then <i>K</i> is <span>\\(C^{1,\\alpha }\\)</span> in a slightly smaller ball. Our result contains and generalizes the 2 dimensional result of <span>Babadjian</span> et al. (J Eur Math Soc 24(7):2443–2492, 2022), with a different and more sophisticate approach inspired by <span>Lemenant</span> (Ann Sc Norm Super Pisa Cl Sci 9(2):351–384, 2010; Ann Sc Norm Super Pisa Cl Sci 10(3):561–609, 2011), using also <span>Labourie</span> (J Geom Anal 31(10):10024–10135, 2021) in order to adapt a part of the argument to Griffith minimizers.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6000,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Rational Mechanics and Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00205-023-01935-z","RegionNum":1,"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
Abstract
In this paper we prove that if (u, K) is an almost-minimizer of the Griffith functional and K is \(\varepsilon \)-close to a plane in some ball \(B\subset {\mathbb {R}}^N\) while separating the ball B in two big parts, then K is \(C^{1,\alpha }\) in a slightly smaller ball. Our result contains and generalizes the 2 dimensional result of Babadjian et al. (J Eur Math Soc 24(7):2443–2492, 2022), with a different and more sophisticate approach inspired by Lemenant (Ann Sc Norm Super Pisa Cl Sci 9(2):351–384, 2010; Ann Sc Norm Super Pisa Cl Sci 10(3):561–609, 2011), using also Labourie (J Geom Anal 31(10):10024–10135, 2021) in order to adapt a part of the argument to Griffith minimizers.
期刊介绍:
The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.