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Asymptotic behavior of the capacity in two-dimensional heterogeneous media 二维非均质介质中容量的渐近行为
Atti della Accademia nazionale dei Lincei Pub Date : 2023-10-06 DOI: 10.4171/rlm/1011
Andrea Braides, Giuseppe Cosma Brusca
{"title":"Asymptotic behavior of the capacity in two-dimensional heterogeneous media","authors":"Andrea Braides, Giuseppe Cosma Brusca","doi":"10.4171/rlm/1011","DOIUrl":"https://doi.org/10.4171/rlm/1011","url":null,"abstract":"We describe the asymptotic behavior of the minimal inhomogeneous two-capacity of small sets in the plane with respect to a fixed open set $Omega$. This problem is governed by two small parameters: $varepsilon$, the size of the inclusion (which is not restrictive to assume to be a ball), and $delta$, the period of the inhomogeneity modeled by oscillating coefficients. We show that this capacity behaves as $C|{logvarepsilon}|^{-1}$. The coefficient $C$ is explicitly computed from the minimum of the oscillating coefficient and the determinant of the corresponding homogenized matrix, through a harmonic mean with a proportion depending on the asymptotic behavior of $|{logdelta}|/|{logvarepsilon}|$.","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135346281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the symmetric rearrangement of the gradient of a Sobolev function Sobolev函数梯度的对称重排
Atti della Accademia nazionale dei Lincei Pub Date : 2023-10-06 DOI: 10.4171/rlm/1013
Vincenzo Amato, Andrea Gentile
{"title":"On the symmetric rearrangement of the gradient of a Sobolev function","authors":"Vincenzo Amato, Andrea Gentile","doi":"10.4171/rlm/1013","DOIUrl":"https://doi.org/10.4171/rlm/1013","url":null,"abstract":"In this paper, we generalize a classical comparison result for solutions to Hamilton–Jacobi equations with Dirichlet boundary conditions, to solutions to Hamilton–Jacobi equations with non-zero boundary trace. As a consequence, we prove the isoperimetric inequality for the torsional rigidity (with Robin boundary conditions) and for other functionals involving such boundary conditions.","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135304184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
An overview on extremals and critical points of the Sobolev inequality in convex cones 凸锥中Sobolev不等式的极值点和临界点综述
Atti della Accademia nazionale dei Lincei Pub Date : 2023-05-30 DOI: 10.4171/rlm/995
Alberto Roncoroni
{"title":"An overview on extremals and critical points of the Sobolev inequality in convex cones","authors":"Alberto Roncoroni","doi":"10.4171/rlm/995","DOIUrl":"https://doi.org/10.4171/rlm/995","url":null,"abstract":"In this survey, we consider the sharp Sobolev inequality in convex cones. We also prove it by using the optimal transport technique. Then we present some results related to the Euler–Lagrange equation of the Sobolev inequality: the so-called critical $p$-Laplace equation. Finally, we discuss some stability result related to the Sobolev inequality.","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135478147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Higher differentiability and existence for a class of problems under p, q subquadratic growth p, q次二次增长下一类问题的高可微性与存在性
Atti della Accademia nazionale dei Lincei Pub Date : 2023-05-30 DOI: 10.4171/rlm/992
Elvira Mascolo, Antonia Passarelli di Napoli
{"title":"Higher differentiability and existence for a class of problems under p, q subquadratic growth","authors":"Elvira Mascolo, Antonia Passarelli di Napoli","doi":"10.4171/rlm/992","DOIUrl":"https://doi.org/10.4171/rlm/992","url":null,"abstract":"","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135641954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
New examples of tangentially degenerate curves 切简并曲线的新例子
Atti della Accademia nazionale dei Lincei Pub Date : 2023-05-30 DOI: 10.4171/rlm/994
Satoru Fukasawa
{"title":"New examples of tangentially degenerate curves","authors":"Satoru Fukasawa","doi":"10.4171/rlm/994","DOIUrl":"https://doi.org/10.4171/rlm/994","url":null,"abstract":"","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135641776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
LS algebras, valuations, and Schubert varieties LS代数，赋值和Schubert变分
Atti della Accademia nazionale dei Lincei Pub Date : 2023-05-30 DOI: 10.4171/rlm/993
Rocco Chirivì, Xin Fang, Peter Littelmann
{"title":"LS algebras, valuations, and Schubert varieties","authors":"Rocco Chirivì, Xin Fang, Peter Littelmann","doi":"10.4171/rlm/993","DOIUrl":"https://doi.org/10.4171/rlm/993","url":null,"abstract":"In this paper, we propose an algebraic approach via Lakshmibai–Seshadri (LS) algebras to establish a link between standard monomial theories, Newton–Okounkov bodies and valuations. This is applied to Schubert varieties, where this approach is compatible with the one using Seshadri stratifications in arXiv:2112.03776, showing that LS paths encode vanishing multiplicities with respect to the web of Schubert varieties.","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135478678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
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